From c2e94ae730e9dbd7d1992aeaf63c41b8fb1d5c4c Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 10:57:07 -0400 Subject: [PATCH 01/64] Created demo.py --- .../demo.py | 49 +++++++++++++++++++ 1 file changed, 49 insertions(+) create mode 100644 demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py new file mode 100644 index 0000000000..d8d2830ba4 --- /dev/null +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py @@ -0,0 +1,49 @@ +r"""Intro to the Quantum Chebyshev +============================================================= + +This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. +They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. + +We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. + + +.. figure:: ../_static/demonstration_assets/qft/socialthumbnail_large_QFT_2024-04-04.png + :align: center + :width: 60% + :target: javascript:void(0) + +What are Chebyshev polynomials? +--------------------------------------- + +`Chebyshev polynomials ` of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as + +.. math:: + T_n(x) \equiv \cos(n \arccos(x))\,, + +where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. +.. math:: + T_0(x) &= 1 \\ + T_1(x) &= x \\ + T_2(x) &= 2x^2-1 \\ + T_3(x) &= 4x^3 - 3x \\ + T_4(x) &= 8x^4 - 8x^2 + 1 \\ + &\ \,\vdots \\ + T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x)\,. + +The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. +The roots of the :math:`T_n(x)` occur at the values +.. math:: + x_n^\mathrm{Ch} = \cos\left(\frac{2k+1}{2n}\pi\right)\,, \quad k=0, ..., n-1\,. + +These are known as the `Chebyshev nodes `. + +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png + :align: center + :width: 60% + :target: javascript:void(0) +""" + +# +# About the authors +# ----------------- +# From b76fe6e35e0058572141f28cf989c127fd6c06db Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 10:58:27 -0400 Subject: [PATCH 02/64] Added image files to demo assets 5 images added to /_static/demonstration_assets/quantum_chebyshef_transform --- .../quantum_chebyshev_transform/amplitudes.png | Bin 0 -> 131206 bytes .../chebyshev_polynomials.png | Bin 0 -> 210846 bytes .../qcht_diagram.png | Bin 0 -> 201122 bytes .../qcht_diagram_4qubits.png | Bin 0 -> 220690 bytes .../squared_overlap.png | Bin 0 -> 106753 bytes 5 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 _static/demonstration_assets/quantum_chebyshev_transform/amplitudes.png create mode 100644 _static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png create mode 100644 _static/demonstration_assets/quantum_chebyshev_transform/qcht_diagram.png create mode 100644 _static/demonstration_assets/quantum_chebyshev_transform/qcht_diagram_4qubits.png create mode 100644 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z*f5Xu_ZfBoCENo%pO!`W$gm+9s29nuV+!eKX>^b%e|%Ld1O!SzS#pb zV3Z#=<*Sgzoq*_6YTttj9WpouRlw(PKly0VbDjTSvFI!<6SB~V#Depu!_Dv_u)yB@ z7#=@%(1uQl7G)FX8lmFgA!Hw}y5WMG?tZcjjFkzPVbN};_WQP@{$sc52H>ULUBZyY z?f@7cNAw$OL;jo`3_a8-pa6$>4N(S=4RNNWLq1oY_V1adTz4FPTDk;-^0KoVfh+T3 zdnTNBp62l9ZK1gXS8DwF$>!jmp`rM8z0)I#Cj2Y+rxyn_C*>a@`Ymr5|KFVa-!dov ek8k*olQX>UI(t&+P1}&^*W&kOMDv7op8XF=CdSeL literal 0 HcmV?d00001 From e84cd35765fe0ddf45d9bbec684cb6fb1dc4486b Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 10:59:36 -0400 Subject: [PATCH 03/64] Created metadata.json Filled with preliminary information. --- .../metadata.json | 28 +++++++++++++++++++ 1 file changed, 28 insertions(+) create mode 100644 demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json new file mode 100644 index 0000000000..119ae6dd35 --- /dev/null +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json @@ -0,0 +1,28 @@ +{ + "title": "Intro to Quantum Chebyshev Transform", + "authors": [ + { + "username": "colinjdale" + } + ], + "dateOfPublication": "2025-05-09T00:00:00+00:00", + "dateOfLastModification": "2025-05-06T00:00:00+00:00", + "categories": [ + "Algorithms", + "Quantum Computing" + ], + "tags": [], + "previewImages": [], + "seoDescription": "Implement the Quantum Chebyshev Transform in PennyLane.", + "doi": "", + "references": [], + "basedOnPapers": ["10.48550/arXiv.2306.17026"], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_qft", + "weight": 1.0 + } + ] +} From 24824aeb9b35a68b257664f186ebf5a8d8c1714d Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 11:29:57 -0400 Subject: [PATCH 04/64] Finished translating the introduction Removed some punctuation from equations. --- .../demo.py | 25 +++++++++++++++++-- 1 file changed, 23 insertions(+), 2 deletions(-) diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py index d8d2830ba4..798caa9bde 100644 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py @@ -18,7 +18,7 @@ `Chebyshev polynomials ` of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as .. math:: - T_n(x) \equiv \cos(n \arccos(x))\,, + T_n(x) \equiv \cos(n \arccos(x)) where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. .. math:: @@ -28,7 +28,7 @@ T_3(x) &= 4x^3 - 3x \\ T_4(x) &= 8x^4 - 8x^2 + 1 \\ &\ \,\vdots \\ - T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x)\,. + T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. The roots of the :math:`T_n(x)` occur at the values @@ -41,6 +41,27 @@ :align: center :width: 60% :target: javascript:void(0) + +The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of $T_N(x)$ in the following way for $k$, $\ell` can be related to the `discrete cosine transform (DCT) ` to leverage the efficiency of the `fast-Fourier-transform `-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. + +The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. +This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. +In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. + +Quantum Chebyshev basis +--------------------------------------- + """ # From 8e5d164af60a7099f73605c1ab5f967541fb7d4f Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 11:55:17 -0400 Subject: [PATCH 05/64] Fixed hyperlink and reference formatting Added reference section with Williams et al. 2023 --- .../demo.py | 83 +++++++++++++++++-- 1 file changed, 76 insertions(+), 7 deletions(-) diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py index 798caa9bde..14b1ec7ca8 100644 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py @@ -1,7 +1,7 @@ r"""Intro to the Quantum Chebyshev ============================================================= -This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. +This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `_ wherein the authors describe a workflow for quantum Chebyshev-based model building. They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. @@ -12,15 +12,17 @@ :width: 60% :target: javascript:void(0) + What are Chebyshev polynomials? --------------------------------------- -`Chebyshev polynomials ` of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as +`Chebyshev polynomials `_ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as .. math:: T_n(x) \equiv \cos(n \arccos(x)) where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. + .. math:: T_0(x) &= 1 \\ T_1(x) &= x \\ @@ -32,10 +34,11 @@ The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. The roots of the :math:`T_n(x)` occur at the values + .. math:: x_n^\mathrm{Ch} = \cos\left(\frac{2k+1}{2n}\pi\right)\,, \quad k=0, ..., n-1\,. -These are known as the `Chebyshev nodes `. +These are known as the `Chebyshev nodes `_. .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png :align: center @@ -43,6 +46,7 @@ :target: javascript:void(0) The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of $T_N(x)$ in the following way for $k$, $\ell` can be related to the `discrete cosine transform (DCT) ` to leverage the efficiency of the `fast-Fourier-transform `-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `_ can be related to the `discrete cosine transform (DCT) `_ to leverage the efficiency of the `fast-Fourier-transform `_-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. + Quantum Chebyshev basis --------------------------------------- +The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{\ket{x_j}\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{\ket{\tau(x_j^\mathrm{Ch})}\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. + +.. math:: + \ket{\tau(x)} = \frac1{2^{N/2}}T_0(x)\ket{0} + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)\ket{k} + +where :math:`\ket{k}` are the computational basis states and :math:`N` is the number of qubits. These states are orthonormal at the Chebyshev nodes due to the orthogonality condition, that is + +.. math:: + \braket{\tau(x_j^\mathrm{Ch}}{\tau(x_{j'}^\mathrm{Ch})} = \delta_{j, j'} + +The squared overlap if one of the variables is not at a node can be derived analytically as + +.. math:: + \abs{\braket{\tau(x_j^\mathrm{Ch})}{\tau(x)}}^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2} + +The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} \ket{\tau(x_j^\mathrm{Ch})}\bra{x_j}`. + + +Designing the transform circuit +--------------------------------------- +Let's start from the end and look at the circuit diagram generated from the code we want to write. +An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`\ket{x}` which starts in the computational basis, shown below as :math:`\ket{\psi}`. We demonstrate for :math:`N=4` non-ancilla qubits. + +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/QChT_diagram_4qubits.png + :align: center + :width: 60% + :target: javascript:void(0) + +The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. +Notice the use of the `quantum Fourier transform (QFT) `_ applied on all qubits. +The QChT is an extended QFT circuit with some added interference and mixing of the elements. + +Let's break down the circuit above into pieces that we will use inside our circuit function. +First, a Hadamard gate is applied to the ancilla, and then a CNOT ladder is applied, controlled on the ancilla. +To start, we will define a function for the CNOT ladder. """ +import pennylane as qml + +# number of qubits (non-ancilla) +N = 4 + +def CNOT_ladder(): + for wire in range(1, N+1): + qml.CNOT([0, wire]) + +############################################# +# After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. + +# Next are phase rotations and shifts. +# In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. +# The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. +# + + + + +############################################# +# References +# ---------- +# +# .. [#williams2023] +# +# Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving and Oleksandr Kyriienk. "Quantum Chebyshev +# transform: mapping, embedding, learning and sampling distributions." `arxiv:2306.17026 +# `__ (2023). # -# About the authors -# ----------------- +# About the author +# ---------------- # From 1095c80164f3f8bd1b96c0c0d85d23a39f99ddd3 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 12:19:16 -0400 Subject: [PATCH 06/64] Fully translated demo into demo.py --- .../demo.py | 171 +++++++++++++++++- 1 file changed, 170 insertions(+), 1 deletion(-) diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py index 14b1ec7ca8..82e131473c 100644 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py @@ -119,12 +119,181 @@ def CNOT_ladder(): # Next are phase rotations and shifts. # In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. # The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. -# +import numpy as np +pi = np.pi + +def rotate_phases(): + """shift the ancilla's phase and rotate the jth qubit by + pi/2^(j+1) in Z""" + qml.PhaseShift(-pi/2**(N+1), wires=0) + qml.RZ(-pi*(2**N-1)/2**(N+1), wires=0) + for wire in range(1, N+1): + qml.RZ(pi/2**(wire+1), wires=wire) + +############################################# +# Now a permutation of the qubits is used to reorder them. +# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. The multicontrolled NOT gate can be implimented using a multicontrolled Pauli X gate. +# Let's see what that looks like. + +def permute_elements(): + """reorders amplitudes of the conditioned states""" + for wire in reversed(range(1, N+1)): + control_wires = [0] + list(range(wire+1,N+1)) + qml.MultiControlledX(wires=(*control_wires, wire)) + +############################################# +# In the above code, we use \texttt{reversed} to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. +# After the permutation is another CNOT ladder, which we already have a function for. +# +# The last part is a phase adjustment of the ancilla qubit: a phase shift of $-\pi/2$, followed by a rotation in $Y$ by $\pi/2$ and a multicontrolled $X$ rotation by $\pi/2$. +# All of the other qubits control the $X$ rotation, but the control is sandwiched by Pauli $X$ operators. We can impliment the multicontrolled $X$ rotation using \texttt{qml.ControlledQubitUnitary}. + +def adjust_phase(): + """adjusts the phase of the ancilla qubit""" + qml.PhaseShift(-pi/2, wires=0) + qml.RY(-pi/2, wires=0) + # first Pauli Xs + for wire in range(1, N+1): + qml.PauliX(wires=wire) + # controlled RX gate + qml.ControlledQubitUnitary(qml.RX(pi/2, wires=0), range(1, N+1)) + # second Pauli Xs + for wire in range(1, N+1): + qml.PauliX(wires=wire) + +############################################# +# All together, we can construct the circuit. +# We have added :class:`qml.BasisState` to initialize the input in any computational basis state with the optional argument ``state``. + +def QChT(): + qml.Hadamard(wires=0) + CNOT_ladder() + qml.QFT(wires=range(N+1)) + rotate_phases() + permute_elements() + CNOT_ladder() + adjust_phases() + +dev = qml.device("default.qubit") +@qml.qnode(dev) +def circuit(state=None): + qml.BasisState(state=state, wires=range(1,N+1)) + QChT() + return qml.state() + +############################################# +# Testing the QChT +# ---------------- +# With our QChT circuit, let's see if the orthonormality described earlier holds. +# To do this, we'll use the computational state :math:`\ket{7}`, which will transform into :math:`\ket{\tau(x_7^\mathrm{Ch})}`. +# Then, we will compute the overlap at the nodes with all other :math:`\ket{\tau(x_j^\mathrm{Ch})}`. + + +j = 7 # initial state in computational basis + +# compute state after transform +total_state = circuit(state=j)#[:2**N] + +# reduce state size, effectively removing the ancilla +state = total_state[:2**N]*np.sqrt(2) + +# compute nodes +def ch_node(j, N): + return np.cos(pi*(2*j+1)/2**(N+1)) +js = list(range(int(len(state)/2))) +nodes = [ch_node(i, N) for i in js] +# compute overlap with other basis states using np.inner() +overlaps = [] +for i in js: + state_i = circuit(state=i)[:2**N]*np.sqrt(2) + overlaps.append(np.inner(state, state_i)) ############################################# +# Now we plot the squared overlaps at the nodes computed from the circuit transformation, and compare to the definition, plotting the squared overlaps at all values of :math:`x`. +# + +import matplotlib.pyplot as plt + +def T_n(x, n): + """Chebyshev polynomial of order n""" + return np.cos(n * np.arccos(x)) + +def overlap_sq(x, xp, N): + """computes the squared overlap""" + numerator = T_n(xp, 2**N+1) * T_n(x, 2**N) - \ + T_n(xp, 2**N) * T_n(x, 2**N+1) + return numerator**2/(2**(2*N))/(xp - x)**2 + +plt.style.use('pennylane.drawer.plot') +fig, ax = plt.subplots(figsize=(6,4)) +ax.set(xlabel="x", ylabel="Squared Overlap") + +# plot squared overlaps computed in circuit +ax.plot(nodes, np.abs(overlaps)**2, marker='o', label='circuit') + +# plot expected squared overlaps +xs = np.linspace(-1, 1, 1000) +ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label='expectation') + +ax.legend() +plt.show() + + +############################################# +# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/squared_overlap.png +# :align: center +# :width: 60% +# :target: javascript:void(0) +# +# Almost orthonormal. We note that the amplitude at the corresponding node approaches :math:`1` if the number of qubits is increased. +# +# Let's also see if the amplitudes of the state in the computational basis agree with expectation. +# To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). + +@qml.qnode(dev) +def circuit(state=None): + qml.BasisState(state=state, wires=range(1,N+1)) + QChT() + return qml.probs(wires=range(1,N+1)) + +probs = circuit(state=j) + +# computational basis indeces +x = range(2**N) + +def tau_amplitudes(x, k, N): + """computes the expected amplitudes of tau""" + if k == 0: + prefactor = 1/2**(N/2) + else: + prefactor = 1/2**((N-1)/2) + return prefactor * np.cos(k*np.arccos(x)) + +fig, ax = plt.subplots(figsize=(6,4)) +ax.plot(x, probs, 'o', label='circuit') +ax.plot(x, [tau_amplitudes(nodes[j], xs, N)**2 for xs in x], label='expectation') +ax.set(xlabel="|k>", ylabel="Probability") +ax.legend() +plt.show() + +############################################# +# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/amplitudes.png +# :align: center +# :width: 60% +# :target: javascript:void(0) +# +# The circuit output probabilities are exactly what we want. +# +# +# Conclusion +# ---------- +# In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. +# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. +# +# # References # ---------- # From 1942bbe36ff8c5febd40698b68625ac1f31d0849 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 6 May 2025 17:32:35 -0400 Subject: [PATCH 07/64] Fixed hyperlinks again. --- .../tutorial_quantum_chebyshev_transform/demo.py | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py index 82e131473c..d4d4961be3 100644 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py @@ -1,7 +1,7 @@ r"""Intro to the Quantum Chebyshev ============================================================= -This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `_ wherein the authors describe a workflow for quantum Chebyshev-based model building. +This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. @@ -16,7 +16,7 @@ What are Chebyshev polynomials? --------------------------------------- -`Chebyshev polynomials `_ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as +`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as .. math:: T_n(x) \equiv \cos(n \arccos(x)) @@ -38,7 +38,7 @@ .. math:: x_n^\mathrm{Ch} = \cos\left(\frac{2k+1}{2n}\pi\right)\,, \quad k=0, ..., n-1\,. -These are known as the `Chebyshev nodes `_. +These are known as the `Chebyshev nodes `__. .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png :align: center @@ -57,7 +57,7 @@ The Chebyshev polynomials have a lot of *nice* properties. Because they are complete, any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded in :math:`T_n(x)` up to order :math`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. \ To do this process numerically for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions \emoji{snail}. -However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `_ can be related to the `discrete cosine transform (DCT) `_ to leverage the efficiency of the `fast-Fourier-transform `_-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. @@ -95,7 +95,7 @@ :target: javascript:void(0) The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. -Notice the use of the `quantum Fourier transform (QFT) `_ applied on all qubits. +Notice the use of the `quantum Fourier transform (QFT) `__ applied on all qubits. The QChT is an extended QFT circuit with some added interference and mixing of the elements. Let's break down the circuit above into pieces that we will use inside our circuit function. From c1085a5a183dde1c98b537cccccfe30805eb0aa0 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 09:23:14 -0400 Subject: [PATCH 08/64] Updated metadata.json Added de Lejarza 2025 paper to references which uses the transform for generative modelling. Not yet referenced in text. --- .../metadata.json | 29 +++++++++++++++++-- 1 file changed, 26 insertions(+), 3 deletions(-) diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json index 119ae6dd35..7e84b169bb 100644 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json @@ -6,16 +6,34 @@ } ], "dateOfPublication": "2025-05-09T00:00:00+00:00", - "dateOfLastModification": "2025-05-06T00:00:00+00:00", + "dateOfLastModification": "2025-05-07T00:00:00+00:00", "categories": [ "Algorithms", - "Quantum Computing" + "Quantum Machine Learning" ], "tags": [], "previewImages": [], "seoDescription": "Implement the Quantum Chebyshev Transform in PennyLane.", "doi": "", - "references": [], + "references": [ + { + "id": "williams2023", + "type": "article", + "title": "Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions", + "authors": "Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving, Oleksandr Kyriienk", + "year": "2023", + "journal": "", + "url": "https://arxiv.org/abs/2306.17026" + }, + { + "id": "delejarza2025", + "type": "arcticle", + "title":"Quantum Chebyshev Probabilistic Models for Fragmentation Functions", + "authors": "Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi", + "year": "2025", + "url": "https://arxiv.org/abs/2503.16073" + } + ], "basedOnPapers": ["10.48550/arXiv.2306.17026"], "referencedByPapers": [], "relatedContent": [ @@ -23,6 +41,11 @@ "type": "demonstration", "id": "tutorial_qft", "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_qcbm", + "weight": 1.0 } ] } From 53fe92f2c08036e6f90344d5026b3b9d678b94a5 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 10:13:13 -0400 Subject: [PATCH 09/64] Added placeholder thumbnail. 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z@ipmQi^!?(lzAyFJAkD`SK{q)A` Date: Wed, 7 May 2025 10:36:57 -0400 Subject: [PATCH 10/64] Copied demo.py and metadata to demonstrations. Copied relevant files to demonstrations, so they are now in both demonstrations and demonstrations_v2. --- ..._quantum_chebyshev_transform.metadata.json | 56 ++++ .../tutorial_quantum_chebyshev_transform.py | 308 ++++++++++++++++++ 2 files changed, 364 insertions(+) create mode 100644 demonstrations/tutorial_quantum_chebyshev_transform.metadata.json create mode 100644 demonstrations/tutorial_quantum_chebyshev_transform.py diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json new file mode 100644 index 0000000000..9d86f2df71 --- /dev/null +++ b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json @@ -0,0 +1,56 @@ +{ + "title": "Intro to Quantum Chebyshev Transform", + "authors": [ + { + "username": "colinjdale" + } + ], + "dateOfPublication": "2025-05-09T00:00:00+00:00", + "dateOfLastModification": "2025-05-07T00:00:00+00:00", + "categories": [ + "Algorithms", + "Quantum Machine Learning" + ], + "tags": [], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png" + } + ], + "seoDescription": "Implement the Quantum Chebyshev Transform in PennyLane.", + "doi": "", + "references": [ + { + "id": "williams2023", + "type": "article", + "title": "Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions", + "authors": "Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving, Oleksandr Kyriienk", + "year": "2023", + "journal": "", + "url": "https://arxiv.org/abs/2306.17026" + }, + { + "id": "delejarza2025", + "type": "arcticle", + "title":"Quantum Chebyshev Probabilistic Models for Fragmentation Functions", + "authors": "Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi", + "year": "2025", + "url": "https://arxiv.org/abs/2503.16073" + } + ], + "basedOnPapers": ["10.48550/arXiv.2306.17026"], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_qft", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_qcbm", + "weight": 1.0 + } + ] +} diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py new file mode 100644 index 0000000000..d4d4961be3 --- /dev/null +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -0,0 +1,308 @@ +r"""Intro to the Quantum Chebyshev +============================================================= + +This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. +They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. + +We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. + + +.. figure:: ../_static/demonstration_assets/qft/socialthumbnail_large_QFT_2024-04-04.png + :align: center + :width: 60% + :target: javascript:void(0) + + +What are Chebyshev polynomials? +--------------------------------------- + +`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as + +.. math:: + T_n(x) \equiv \cos(n \arccos(x)) + +where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. + +.. math:: + T_0(x) &= 1 \\ + T_1(x) &= x \\ + T_2(x) &= 2x^2-1 \\ + T_3(x) &= 4x^3 - 3x \\ + T_4(x) &= 8x^4 - 8x^2 + 1 \\ + &\ \,\vdots \\ + T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) + +The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. +The roots of the :math:`T_n(x)` occur at the values + +.. math:: + x_n^\mathrm{Ch} = \cos\left(\frac{2k+1}{2n}\pi\right)\,, \quad k=0, ..., n-1\,. + +These are known as the `Chebyshev nodes `__. + +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png + :align: center + :width: 60% + :target: javascript:void(0) + +The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of $T_N(x)$ in the following way for $k$, $\ell`__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. + +The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. +This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. +In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. + + +Quantum Chebyshev basis +--------------------------------------- +The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{\ket{x_j}\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{\ket{\tau(x_j^\mathrm{Ch})}\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. + +.. math:: + \ket{\tau(x)} = \frac1{2^{N/2}}T_0(x)\ket{0} + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)\ket{k} + +where :math:`\ket{k}` are the computational basis states and :math:`N` is the number of qubits. These states are orthonormal at the Chebyshev nodes due to the orthogonality condition, that is + +.. math:: + \braket{\tau(x_j^\mathrm{Ch}}{\tau(x_{j'}^\mathrm{Ch})} = \delta_{j, j'} + +The squared overlap if one of the variables is not at a node can be derived analytically as + +.. math:: + \abs{\braket{\tau(x_j^\mathrm{Ch})}{\tau(x)}}^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2} + +The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} \ket{\tau(x_j^\mathrm{Ch})}\bra{x_j}`. + + +Designing the transform circuit +--------------------------------------- +Let's start from the end and look at the circuit diagram generated from the code we want to write. +An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`\ket{x}` which starts in the computational basis, shown below as :math:`\ket{\psi}`. We demonstrate for :math:`N=4` non-ancilla qubits. + +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/QChT_diagram_4qubits.png + :align: center + :width: 60% + :target: javascript:void(0) + +The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. +Notice the use of the `quantum Fourier transform (QFT) `__ applied on all qubits. +The QChT is an extended QFT circuit with some added interference and mixing of the elements. + +Let's break down the circuit above into pieces that we will use inside our circuit function. +First, a Hadamard gate is applied to the ancilla, and then a CNOT ladder is applied, controlled on the ancilla. +To start, we will define a function for the CNOT ladder. + +""" + +import pennylane as qml + +# number of qubits (non-ancilla) +N = 4 + +def CNOT_ladder(): + for wire in range(1, N+1): + qml.CNOT([0, wire]) + +############################################# +# After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. + +# Next are phase rotations and shifts. +# In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. +# The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. + +import numpy as np +pi = np.pi + +def rotate_phases(): + """shift the ancilla's phase and rotate the jth qubit by + pi/2^(j+1) in Z""" + qml.PhaseShift(-pi/2**(N+1), wires=0) + qml.RZ(-pi*(2**N-1)/2**(N+1), wires=0) + for wire in range(1, N+1): + qml.RZ(pi/2**(wire+1), wires=wire) + +############################################# +# Now a permutation of the qubits is used to reorder them. +# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. The multicontrolled NOT gate can be implimented using a multicontrolled Pauli X gate. +# Let's see what that looks like. + +def permute_elements(): + """reorders amplitudes of the conditioned states""" + for wire in reversed(range(1, N+1)): + control_wires = [0] + list(range(wire+1,N+1)) + qml.MultiControlledX(wires=(*control_wires, wire)) + +############################################# +# In the above code, we use \texttt{reversed} to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. +# After the permutation is another CNOT ladder, which we already have a function for. +# +# The last part is a phase adjustment of the ancilla qubit: a phase shift of $-\pi/2$, followed by a rotation in $Y$ by $\pi/2$ and a multicontrolled $X$ rotation by $\pi/2$. +# All of the other qubits control the $X$ rotation, but the control is sandwiched by Pauli $X$ operators. We can impliment the multicontrolled $X$ rotation using \texttt{qml.ControlledQubitUnitary}. + +def adjust_phase(): + """adjusts the phase of the ancilla qubit""" + qml.PhaseShift(-pi/2, wires=0) + qml.RY(-pi/2, wires=0) + # first Pauli Xs + for wire in range(1, N+1): + qml.PauliX(wires=wire) + # controlled RX gate + qml.ControlledQubitUnitary(qml.RX(pi/2, wires=0), range(1, N+1)) + # second Pauli Xs + for wire in range(1, N+1): + qml.PauliX(wires=wire) + +############################################# +# All together, we can construct the circuit. +# We have added :class:`qml.BasisState` to initialize the input in any computational basis state with the optional argument ``state``. + +def QChT(): + qml.Hadamard(wires=0) + CNOT_ladder() + qml.QFT(wires=range(N+1)) + rotate_phases() + permute_elements() + CNOT_ladder() + adjust_phases() + +dev = qml.device("default.qubit") +@qml.qnode(dev) +def circuit(state=None): + qml.BasisState(state=state, wires=range(1,N+1)) + QChT() + return qml.state() + +############################################# +# Testing the QChT +# ---------------- +# With our QChT circuit, let's see if the orthonormality described earlier holds. +# To do this, we'll use the computational state :math:`\ket{7}`, which will transform into :math:`\ket{\tau(x_7^\mathrm{Ch})}`. +# Then, we will compute the overlap at the nodes with all other :math:`\ket{\tau(x_j^\mathrm{Ch})}`. + + +j = 7 # initial state in computational basis + +# compute state after transform +total_state = circuit(state=j)#[:2**N] + +# reduce state size, effectively removing the ancilla +state = total_state[:2**N]*np.sqrt(2) + +# compute nodes +def ch_node(j, N): + return np.cos(pi*(2*j+1)/2**(N+1)) + +js = list(range(int(len(state)/2))) +nodes = [ch_node(i, N) for i in js] + +# compute overlap with other basis states using np.inner() +overlaps = [] +for i in js: + state_i = circuit(state=i)[:2**N]*np.sqrt(2) + overlaps.append(np.inner(state, state_i)) + +############################################# +# Now we plot the squared overlaps at the nodes computed from the circuit transformation, and compare to the definition, plotting the squared overlaps at all values of :math:`x`. +# + +import matplotlib.pyplot as plt + +def T_n(x, n): + """Chebyshev polynomial of order n""" + return np.cos(n * np.arccos(x)) + +def overlap_sq(x, xp, N): + """computes the squared overlap""" + numerator = T_n(xp, 2**N+1) * T_n(x, 2**N) - \ + T_n(xp, 2**N) * T_n(x, 2**N+1) + return numerator**2/(2**(2*N))/(xp - x)**2 + +plt.style.use('pennylane.drawer.plot') +fig, ax = plt.subplots(figsize=(6,4)) +ax.set(xlabel="x", ylabel="Squared Overlap") + +# plot squared overlaps computed in circuit +ax.plot(nodes, np.abs(overlaps)**2, marker='o', label='circuit') + +# plot expected squared overlaps +xs = np.linspace(-1, 1, 1000) +ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label='expectation') + +ax.legend() +plt.show() + + +############################################# +# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/squared_overlap.png +# :align: center +# :width: 60% +# :target: javascript:void(0) +# +# Almost orthonormal. We note that the amplitude at the corresponding node approaches :math:`1` if the number of qubits is increased. +# +# Let's also see if the amplitudes of the state in the computational basis agree with expectation. +# To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). + +@qml.qnode(dev) +def circuit(state=None): + qml.BasisState(state=state, wires=range(1,N+1)) + QChT() + return qml.probs(wires=range(1,N+1)) + +probs = circuit(state=j) + +# computational basis indeces +x = range(2**N) + +def tau_amplitudes(x, k, N): + """computes the expected amplitudes of tau""" + if k == 0: + prefactor = 1/2**(N/2) + else: + prefactor = 1/2**((N-1)/2) + return prefactor * np.cos(k*np.arccos(x)) + +fig, ax = plt.subplots(figsize=(6,4)) +ax.plot(x, probs, 'o', label='circuit') +ax.plot(x, [tau_amplitudes(nodes[j], xs, N)**2 for xs in x], label='expectation') +ax.set(xlabel="|k>", ylabel="Probability") +ax.legend() +plt.show() + +############################################# +# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/amplitudes.png +# :align: center +# :width: 60% +# :target: javascript:void(0) +# +# The circuit output probabilities are exactly what we want. +# +# +# Conclusion +# ---------- +# In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. +# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. +# +# +# References +# ---------- +# +# .. [#williams2023] +# +# Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving and Oleksandr Kyriienk. "Quantum Chebyshev +# transform: mapping, embedding, learning and sampling distributions." `arxiv:2306.17026 +# `__ (2023). +# +# About the author +# ---------------- +# From 25ddd90186264f52afbf39edc597a233264648e6 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 10:44:11 -0400 Subject: [PATCH 11/64] Typo in metadata Fixed typo in metadata in both demonstrations folders --- .../tutorial_quantum_chebyshev_transform.metadata.json | 2 +- .../tutorial_quantum_chebyshev_transform/metadata.json | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json index 9d86f2df71..0d2992d503 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json +++ b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json @@ -32,7 +32,7 @@ }, { "id": "delejarza2025", - "type": "arcticle", + "type": "article", "title":"Quantum Chebyshev Probabilistic Models for Fragmentation Functions", "authors": "Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi", "year": "2025", diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json index 9d86f2df71..0d2992d503 100644 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json +++ b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json @@ -32,7 +32,7 @@ }, { "id": "delejarza2025", - "type": "arcticle", + "type": "article", "title":"Quantum Chebyshev Probabilistic Models for Fragmentation Functions", "authors": "Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi", "year": "2025", From 27fc22399f6b3d821398c3e5c22491d86534ac9d Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 11:13:32 -0400 Subject: [PATCH 12/64] Removed demo and metadata from demonstrations_v2 So that I don't have to edit both, and since I don't know what demonstrations_v2 is. --- .../demo.py | 308 ------------------ .../metadata.json | 56 ---- 2 files changed, 364 deletions(-) delete mode 100644 demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py delete mode 100644 demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py b/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py deleted file mode 100644 index d4d4961be3..0000000000 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/demo.py +++ /dev/null @@ -1,308 +0,0 @@ -r"""Intro to the Quantum Chebyshev -============================================================= - -This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. -They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. - -We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. - - -.. figure:: ../_static/demonstration_assets/qft/socialthumbnail_large_QFT_2024-04-04.png - :align: center - :width: 60% - :target: javascript:void(0) - - -What are Chebyshev polynomials? ---------------------------------------- - -`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as - -.. math:: - T_n(x) \equiv \cos(n \arccos(x)) - -where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. - -.. math:: - T_0(x) &= 1 \\ - T_1(x) &= x \\ - T_2(x) &= 2x^2-1 \\ - T_3(x) &= 4x^3 - 3x \\ - T_4(x) &= 8x^4 - 8x^2 + 1 \\ - &\ \,\vdots \\ - T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) - -The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. -The roots of the :math:`T_n(x)` occur at the values - -.. math:: - x_n^\mathrm{Ch} = \cos\left(\frac{2k+1}{2n}\pi\right)\,, \quad k=0, ..., n-1\,. - -These are known as the `Chebyshev nodes `__. - -.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png - :align: center - :width: 60% - :target: javascript:void(0) - -The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of $T_N(x)$ in the following way for $k$, $\ell`__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. - -The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. -This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. -In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. - - -Quantum Chebyshev basis ---------------------------------------- -The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{\ket{x_j}\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{\ket{\tau(x_j^\mathrm{Ch})}\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. - -.. math:: - \ket{\tau(x)} = \frac1{2^{N/2}}T_0(x)\ket{0} + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)\ket{k} - -where :math:`\ket{k}` are the computational basis states and :math:`N` is the number of qubits. These states are orthonormal at the Chebyshev nodes due to the orthogonality condition, that is - -.. math:: - \braket{\tau(x_j^\mathrm{Ch}}{\tau(x_{j'}^\mathrm{Ch})} = \delta_{j, j'} - -The squared overlap if one of the variables is not at a node can be derived analytically as - -.. math:: - \abs{\braket{\tau(x_j^\mathrm{Ch})}{\tau(x)}}^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2} - -The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} \ket{\tau(x_j^\mathrm{Ch})}\bra{x_j}`. - - -Designing the transform circuit ---------------------------------------- -Let's start from the end and look at the circuit diagram generated from the code we want to write. -An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`\ket{x}` which starts in the computational basis, shown below as :math:`\ket{\psi}`. We demonstrate for :math:`N=4` non-ancilla qubits. - -.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/QChT_diagram_4qubits.png - :align: center - :width: 60% - :target: javascript:void(0) - -The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. -Notice the use of the `quantum Fourier transform (QFT) `__ applied on all qubits. -The QChT is an extended QFT circuit with some added interference and mixing of the elements. - -Let's break down the circuit above into pieces that we will use inside our circuit function. -First, a Hadamard gate is applied to the ancilla, and then a CNOT ladder is applied, controlled on the ancilla. -To start, we will define a function for the CNOT ladder. - -""" - -import pennylane as qml - -# number of qubits (non-ancilla) -N = 4 - -def CNOT_ladder(): - for wire in range(1, N+1): - qml.CNOT([0, wire]) - -############################################# -# After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. - -# Next are phase rotations and shifts. -# In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. -# The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. - -import numpy as np -pi = np.pi - -def rotate_phases(): - """shift the ancilla's phase and rotate the jth qubit by - pi/2^(j+1) in Z""" - qml.PhaseShift(-pi/2**(N+1), wires=0) - qml.RZ(-pi*(2**N-1)/2**(N+1), wires=0) - for wire in range(1, N+1): - qml.RZ(pi/2**(wire+1), wires=wire) - -############################################# -# Now a permutation of the qubits is used to reorder them. -# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. The multicontrolled NOT gate can be implimented using a multicontrolled Pauli X gate. -# Let's see what that looks like. - -def permute_elements(): - """reorders amplitudes of the conditioned states""" - for wire in reversed(range(1, N+1)): - control_wires = [0] + list(range(wire+1,N+1)) - qml.MultiControlledX(wires=(*control_wires, wire)) - -############################################# -# In the above code, we use \texttt{reversed} to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. -# After the permutation is another CNOT ladder, which we already have a function for. -# -# The last part is a phase adjustment of the ancilla qubit: a phase shift of $-\pi/2$, followed by a rotation in $Y$ by $\pi/2$ and a multicontrolled $X$ rotation by $\pi/2$. -# All of the other qubits control the $X$ rotation, but the control is sandwiched by Pauli $X$ operators. We can impliment the multicontrolled $X$ rotation using \texttt{qml.ControlledQubitUnitary}. - -def adjust_phase(): - """adjusts the phase of the ancilla qubit""" - qml.PhaseShift(-pi/2, wires=0) - qml.RY(-pi/2, wires=0) - # first Pauli Xs - for wire in range(1, N+1): - qml.PauliX(wires=wire) - # controlled RX gate - qml.ControlledQubitUnitary(qml.RX(pi/2, wires=0), range(1, N+1)) - # second Pauli Xs - for wire in range(1, N+1): - qml.PauliX(wires=wire) - -############################################# -# All together, we can construct the circuit. -# We have added :class:`qml.BasisState` to initialize the input in any computational basis state with the optional argument ``state``. - -def QChT(): - qml.Hadamard(wires=0) - CNOT_ladder() - qml.QFT(wires=range(N+1)) - rotate_phases() - permute_elements() - CNOT_ladder() - adjust_phases() - -dev = qml.device("default.qubit") -@qml.qnode(dev) -def circuit(state=None): - qml.BasisState(state=state, wires=range(1,N+1)) - QChT() - return qml.state() - -############################################# -# Testing the QChT -# ---------------- -# With our QChT circuit, let's see if the orthonormality described earlier holds. -# To do this, we'll use the computational state :math:`\ket{7}`, which will transform into :math:`\ket{\tau(x_7^\mathrm{Ch})}`. -# Then, we will compute the overlap at the nodes with all other :math:`\ket{\tau(x_j^\mathrm{Ch})}`. - - -j = 7 # initial state in computational basis - -# compute state after transform -total_state = circuit(state=j)#[:2**N] - -# reduce state size, effectively removing the ancilla -state = total_state[:2**N]*np.sqrt(2) - -# compute nodes -def ch_node(j, N): - return np.cos(pi*(2*j+1)/2**(N+1)) - -js = list(range(int(len(state)/2))) -nodes = [ch_node(i, N) for i in js] - -# compute overlap with other basis states using np.inner() -overlaps = [] -for i in js: - state_i = circuit(state=i)[:2**N]*np.sqrt(2) - overlaps.append(np.inner(state, state_i)) - -############################################# -# Now we plot the squared overlaps at the nodes computed from the circuit transformation, and compare to the definition, plotting the squared overlaps at all values of :math:`x`. -# - -import matplotlib.pyplot as plt - -def T_n(x, n): - """Chebyshev polynomial of order n""" - return np.cos(n * np.arccos(x)) - -def overlap_sq(x, xp, N): - """computes the squared overlap""" - numerator = T_n(xp, 2**N+1) * T_n(x, 2**N) - \ - T_n(xp, 2**N) * T_n(x, 2**N+1) - return numerator**2/(2**(2*N))/(xp - x)**2 - -plt.style.use('pennylane.drawer.plot') -fig, ax = plt.subplots(figsize=(6,4)) -ax.set(xlabel="x", ylabel="Squared Overlap") - -# plot squared overlaps computed in circuit -ax.plot(nodes, np.abs(overlaps)**2, marker='o', label='circuit') - -# plot expected squared overlaps -xs = np.linspace(-1, 1, 1000) -ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label='expectation') - -ax.legend() -plt.show() - - -############################################# -# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/squared_overlap.png -# :align: center -# :width: 60% -# :target: javascript:void(0) -# -# Almost orthonormal. We note that the amplitude at the corresponding node approaches :math:`1` if the number of qubits is increased. -# -# Let's also see if the amplitudes of the state in the computational basis agree with expectation. -# To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). - -@qml.qnode(dev) -def circuit(state=None): - qml.BasisState(state=state, wires=range(1,N+1)) - QChT() - return qml.probs(wires=range(1,N+1)) - -probs = circuit(state=j) - -# computational basis indeces -x = range(2**N) - -def tau_amplitudes(x, k, N): - """computes the expected amplitudes of tau""" - if k == 0: - prefactor = 1/2**(N/2) - else: - prefactor = 1/2**((N-1)/2) - return prefactor * np.cos(k*np.arccos(x)) - -fig, ax = plt.subplots(figsize=(6,4)) -ax.plot(x, probs, 'o', label='circuit') -ax.plot(x, [tau_amplitudes(nodes[j], xs, N)**2 for xs in x], label='expectation') -ax.set(xlabel="|k>", ylabel="Probability") -ax.legend() -plt.show() - -############################################# -# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/amplitudes.png -# :align: center -# :width: 60% -# :target: javascript:void(0) -# -# The circuit output probabilities are exactly what we want. -# -# -# Conclusion -# ---------- -# In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. -# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. -# -# -# References -# ---------- -# -# .. [#williams2023] -# -# Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving and Oleksandr Kyriienk. "Quantum Chebyshev -# transform: mapping, embedding, learning and sampling distributions." `arxiv:2306.17026 -# `__ (2023). -# -# About the author -# ---------------- -# diff --git a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json b/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json deleted file mode 100644 index 0d2992d503..0000000000 --- a/demonstrations_v2/tutorial_quantum_chebyshev_transform/metadata.json +++ /dev/null @@ -1,56 +0,0 @@ -{ - "title": "Intro to Quantum Chebyshev Transform", - "authors": [ - { - "username": "colinjdale" - } - ], - "dateOfPublication": "2025-05-09T00:00:00+00:00", - "dateOfLastModification": "2025-05-07T00:00:00+00:00", - "categories": [ - "Algorithms", - "Quantum Machine Learning" - ], - "tags": [], - "previewImages": [ - { - "type": "thumbnail", - "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png" - } - ], - "seoDescription": "Implement the Quantum Chebyshev Transform in PennyLane.", - "doi": "", - "references": [ - { - "id": "williams2023", - "type": "article", - "title": "Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions", - "authors": "Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving, Oleksandr Kyriienk", - "year": "2023", - "journal": "", - "url": "https://arxiv.org/abs/2306.17026" - }, - { - "id": "delejarza2025", - "type": "article", - "title":"Quantum Chebyshev Probabilistic Models for Fragmentation Functions", - "authors": "Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi", - "year": "2025", - "url": "https://arxiv.org/abs/2503.16073" - } - ], - "basedOnPapers": ["10.48550/arXiv.2306.17026"], - "referencedByPapers": [], - "relatedContent": [ - { - "type": "demonstration", - "id": "tutorial_qft", - "weight": 1.0 - }, - { - "type": "demonstration", - "id": "tutorial_qcbm", - "weight": 1.0 - } - ] -} From 8d7669bff396262ca2790869e04ae89fec3d873b Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 11:14:08 -0400 Subject: [PATCH 13/64] Fixed some math mode text, and a function definition typo. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index d4d4961be3..502648573a 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -143,13 +143,13 @@ def permute_elements(): qml.MultiControlledX(wires=(*control_wires, wire)) ############################################# -# In the above code, we use \texttt{reversed} to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. +# In the above code, we use ``reversed`` to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. # After the permutation is another CNOT ladder, which we already have a function for. # -# The last part is a phase adjustment of the ancilla qubit: a phase shift of $-\pi/2$, followed by a rotation in $Y$ by $\pi/2$ and a multicontrolled $X$ rotation by $\pi/2$. -# All of the other qubits control the $X$ rotation, but the control is sandwiched by Pauli $X$ operators. We can impliment the multicontrolled $X$ rotation using \texttt{qml.ControlledQubitUnitary}. +# The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. +# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. We can impliment the multicontrolled :math:`X` rotation using :class:`qml.ControlledQubitUnitary`. -def adjust_phase(): +def adjust_phases(): """adjusts the phase of the ancilla qubit""" qml.PhaseShift(-pi/2, wires=0) qml.RY(-pi/2, wires=0) @@ -275,7 +275,7 @@ def tau_amplitudes(x, k, N): fig, ax = plt.subplots(figsize=(6,4)) ax.plot(x, probs, 'o', label='circuit') ax.plot(x, [tau_amplitudes(nodes[j], xs, N)**2 for xs in x], label='expectation') -ax.set(xlabel="|k>", ylabel="Probability") +ax.set(xlabel=r"|k$\rangle$", ylabel="Probability") ax.legend() plt.show() From 1b5b50c0c58d565355e30a2fa2d370748b5a170e Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 11:34:32 -0400 Subject: [PATCH 14/64] Changed the depreciated qml.ControlledQubitUnitary ... to qml.ctrl on the required unitary. Edited text to properly explain the new implimentation of the multicontrolled RX. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 502648573a..59d73b5377 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -133,7 +133,8 @@ def rotate_phases(): ############################################# # Now a permutation of the qubits is used to reorder them. -# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. The multicontrolled NOT gate can be implimented using a multicontrolled Pauli X gate. +# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. +# The multicontrolled NOT gate can be implimented using a multicontrolled Pauli X gate. # Let's see what that looks like. def permute_elements(): @@ -147,7 +148,7 @@ def permute_elements(): # After the permutation is another CNOT ladder, which we already have a function for. # # The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. -# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. We can impliment the multicontrolled :math:`X` rotation using :class:`qml.ControlledQubitUnitary`. +# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. We can impliment the multicontrolled :math:`X` rotation using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX``, and the control wires as the second argument of ``qml.ctrl``. def adjust_phases(): """adjusts the phase of the ancilla qubit""" @@ -157,7 +158,7 @@ def adjust_phases(): for wire in range(1, N+1): qml.PauliX(wires=wire) # controlled RX gate - qml.ControlledQubitUnitary(qml.RX(pi/2, wires=0), range(1, N+1)) + qml.ctrl(qml.RX(pi/2, wires=0), range(1, N+1)) # second Pauli Xs for wire in range(1, N+1): qml.PauliX(wires=wire) @@ -202,7 +203,7 @@ def circuit(state=None): def ch_node(j, N): return np.cos(pi*(2*j+1)/2**(N+1)) -js = list(range(int(len(state)/2))) +js = list(range(int(len(state)))) nodes = [ch_node(i, N) for i in js] # compute overlap with other basis states using np.inner() From d09e721a81038ede513a6e53acb852d15ad415bb Mon Sep 17 00:00:00 2001 From: "Colin J. 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Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 12:25:17 -0400 Subject: [PATCH 16/64] Fixed LaTeX macros and redundant figures Removed redundant figures, fixed latex macros, and emojis. Some typoes. --- .../tutorial_quantum_chebyshev_transform.py | 56 +++++++++---------- 1 file changed, 27 insertions(+), 29 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 59d73b5377..7c4d9d12f1 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -2,7 +2,8 @@ ============================================================= This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. -They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. +They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. +Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. @@ -13,6 +14,7 @@ :target: javascript:void(0) + What are Chebyshev polynomials? --------------------------------------- @@ -32,7 +34,8 @@ &\ \,\vdots \\ T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) -The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. +The recursion relation in the last line can be used to compute the next orders. +Observe that odd and even order :math:`T_n` are odd and even functions, respectively. The roots of the :math:`T_n(x)` occur at the values .. math:: @@ -55,41 +58,42 @@ N/2 & k = \ell \neq 0 \end{cases} -The Chebyshev polynomials have a lot of *nice* properties. Because they are complete, any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded in :math:`T_n(x)` up to order :math`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. \ -To do this process numerically for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions \emoji{snail}. -However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations \emoji{rocket}. +The Chebyshev polynomials have a lot of *nice* properties. Because they are complete, any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. \ +To do this process numerically for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. -This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. +This non-uniform sampling has more resolution at the boundary, but less in the middle. +This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. Quantum Chebyshev basis --------------------------------------- -The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{\ket{x_j}\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{\ket{\tau(x_j^\mathrm{Ch})}\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. +The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau\rangle(x_j^\mathrm{Ch})}\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. .. math:: - \ket{\tau(x)} = \frac1{2^{N/2}}T_0(x)\ket{0} + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)\ket{k} + |\tau(x)\rangle = \frac1{2^{N/2}}T_0(x)|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)|k\rangle -where :math:`\ket{k}` are the computational basis states and :math:`N` is the number of qubits. These states are orthonormal at the Chebyshev nodes due to the orthogonality condition, that is +where :math:`|k\rangle` are the computational basis states and :math:`N` is the number of qubits. These states are orthonormal at the Chebyshev nodes due to the orthogonality condition, that is .. math:: - \braket{\tau(x_j^\mathrm{Ch}}{\tau(x_{j'}^\mathrm{Ch})} = \delta_{j, j'} + \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \delta_{j, j'} The squared overlap if one of the variables is not at a node can be derived analytically as .. math:: - \abs{\braket{\tau(x_j^\mathrm{Ch})}{\tau(x)}}^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2} + |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2} -The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} \ket{\tau(x_j^\mathrm{Ch})}\bra{x_j}`. +The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} |\tau(x_j^\mathrm{Ch})\rangle\langle x_j|`. Designing the transform circuit --------------------------------------- Let's start from the end and look at the circuit diagram generated from the code we want to write. -An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`\ket{x}` which starts in the computational basis, shown below as :math:`\ket{\psi}`. We demonstrate for :math:`N=4` non-ancilla qubits. +An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`|x\rangle` which starts in the computational basis, shown below as :math:`|\psi\rangle`. We demonstrate for :math:`N=4` non-ancilla qubits. -.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/QChT_diagram_4qubits.png +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/qcht_diagram_4qubits.png :align: center :width: 60% :target: javascript:void(0) @@ -115,7 +119,7 @@ def CNOT_ladder(): ############################################# # After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. - +# # Next are phase rotations and shifts. # In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. # The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. @@ -148,7 +152,8 @@ def permute_elements(): # After the permutation is another CNOT ladder, which we already have a function for. # # The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. -# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. We can impliment the multicontrolled :math:`X` rotation using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX``, and the control wires as the second argument of ``qml.ctrl``. +# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. +# We can impliment the multicontrolled :math:`X` rotation using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX``, and the control wires as the second argument of ``qml.ctrl``. def adjust_phases(): """adjusts the phase of the ancilla qubit""" @@ -187,8 +192,8 @@ def circuit(state=None): # Testing the QChT # ---------------- # With our QChT circuit, let's see if the orthonormality described earlier holds. -# To do this, we'll use the computational state :math:`\ket{7}`, which will transform into :math:`\ket{\tau(x_7^\mathrm{Ch})}`. -# Then, we will compute the overlap at the nodes with all other :math:`\ket{\tau(x_j^\mathrm{Ch})}`. +# To do this, we'll use the computational state :math:`|7\rangle`, which will transform into :math:`|{\tau(x_7^\mathrm{Ch})\rangle`. +# Then, we will compute the overlap at the nodes with all other :math:`|\tau(x_j^\mathrm{Ch})\rangle`. j = 7 # initial state in computational basis @@ -244,11 +249,6 @@ def overlap_sq(x, xp, N): ############################################# -# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/squared_overlap.png -# :align: center -# :width: 60% -# :target: javascript:void(0) -# # Almost orthonormal. We note that the amplitude at the corresponding node approaches :math:`1` if the number of qubits is increased. # # Let's also see if the amplitudes of the state in the computational basis agree with expectation. @@ -261,6 +261,8 @@ def circuit(state=None): return qml.probs(wires=range(1,N+1)) probs = circuit(state=j) +fig = plt.figure(figsize=(8,4)) +qml.draw_mpl(circuit, decimals=2, style='pennylane', fontsize='x-small', fig=fig)(state=j) # computational basis indeces x = range(2**N) @@ -281,18 +283,14 @@ def tau_amplitudes(x, k, N): plt.show() ############################################# -# .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/amplitudes.png -# :align: center -# :width: 60% -# :target: javascript:void(0) -# # The circuit output probabilities are exactly what we want. # # # Conclusion # ---------- # In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. -# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. +# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. +# One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. # # # References From 7eeef0166842ef4de89c56e8ec4ae5887c9ebd4c Mon Sep 17 00:00:00 2001 From: "Colin J. 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z*f5Xu_ZfBoCENo%pO!`W$gm+9s29nuV+!eKX>^b%e|%Ld1O!SzS#pb zV3Z#=<*Sgzoq*_6YTttj9WpouRlw(PKly0VbDjTSvFI!<6SB~V#Depu!_Dv_u)yB@ z7#=@%(1uQl7G)FX8lmFgA!Hw}y5WMG?tZcjjFkzPVbN};_WQP@{$sc52H>ULUBZyY z?f@7cNAw$OL;jo`3_a8-pa6$>4N(S=4RNNWLq1oY_V1adTz4FPTDk;-^0KoVfh+T3 zdnTNBp62l9ZK1gXS8DwF$>!jmp`rM8z0)I#Cj2Y+rxyn_C*>a@`Ymr5|KFVa-!dov ek8k*olQX>UI(t&+P1}&^*W&kOMDv7op8XF=CdSeL From 186da0935b3bdbe792c1ae31c9bfdc728fa7c9a2 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 13:57:30 -0400 Subject: [PATCH 18/64] Fixed some errors. Fixed the title. Removed QFT figure at the beginning. Added reference to de Lejarza 2025 paper in the introduction. --- .../tutorial_quantum_chebyshev_transform.py | 54 ++++++++++--------- 1 file changed, 29 insertions(+), 25 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 7c4d9d12f1..00010882cd 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -1,20 +1,13 @@ -r"""Intro to the Quantum Chebyshev +r"""Intro to the Quantum Chebyshev Transform ============================================================= -This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ wherein the authors describe a workflow for quantum Chebyshev-based model building. -They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. -Crucial to their implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. +This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. +They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. +Crucial to the implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. -.. figure:: ../_static/demonstration_assets/qft/socialthumbnail_large_QFT_2024-04-04.png - :align: center - :width: 60% - :target: javascript:void(0) - - - What are Chebyshev polynomials? --------------------------------------- @@ -70,7 +63,7 @@ Quantum Chebyshev basis --------------------------------------- -The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau\rangle(x_j^\mathrm{Ch})}\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. +The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau\rangle(x_j^\mathrm{Ch})\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. .. math:: |\tau(x)\rangle = \frac1{2^{N/2}}T_0(x)|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)|k\rangle @@ -95,7 +88,7 @@ .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/qcht_diagram_4qubits.png :align: center - :width: 60% + :width: 100% :target: javascript:void(0) The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. @@ -138,7 +131,7 @@ def rotate_phases(): ############################################# # Now a permutation of the qubits is used to reorder them. # This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. -# The multicontrolled NOT gate can be implimented using a multicontrolled Pauli X gate. +# The multicontrolled NOT gate can be implemented using a multicontrolled Pauli X gate. # Let's see what that looks like. def permute_elements(): @@ -153,7 +146,7 @@ def permute_elements(): # # The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. # All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. -# We can impliment the multicontrolled :math:`X` rotation using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX``, and the control wires as the second argument of ``qml.ctrl``. +# We can implement the multicontrolled :math:`X` rotation by using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX`` and the control wires as the second argument of ``qml.ctrl``. def adjust_phases(): """adjusts the phase of the ancilla qubit""" @@ -183,23 +176,35 @@ def QChT(): dev = qml.device("default.qubit") @qml.qnode(dev) -def circuit(state=None): +def circuit(state=0): qml.BasisState(state=state, wires=range(1,N+1)) QChT() return qml.state() +############################################# +# Finally, we can reproduce the circuit diagram shown at the beginning of this section using ``qml.draw_mpl``. + +def circuit_to_draw(): + qml.BasisState(state=0, wires=range(1,N+1)) + QChT() + +fig, ax = qml.draw_mpl(circuit_to_draw, decimals=2, style='pennylane')() +fig.show() + +############################################# +# Note the new function is defined only to remove the returned ``qml.state``, simplifying the circuit drawn. + ############################################# # Testing the QChT # ---------------- # With our QChT circuit, let's see if the orthonormality described earlier holds. -# To do this, we'll use the computational state :math:`|7\rangle`, which will transform into :math:`|{\tau(x_7^\mathrm{Ch})\rangle`. +# To do this, we'll use the computational state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. # Then, we will compute the overlap at the nodes with all other :math:`|\tau(x_j^\mathrm{Ch})\rangle`. - j = 7 # initial state in computational basis # compute state after transform -total_state = circuit(state=j)#[:2**N] +total_state = circuit(state=j) # reduce state size, effectively removing the ancilla state = total_state[:2**N]*np.sqrt(2) @@ -219,7 +224,6 @@ def ch_node(j, N): ############################################# # Now we plot the squared overlaps at the nodes computed from the circuit transformation, and compare to the definition, plotting the squared overlaps at all values of :math:`x`. -# import matplotlib.pyplot as plt @@ -261,8 +265,6 @@ def circuit(state=None): return qml.probs(wires=range(1,N+1)) probs = circuit(state=j) -fig = plt.figure(figsize=(8,4)) -qml.draw_mpl(circuit, decimals=2, style='pennylane', fontsize='x-small', fig=fig)(state=j) # computational basis indeces x = range(2**N) @@ -298,9 +300,11 @@ def tau_amplitudes(x, k, N): # # .. [#williams2023] # -# Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving and Oleksandr Kyriienk. "Quantum Chebyshev -# transform: mapping, embedding, learning and sampling distributions." `arxiv:2306.17026 -# `__ (2023). +# Chelsea A. Williams, Annie E. Paine, Hsin-Yu Wu, Vincent E. Elfving and Oleksandr Kyriienk. "Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions." `arxiv:2306.17026 `__ (2023). +# +# .. [#delejarza2025] +# +# Jorge J. Martínez de Lejarza, Hsin-Yu Wu, Oleksandr Kyriienko, Germán Rodrigo, Michele Grossi. "Quantum Chebyshev probabilistic models for fragmentation functions." `arxiv:2503.16073 `__ (2025). # # About the author # ---------------- From e9e790f07af7ed9f5878270238b647de857df8be Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 14:12:21 -0400 Subject: [PATCH 19/64] Spell and grammar check --- demonstrations/tutorial_quantum_chebyshev_transform.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 00010882cd..97a5f1e6a3 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -223,11 +223,11 @@ def ch_node(j, N): overlaps.append(np.inner(state, state_i)) ############################################# -# Now we plot the squared overlaps at the nodes computed from the circuit transformation, and compare to the definition, plotting the squared overlaps at all values of :math:`x`. +# Now we plot the squared overlaps at the nodes computed from the circuit transformation and compare to the definition, plotting the squared overlaps at all values of :math:`x`. import matplotlib.pyplot as plt -def T_n(x, n): +def T_n(x, n): """Chebyshev polynomial of order n""" return np.cos(n * np.arccos(x)) @@ -266,11 +266,11 @@ def circuit(state=None): probs = circuit(state=j) -# computational basis indeces +# computational basis indices x = range(2**N) def tau_amplitudes(x, k, N): - """computes the expected amplitudes of tau""" + """computes the expected amplitud es of tau""" if k == 0: prefactor = 1/2**(N/2) else: From cb797c8867b279f7bc6e9e0f178d52c4d5b802bc Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 15:23:01 -0400 Subject: [PATCH 20/64] Ran black on demo.py --- .../tutorial_quantum_chebyshev_transform.py | 140 ++++++++++-------- 1 file changed, 82 insertions(+), 58 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 97a5f1e6a3..c95fe15409 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -106,98 +106,114 @@ # number of qubits (non-ancilla) N = 4 + def CNOT_ladder(): - for wire in range(1, N+1): + for wire in range(1, N + 1): qml.CNOT([0, wire]) + ############################################# -# After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. +# After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. # -# Next are phase rotations and shifts. -# In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. +# Next are phase rotations and shifts. +# In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. # The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. import numpy as np + pi = np.pi + def rotate_phases(): - """shift the ancilla's phase and rotate the jth qubit by - pi/2^(j+1) in Z""" - qml.PhaseShift(-pi/2**(N+1), wires=0) - qml.RZ(-pi*(2**N-1)/2**(N+1), wires=0) - for wire in range(1, N+1): - qml.RZ(pi/2**(wire+1), wires=wire) + """shift the ancilla's phase and rotate the jth qubit by + pi/2^(j+1) in Z""" + qml.PhaseShift(-pi / 2 ** (N + 1), wires=0) + qml.RZ(-pi * (2**N - 1) / 2 ** (N + 1), wires=0) + for wire in range(1, N + 1): + qml.RZ(pi / 2 ** (wire + 1), wires=wire) + ############################################# -# Now a permutation of the qubits is used to reorder them. -# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. -# The multicontrolled NOT gate can be implemented using a multicontrolled Pauli X gate. +# Now a permutation of the qubits is used to reorder them. +# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. +# The multicontrolled NOT gate can be implemented using a multicontrolled Pauli X gate. # Let's see what that looks like. + def permute_elements(): """reorders amplitudes of the conditioned states""" - for wire in reversed(range(1, N+1)): - control_wires = [0] + list(range(wire+1,N+1)) + for wire in reversed(range(1, N + 1)): + control_wires = [0] + list(range(wire + 1, N + 1)) qml.MultiControlledX(wires=(*control_wires, wire)) + ############################################# -# In the above code, we use ``reversed`` to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. +# In the above code, we use ``reversed`` to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. # After the permutation is another CNOT ladder, which we already have a function for. -# -# The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. -# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. +# +# The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. +# All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. # We can implement the multicontrolled :math:`X` rotation by using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX`` and the control wires as the second argument of ``qml.ctrl``. + def adjust_phases(): """adjusts the phase of the ancilla qubit""" - qml.PhaseShift(-pi/2, wires=0) - qml.RY(-pi/2, wires=0) + qml.PhaseShift(-pi / 2, wires=0) + qml.RY(-pi / 2, wires=0) # first Pauli Xs - for wire in range(1, N+1): + for wire in range(1, N + 1): qml.PauliX(wires=wire) # controlled RX gate - qml.ctrl(qml.RX(pi/2, wires=0), range(1, N+1)) + qml.ctrl(qml.RX(pi / 2, wires=0), range(1, N + 1)) # second Pauli Xs - for wire in range(1, N+1): + for wire in range(1, N + 1): qml.PauliX(wires=wire) + ############################################# -# All together, we can construct the circuit. +# All together, we can construct the circuit. # We have added :class:`qml.BasisState` to initialize the input in any computational basis state with the optional argument ``state``. + def QChT(): qml.Hadamard(wires=0) CNOT_ladder() - qml.QFT(wires=range(N+1)) + qml.QFT(wires=range(N + 1)) rotate_phases() permute_elements() CNOT_ladder() adjust_phases() + dev = qml.device("default.qubit") + + @qml.qnode(dev) def circuit(state=0): - qml.BasisState(state=state, wires=range(1,N+1)) + qml.BasisState(state=state, wires=range(1, N + 1)) QChT() return qml.state() + ############################################# # Finally, we can reproduce the circuit diagram shown at the beginning of this section using ``qml.draw_mpl``. + def circuit_to_draw(): - qml.BasisState(state=0, wires=range(1,N+1)) + qml.BasisState(state=0, wires=range(1, N + 1)) QChT() -fig, ax = qml.draw_mpl(circuit_to_draw, decimals=2, style='pennylane')() + +fig, ax = qml.draw_mpl(circuit_to_draw, decimals=2, style="pennylane")() fig.show() ############################################# -# Note the new function is defined only to remove the returned ``qml.state``, simplifying the circuit drawn. +# Note the new function is defined only to remove the returned ``qml.state``, simplifying the circuit drawn. ############################################# # Testing the QChT # ---------------- -# With our QChT circuit, let's see if the orthonormality described earlier holds. +# With our QChT circuit, let's see if the orthonormality described earlier holds. # To do this, we'll use the computational state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. # Then, we will compute the overlap at the nodes with all other :math:`|\tau(x_j^\mathrm{Ch})\rangle`. @@ -207,11 +223,13 @@ def circuit_to_draw(): total_state = circuit(state=j) # reduce state size, effectively removing the ancilla -state = total_state[:2**N]*np.sqrt(2) +state = total_state[: 2**N] * np.sqrt(2) + # compute nodes def ch_node(j, N): - return np.cos(pi*(2*j+1)/2**(N+1)) + return np.cos(pi * (2 * j + 1) / 2 ** (N + 1)) + js = list(range(int(len(state)))) nodes = [ch_node(i, N) for i in js] @@ -219,7 +237,7 @@ def ch_node(j, N): # compute overlap with other basis states using np.inner() overlaps = [] for i in js: - state_i = circuit(state=i)[:2**N]*np.sqrt(2) + state_i = circuit(state=i)[: 2**N] * np.sqrt(2) overlaps.append(np.inner(state, state_i)) ############################################# @@ -227,26 +245,28 @@ def ch_node(j, N): import matplotlib.pyplot as plt -def T_n(x, n): + +def T_n(x, n): """Chebyshev polynomial of order n""" return np.cos(n * np.arccos(x)) + def overlap_sq(x, xp, N): """computes the squared overlap""" - numerator = T_n(xp, 2**N+1) * T_n(x, 2**N) - \ - T_n(xp, 2**N) * T_n(x, 2**N+1) - return numerator**2/(2**(2*N))/(xp - x)**2 - -plt.style.use('pennylane.drawer.plot') -fig, ax = plt.subplots(figsize=(6,4)) + numerator = T_n(xp, 2**N + 1) * T_n(x, 2**N) - T_n(xp, 2**N) * T_n(x, 2**N + 1) + return numerator**2 / (2 ** (2 * N)) / (xp - x) ** 2 + + +plt.style.use("pennylane.drawer.plot") +fig, ax = plt.subplots(figsize=(6, 4)) ax.set(xlabel="x", ylabel="Squared Overlap") # plot squared overlaps computed in circuit -ax.plot(nodes, np.abs(overlaps)**2, marker='o', label='circuit') +ax.plot(nodes, np.abs(overlaps) ** 2, marker="o", label="circuit") # plot expected squared overlaps xs = np.linspace(-1, 1, 1000) -ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label='expectation') +ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label="expectation") ax.legend() plt.show() @@ -254,32 +274,36 @@ def overlap_sq(x, xp, N): ############################################# # Almost orthonormal. We note that the amplitude at the corresponding node approaches :math:`1` if the number of qubits is increased. -# -# Let's also see if the amplitudes of the state in the computational basis agree with expectation. +# +# Let's also see if the amplitudes of the state in the computational basis agree with expectation. # To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). + @qml.qnode(dev) def circuit(state=None): - qml.BasisState(state=state, wires=range(1,N+1)) + qml.BasisState(state=state, wires=range(1, N + 1)) QChT() - return qml.probs(wires=range(1,N+1)) + return qml.probs(wires=range(1, N + 1)) + probs = circuit(state=j) # computational basis indices x = range(2**N) + def tau_amplitudes(x, k, N): """computes the expected amplitud es of tau""" - if k == 0: - prefactor = 1/2**(N/2) - else: - prefactor = 1/2**((N-1)/2) - return prefactor * np.cos(k*np.arccos(x)) - -fig, ax = plt.subplots(figsize=(6,4)) -ax.plot(x, probs, 'o', label='circuit') -ax.plot(x, [tau_amplitudes(nodes[j], xs, N)**2 for xs in x], label='expectation') + if k == 0: + prefactor = 1 / 2 ** (N / 2) + else: + prefactor = 1 / 2 ** ((N - 1) / 2) + return prefactor * np.cos(k * np.arccos(x)) + + +fig, ax = plt.subplots(figsize=(6, 4)) +ax.plot(x, probs, "o", label="circuit") +ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"|k$\rangle$", ylabel="Probability") ax.legend() plt.show() @@ -290,8 +314,8 @@ def tau_amplitudes(x, k, N): # # Conclusion # ---------- -# In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. -# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. +# In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. +# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. # One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. # # From 0bd713b6179d0fcb280773f53aeda78e4fe43f38 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 15:45:54 -0400 Subject: [PATCH 21/64] Tried removed figsize arguments when generating plots. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index c95fe15409..d065c73eea 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -258,7 +258,7 @@ def overlap_sq(x, xp, N): plt.style.use("pennylane.drawer.plot") -fig, ax = plt.subplots(figsize=(6, 4)) +fig, ax = plt.subplots() ax.set(xlabel="x", ylabel="Squared Overlap") # plot squared overlaps computed in circuit @@ -301,7 +301,7 @@ def tau_amplitudes(x, k, N): return prefactor * np.cos(k * np.arccos(x)) -fig, ax = plt.subplots(figsize=(6, 4)) +fig, ax = plt.subplots() ax.plot(x, probs, "o", label="circuit") ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"|k$\rangle$", ylabel="Probability") From 8eb61678bf4cb3c6a8c5b2dbaeff862d0ca5ba3a Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Wed, 7 May 2025 15:53:10 -0400 Subject: [PATCH 22/64] Small changes. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index d065c73eea..8d0d95f6fe 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -304,7 +304,7 @@ def tau_amplitudes(x, k, N): fig, ax = plt.subplots() ax.plot(x, probs, "o", label="circuit") ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") -ax.set(xlabel=r"|k$\rangle$", ylabel="Probability") +ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") ax.legend() plt.show() @@ -314,7 +314,7 @@ def tau_amplitudes(x, k, N): # # Conclusion # ---------- -# In this tutorial, we've gone through how to implement the QChT from a recent paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. +# In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. # Further work could test the phase of the output to make sure it matches what we expect the QChT to output. # One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. # From 1a87f524b95d96cd7f47f7a009b3820746d6327b Mon Sep 17 00:00:00 2001 From: "Colin J. 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zY?NFK6}LN9&?f)&U=_S~^(qeD*;pRFm{lh;%f+az%N8m>5)dm$ip{fKCMeZ9c@0LO?MsrrYj$!hN z&pbs$olaoul^p!NBW5gxS?T~7u@sQ3i93T>njB&gP6W=LziuTs$80nLqz8Ax3<$JC z=Z&_uHkaOnyXWC-ar5TQc5vh04mRyU%MS~ho13?TvumV^r9ow^rg~RZj0}|K2!iSf z#SpD|9&XQ(ayw{=kWsdZ@AnaK=~5>)X+88TY+ovyNcL(oBgu>s+bcUNud|-6DoYlv z`3v#Eb~q_?6kh+$4pVH$7}i}wplAfi5pcEEyHaY|dHksM)Dv4+jg2z-`S~#Ama*zp zW-e`wzYT#2Dn2PGsks_KuEwjdZnO z#(ZgXM*C&kXVg4E`Q;P%aNxnu3?2XVFY=pjjQ!U~|M~METH9Z?1Bd?aQ2_M(@;f47 z2tI$=LT&@w|FU^LLTmoZwjp6|N{BFXt2DzN`;6uo4-w*jZ7+pdggpc6YTv>A+9&3q z0vKQp&}Z;KWKKp2=`bgDXXr4eNeJ0t4j$&Tswp~D Date: Thu, 8 May 2025 12:50:00 -0400 Subject: [PATCH 25/64] Changed np.inner to np.vdot np.inner doesn't complex conjugate!!! Oops. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index c1e1f326d8..d848e8d679 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -238,7 +238,7 @@ def ch_node(j, N): overlaps = [] for i in js: state_i = circuit(state=i)[: 2**N] * np.sqrt(2) - overlaps.append(np.inner(state, state_i)) + overlaps.append(np.vdot(state, state_i)) ############################################# # Now we plot the squared overlaps at the nodes computed from the circuit transformation and compare to the definition, plotting the squared overlaps at all values of :math:`x`. From 002ba366046134cfdd01acf9a30913a63f20a895 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Thu, 8 May 2025 12:50:36 -0400 Subject: [PATCH 26/64] edited descriptive comment to match last commit --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index d848e8d679..e691e75884 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -234,7 +234,7 @@ def ch_node(j, N): js = list(range(int(len(state)))) nodes = [ch_node(i, N) for i in js] -# compute overlap with other basis states using np.inner() +# compute overlap with other basis states using np.vdot() overlaps = [] for i in js: state_i = circuit(state=i)[: 2**N] * np.sqrt(2) From cc926ea3920126835aa120564e4b1a14655a1b96 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 9 May 2025 09:46:49 -0400 Subject: [PATCH 27/64] Added figure captions. Struggled to get a generated caption for the drawn circuit diagram. Not sure how to extend figure to make room for a caption. --- .../tutorial_quantum_chebyshev_transform.py | 22 ++++++++++++++----- 1 file changed, 16 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index e691e75884..74ab1dc88c 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -41,6 +41,8 @@ :width: 60% :target: javascript:void(0) + Figure 1. The first six Chebyshev polynomials, along with their corresponding nodes. + The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of $T_N(x)$ in the following way for $k$, $\ell`__ applied on all qubits. The QChT is an extended QFT circuit with some added interference and mixing of the elements. @@ -241,7 +245,8 @@ def ch_node(j, N): overlaps.append(np.vdot(state, state_i)) ############################################# -# Now we plot the squared overlaps at the nodes computed from the circuit transformation and compare to the definition, plotting the squared overlaps at all values of :math:`x`. +# Now we plot the squared overlaps at the nodes of the chosen state, and all other basis states. +# We compare this to the definition, for which we plot the squared overlaps at all values of :math:`x`. import matplotlib.pyplot as plt @@ -258,8 +263,10 @@ def overlap_sq(x, xp, N): plt.style.use("pennylane.drawer.plot") -fig, ax = plt.subplots() -ax.set(xlabel="x", ylabel="Squared Overlap") + +fig = plt.figure(figsize=(6.4, 2.4)) +ax = fig.add_axes((0.15, 0.3, 0.8, 0.65)) +ax.set(xlabel=r"x", ylabel="Square Overlap") # plot squared overlaps computed in circuit ax.plot(nodes, np.abs(overlaps) ** 2, marker="o", label="circuit") @@ -269,12 +276,12 @@ def overlap_sq(x, xp, N): ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label="expectation") ax.legend() +fig.text(0.5, 0.05, "Figure 4. Squared overlap of Chebyshev basis states computed from the circuit compared to the definition.", + horizontalalignment='center', size='x-small', weight='normal') plt.show() ############################################# -# Almost orthonormal. We note that the amplitude at the corresponding node approaches :math:`1` if the number of qubits is increased. -# # Let's also see if the amplitudes of the state in the computational basis agree with expectation. # To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). @@ -301,11 +308,14 @@ def tau_amplitudes(x, k, N): return prefactor * np.cos(k * np.arccos(x)) -fig, ax = plt.subplots() +fig = plt.figure(figsize=(6.4, 4.8)) +ax = fig.add_axes((0.15, 0.25, 0.8, 0.7)) ax.plot(x, probs, "o", label="circuit") ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") ax.legend() +fig.text(0.5, 0.05, "Figure 5. Squared overlap of Chebyshev basis state with computational basis states.", + horizontalalignment='center', size='x-small', weight='normal') plt.show() ############################################# From 7bf1aa0ac8252b8b896633ff5c37248f138af8aa Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 9 May 2025 09:56:19 -0400 Subject: [PATCH 28/64] Added alt text to static images. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 ++ 1 file changed, 2 insertions(+) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 74ab1dc88c..8337c530ec 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -40,6 +40,7 @@ :align: center :width: 60% :target: javascript:void(0) + :alt: Plot of first six Chebyshev polynomials, with nodes denser near the boundary. Figure 1. The first six Chebyshev polynomials, along with their corresponding nodes. @@ -92,6 +93,7 @@ :align: center :width: 100% :target: javascript:void(0) + :alt: Quantum Chebyshev Transform circuit diagram drawn using PennyLane Figure 2. Quantum Chebyshev Transform circuit. From d5e13ab6728cb30a6470ae31c2e4267f66004d43 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 9 May 2025 10:52:08 -0400 Subject: [PATCH 29/64] Used black but then modified result for fig.text --- .../tutorial_quantum_chebyshev_transform.py | 12 +++++++----- 1 file changed, 7 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 8337c530ec..03f15e16a1 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -247,7 +247,7 @@ def ch_node(j, N): overlaps.append(np.vdot(state, state_i)) ############################################# -# Now we plot the squared overlaps at the nodes of the chosen state, and all other basis states. +# Now we plot the squared overlaps at the nodes of the chosen state, and all other basis states. # We compare this to the definition, for which we plot the squared overlaps at all values of :math:`x`. import matplotlib.pyplot as plt @@ -278,8 +278,9 @@ def overlap_sq(x, xp, N): ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label="expectation") ax.legend() -fig.text(0.5, 0.05, "Figure 4. Squared overlap of Chebyshev basis states computed from the circuit compared to the definition.", - horizontalalignment='center', size='x-small', weight='normal') +fig.text(0.5, 0.05, + "Figure 4. Squared overlap of Chebyshev basis states computed from the circuit compared to the definition.", + horizontalalignment="center", size="x-small", weight="normal") plt.show() @@ -316,8 +317,9 @@ def tau_amplitudes(x, k, N): ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") ax.legend() -fig.text(0.5, 0.05, "Figure 5. Squared overlap of Chebyshev basis state with computational basis states.", - horizontalalignment='center', size='x-small', weight='normal') +fig.text(0.5, 0.05, + "Figure 5. Squared overlap of Chebyshev basis state with computational basis states.", + horizontalalignment="center", size="x-small", weight="normal") plt.show() ############################################# From 99203056d2373ec238f87230f2f7e67a039a89a5 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 9 May 2025 11:06:40 -0400 Subject: [PATCH 30/64] Modified captions --- .../tutorial_quantum_chebyshev_transform.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 03f15e16a1..e442fff6a6 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -278,9 +278,9 @@ def overlap_sq(x, xp, N): ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label="expectation") ax.legend() -fig.text(0.5, 0.05, - "Figure 4. Squared overlap of Chebyshev basis states computed from the circuit compared to the definition.", - horizontalalignment="center", size="x-small", weight="normal") +fig.text(0.5, 0.05, + "Figure 4. Squared overlap of Chebyshev basis states.", + horizontalalignment="center", size="small", weight="normal") plt.show() @@ -312,14 +312,14 @@ def tau_amplitudes(x, k, N): fig = plt.figure(figsize=(6.4, 4.8)) -ax = fig.add_axes((0.15, 0.25, 0.8, 0.7)) +ax = fig.add_axes((0.15, 0.23, 0.80, 0.72)) ax.plot(x, probs, "o", label="circuit") ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") ax.legend() -fig.text(0.5, 0.05, +fig.text(0.5, 0.05, "Figure 5. Squared overlap of Chebyshev basis state with computational basis states.", - horizontalalignment="center", size="x-small", weight="normal") + horizontalalignment="center", size="small", weight="normal") plt.show() ############################################# From 888bf10f5fb534c41cb9141b2f550c5bb4ea0085 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 9 May 2025 15:52:21 -0400 Subject: [PATCH 31/64] Fixed math mode mistake --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index e442fff6a6..8e156b9c2e 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -44,7 +44,7 @@ Figure 1. The first six Chebyshev polynomials, along with their corresponding nodes. -The nodes are plotted above along with the corresponding polynomials. 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Subject: [PATCH 33/64] Edited text, better describing the basis It was not explicit before what the basis states were, rather a state for any parameter x was defined. --- .../tutorial_quantum_chebyshev_transform.py | 42 +++++++++++-------- 1 file changed, 25 insertions(+), 17 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 8e156b9c2e..c76e51cba7 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -11,10 +11,10 @@ What are Chebyshev polynomials? --------------------------------------- -`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as +`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as .. math:: - T_n(x) \equiv \cos(n \arccos(x)) + T_n(x) \equiv \cos(n \arccos(x))\,, where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. @@ -25,7 +25,7 @@ T_3(x) &= 4x^3 - 3x \\ T_4(x) &= 8x^4 - 8x^2 + 1 \\ &\ \,\vdots \\ - T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) + T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x)\,. The recursion relation in the last line can be used to compute the next orders. Observe that odd and even order :math:`T_n` are odd and even functions, respectively. @@ -51,11 +51,11 @@ \begin{cases} 0 & k \neq \ell\,,\\ N & k = \ell = 0\,,\\ - N/2 & k = \ell \neq 0 + N/2 & k = \ell \neq 0\,. \end{cases} -The Chebyshev polynomials have a lot of *nice* properties. Because they are complete, any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. \ -To do this process numerically for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. +The Chebyshev polynomials have a lot of *nice* properties. Because they are complete, any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. +To do this process numerically on a classical computer for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. @@ -63,23 +63,22 @@ This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. +Next, we will describe the quantum analogue of this transformation. + Quantum Chebyshev basis --------------------------------------- -The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau\rangle(x_j^\mathrm{Ch})\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. +The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. +The :math:`j`th Chebyshev basis state using :math:`N` qubits is then .. math:: - |\tau(x)\rangle = \frac1{2^{N/2}}T_0(x)|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x)|k\rangle + |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, -where :math:`|k\rangle` are the computational basis states and :math:`N` is the number of qubits. These states are orthonormal at the Chebyshev nodes due to the orthogonality condition, that is +where :math:`|k\rangle` are the computational basis states. +These states are orthonormal due to the orthgonality of the Chebyshev polynomials, that is .. math:: - \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \delta_{j, j'} - -The squared overlap if one of the variables is not at a node can be derived analytically as - -.. math:: - |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2} + \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \delta_{j, j'}\,. The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} |\tau(x_j^\mathrm{Ch})\rangle\langle x_j|`. @@ -239,6 +238,7 @@ def ch_node(j, N): js = list(range(int(len(state)))) nodes = [ch_node(i, N) for i in js] +print(nodes) # compute overlap with other basis states using np.vdot() overlaps = [] @@ -249,6 +249,12 @@ def ch_node(j, N): ############################################# # Now we plot the squared overlaps at the nodes of the chosen state, and all other basis states. # We compare this to the definition, for which we plot the squared overlaps at all values of :math:`x`. +# This can be derived analytically as +# +#.. math:: +# |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2}\,,# +# +# where :math:`\tau(x)` is like one of the Chebyshev basis states defined earlier, but generalized to all :math:`x` rather than just the nodes. import matplotlib.pyplot as plt @@ -279,12 +285,14 @@ def overlap_sq(x, xp, N): ax.legend() fig.text(0.5, 0.05, - "Figure 4. Squared overlap of Chebyshev basis states.", + "Figure 3. Squared overlap of Chebyshev basis states.", horizontalalignment="center", size="small", weight="normal") plt.show() ############################################# +# We can see that the squared overlap of the chosen basis state :math:`|\tau(x_7^\mathrm{Ch})\rangle` with the other states is 0, unless :math:`x=x_7^\mathrm{Ch}\approx 0.1`, then the overlap is 1. +# # Let's also see if the amplitudes of the state in the computational basis agree with expectation. # To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). @@ -318,7 +326,7 @@ def tau_amplitudes(x, k, N): ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") ax.legend() fig.text(0.5, 0.05, - "Figure 5. Squared overlap of Chebyshev basis state with computational basis states.", + "Figure 4. Squared overlap of Chebyshev basis state with computational basis states.", horizontalalignment="center", size="small", weight="normal") plt.show() From 7531181a36feb548499c8123be1896ce68b70566 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 12 May 2025 12:36:21 -0400 Subject: [PATCH 34/64] A couple typos --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index c76e51cba7..cbd6bdf00f 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -252,7 +252,7 @@ def ch_node(j, N): # This can be derived analytically as # #.. math:: -# |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2}\,,# +# |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2}\,, # # where :math:`\tau(x)` is like one of the Chebyshev basis states defined earlier, but generalized to all :math:`x` rather than just the nodes. From dda0b41a668069d033bc1d95d26550dd5eec7e6e Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 12 May 2025 13:23:37 -0400 Subject: [PATCH 35/64] Fixed math mode error and made more edits. --- .../tutorial_quantum_chebyshev_transform.py | 25 ++++++++----------- 1 file changed, 10 insertions(+), 15 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index cbd6bdf00f..a0f66c813b 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -69,7 +69,7 @@ Quantum Chebyshev basis --------------------------------------- The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. -The :math:`j`th Chebyshev basis state using :math:`N` qubits is then +The jth Chebyshev basis state using :math:`N` qubits is .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, @@ -213,7 +213,7 @@ def circuit_to_draw(): fig.show() ############################################# -# Note the new function is defined only to remove the returned ``qml.state``, simplifying the drawn circuit. +# Note we define a new function for the circuit to simplify the drawing, removing the returned ``qml.state``. ############################################# # Testing the QChT @@ -238,23 +238,18 @@ def ch_node(j, N): js = list(range(int(len(state)))) nodes = [ch_node(i, N) for i in js] -print(nodes) # compute overlap with other basis states using np.vdot() -overlaps = [] -for i in js: - state_i = circuit(state=i)[: 2**N] * np.sqrt(2) - overlaps.append(np.vdot(state, state_i)) +overlaps = [np.vdot(state, circuit(state=i)[: 2**N] * np.sqrt(2)) for i in js] ############################################# -# Now we plot the squared overlaps at the nodes of the chosen state, and all other basis states. -# We compare this to the definition, for which we plot the squared overlaps at all values of :math:`x`. -# This can be derived analytically as +# We compare these circuit calculated overlaps to the definition, for which we plot the squared overlaps at all values of :math:`x`. +# This continuous overlap function can be derived analytically as # -#.. math:: +# .. math:: # |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2}\,, # -# where :math:`\tau(x)` is like one of the Chebyshev basis states defined earlier, but generalized to all :math:`x` rather than just the nodes. +# where :math:`\tau(x)` is a generalization of one of Chebyshev basis states defined earlier, where :math:`x` can be any value in :math:`[-1,1] rather than just one of the nodes. import matplotlib.pyplot as plt @@ -273,7 +268,7 @@ def overlap_sq(x, xp, N): plt.style.use("pennylane.drawer.plot") fig = plt.figure(figsize=(6.4, 2.4)) -ax = fig.add_axes((0.15, 0.3, 0.8, 0.65)) +ax = fig.add_axes((0.15, 0.3, 0.8, 0.65)) # make room for caption ax.set(xlabel=r"x", ylabel="Square Overlap") # plot squared overlaps computed in circuit @@ -291,7 +286,7 @@ def overlap_sq(x, xp, N): ############################################# -# We can see that the squared overlap of the chosen basis state :math:`|\tau(x_7^\mathrm{Ch})\rangle` with the other states is 0, unless :math:`x=x_7^\mathrm{Ch}\approx 0.1`, then the overlap is 1. +# We can see that the squared overlap between the basis states and the :math:`j=7` state :math:`|\tau(x_7^\mathrm{Ch})\rangle` is 0, unless :math:`x=x_7^\mathrm{Ch}\approx 0.1`, then the overlap is 1. # # Let's also see if the amplitudes of the state in the computational basis agree with expectation. # To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). @@ -320,7 +315,7 @@ def tau_amplitudes(x, k, N): fig = plt.figure(figsize=(6.4, 4.8)) -ax = fig.add_axes((0.15, 0.23, 0.80, 0.72)) +ax = fig.add_axes((0.15, 0.23, 0.80, 0.72)) # make room for caption ax.plot(x, probs, "o", label="circuit") ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") From 51d8cb22dfcddfefe2c75b05e90980165e9db40b Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 12 May 2025 16:23:16 -0400 Subject: [PATCH 36/64] Increased Fig 1. size, fixed math mod error. --- .../tutorial_quantum_chebyshev_transform.py | 22 ++++++++++++------- 1 file changed, 14 insertions(+), 8 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index a0f66c813b..947f8dc9ee 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -38,7 +38,7 @@ .. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/chebyshev_polynomials.png :align: center - :width: 60% + :width: 100% :target: javascript:void(0) :alt: Plot of first six Chebyshev polynomials, with nodes denser near the boundary. @@ -246,10 +246,10 @@ def ch_node(j, N): # We compare these circuit calculated overlaps to the definition, for which we plot the squared overlaps at all values of :math:`x`. # This continuous overlap function can be derived analytically as # -# .. math:: +# .. math:: # |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2}\,, # -# where :math:`\tau(x)` is a generalization of one of Chebyshev basis states defined earlier, where :math:`x` can be any value in :math:`[-1,1] rather than just one of the nodes. +# where :math:`\tau(x)` is a generalization of one of Chebyshev basis states defined earlier, where :math:`x` can be any value in :math:`[-1,1]` rather than just one of the nodes. import matplotlib.pyplot as plt @@ -279,15 +279,18 @@ def overlap_sq(x, xp, N): ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label="expectation") ax.legend() -fig.text(0.5, 0.05, +fig.text(0.5, 0.05, "Figure 3. Squared overlap of Chebyshev basis states.", - horizontalalignment="center", size="small", weight="normal") + horizontalalignment="center", + size="small", + weight="normal", +) plt.show() ############################################# # We can see that the squared overlap between the basis states and the :math:`j=7` state :math:`|\tau(x_7^\mathrm{Ch})\rangle` is 0, unless :math:`x=x_7^\mathrm{Ch}\approx 0.1`, then the overlap is 1. -# +# # Let's also see if the amplitudes of the state in the computational basis agree with expectation. # To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). @@ -320,9 +323,12 @@ def tau_amplitudes(x, k, N): ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") ax.legend() -fig.text(0.5, 0.05, +fig.text(0.5, 0.05, "Figure 4. Squared overlap of Chebyshev basis state with computational basis states.", - horizontalalignment="center", size="small", weight="normal") + horizontalalignment="center", + size="small", + weight="normal", +) plt.show() ############################################# From 2954e8d1cc550f4cb7e0ae48063a5195e419dc68 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 16 May 2025 09:52:15 -0400 Subject: [PATCH 37/64] Changed title --- .../tutorial_quantum_chebyshev_transform.metadata.json | 2 +- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json index 0d2992d503..2f3fcfbab5 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json +++ b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json @@ -1,5 +1,5 @@ { - "title": "Intro to Quantum Chebyshev Transform", + "title": "The Quantum Chebyshev Transform", "authors": [ { "username": "colinjdale" diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 947f8dc9ee..8322758040 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -1,4 +1,4 @@ -r"""Intro to the Quantum Chebyshev Transform +r"""The Quantum Chebyshev Transform ============================================================= This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. From 01e92b6fd5ba8fcdb66ee8eeb789776050f651bb Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 16 May 2025 09:58:52 -0400 Subject: [PATCH 38/64] Fixed the technical issue. Had some gate orders swapped. Was applying U1 and U2 backwards. --- .../tutorial_quantum_chebyshev_transform.py | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 8322758040..4ad6e87cbe 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -132,8 +132,8 @@ def CNOT_ladder(): def rotate_phases(): """shift the ancilla's phase and rotate the jth qubit by pi/2^(j+1) in Z""" - qml.PhaseShift(-pi / 2 ** (N + 1), wires=0) qml.RZ(-pi * (2**N - 1) / 2 ** (N + 1), wires=0) + qml.PhaseShift(-pi / 2 ** (N + 1), wires=0) for wire in range(1, N + 1): qml.RZ(pi / 2 ** (wire + 1), wires=wire) @@ -163,8 +163,8 @@ def permute_elements(): def adjust_phases(): """adjusts the phase of the ancilla qubit""" - qml.PhaseShift(-pi / 2, wires=0) qml.RY(-pi / 2, wires=0) + qml.PhaseShift(-pi / 2, wires=0) # first Pauli Xs for wire in range(1, N + 1): qml.PauliX(wires=wire) @@ -227,9 +227,12 @@ def circuit_to_draw(): # compute state after transform total_state = circuit(state=j) +print(total_state) + # reduce state size, effectively removing the ancilla -state = total_state[: 2**N] * np.sqrt(2) +state = total_state[: 2**N] +print(state) # compute nodes def ch_node(j, N): @@ -240,7 +243,7 @@ def ch_node(j, N): nodes = [ch_node(i, N) for i in js] # compute overlap with other basis states using np.vdot() -overlaps = [np.vdot(state, circuit(state=i)[: 2**N] * np.sqrt(2)) for i in js] +overlaps = [np.vdot(state, circuit(state=i)[: 2**N]) for i in js] ############################################# # We compare these circuit calculated overlaps to the definition, for which we plot the squared overlaps at all values of :math:`x`. From 3828dc97f06cbef5bad240d51d325fda03c29590 Mon Sep 17 00:00:00 2001 From: "Colin J. 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zV_NcyaiuGV2rb{ zXX7%-KUqi-tsGUP@(@B;R)LZ``+imu(VS5a?98Tj_o|7{k;var_))s+%u&q{IqF$q zganu;@|hgBj@X?ht|*zcqSW+CiRl$zI?J<`aMXduQAhcA(Mg;4*NENy-Z;3=YcOBP zM>DGZu4&BkEDR=^Q{W}4iZcSayh9=%78^X0IWJ)-_yXwdtgI}|??Bv8P@yjHA>a}6 zr78j!7q%O_5~=wik^D}yJ0|bLHkq?%(^jRXw&AC5)ZEHJ&f@D*A~&b*=2T1WFQewE z>zy_-xo@6*Ez11%v2Xv Date: Fri, 16 May 2025 12:43:14 -0400 Subject: [PATCH 40/64] Changed the title again. --- .../tutorial_quantum_chebyshev_transform.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json index 2f3fcfbab5..8ae9c851eb 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json +++ b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json @@ -1,5 +1,5 @@ { - "title": "The Quantum Chebyshev Transform", + "title": "Quantum Chebyshev Transform", "authors": [ { "username": "colinjdale" From 0dcdc18c4fc4014098d9ca60e084360432057ab3 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 16 May 2025 13:11:43 -0400 Subject: [PATCH 41/64] Lots of edits. Plotting state amplitudes instead of probabilities. Swapped plotting order. --- .../tutorial_quantum_chebyshev_transform.py | 151 ++++++++---------- 1 file changed, 70 insertions(+), 81 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 4ad6e87cbe..2b2569dd6a 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -1,4 +1,4 @@ -r"""The Quantum Chebyshev Transform +r"""Quantum Chebyshev Transform ============================================================= This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. @@ -68,14 +68,14 @@ Quantum Chebyshev basis --------------------------------------- -The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps :math:`2^N` computational states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}` which have amplitudes given by the Chebyshev polynomials of the first kind. +The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. The jth Chebyshev basis state using :math:`N` qubits is .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, where :math:`|k\rangle` are the computational basis states. -These states are orthonormal due to the orthgonality of the Chebyshev polynomials, that is +These states are orthonormal due to the orthgonality of the Chebyshev polynomials and the normalization convention defined above, that is .. math:: \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \delta_{j, j'}\,. @@ -86,9 +86,9 @@ Designing the transform circuit --------------------------------------- Let's start from the end and look at the circuit diagram generated from the code we want to write. -An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`|x\rangle` which starts in the computational basis, shown below as :math:`|\psi\rangle`. We demonstrate for :math:`N=4` non-ancilla qubits. +An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`|x\rangle` which starts in the computational basis, shown below as :math:`|\psi\rangle`. We demonstrate for :math:`N=4` non-ancilla qubits. -.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/qcht_diagram_4qubits.png +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/qcht_circuit_diagram.png :align: center :width: 100% :target: javascript:void(0) @@ -98,9 +98,10 @@ The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. Notice the use of the `quantum Fourier transform (QFT) `__ applied on all qubits. -The QChT is an extended QFT circuit with some added interference and mixing of the elements. - +The QChT is an extended QFT circuit with some added interference and mixing of the elements. +Note the ancilla starts and ends in the state :math:`|0\rangle`, and the amplitudes of the transformed state are all real valued. Let's break down the circuit above into pieces that we will use inside our circuit function. + First, a Hadamard gate is applied to the ancilla, and then a CNOT ladder is applied, controlled on the ancilla. To start, we will define a function for the CNOT ladder. @@ -121,8 +122,8 @@ def CNOT_ladder(): # After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. # # Next are phase rotations and shifts. -# In particular, there is a phase shift on the ancilla by :math:`-\pi/2^{(N+1)}` followed by a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}`. -# The other other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. +# For the ancilla, there is a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}` followed by a phase shift of :math:`-\pi/2^{(N+1)}` . +# The other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. import numpy as np @@ -130,8 +131,8 @@ def CNOT_ladder(): def rotate_phases(): - """shift the ancilla's phase and rotate the jth qubit by - pi/2^(j+1) in Z""" + """Rotates and shifts the phase of the ancilla and rotates the jth qubit by + pi/2^(j+1) in Z.""" qml.RZ(-pi * (2**N - 1) / 2 ** (N + 1), wires=0) qml.PhaseShift(-pi / 2 ** (N + 1), wires=0) for wire in range(1, N + 1): @@ -146,7 +147,7 @@ def rotate_phases(): def permute_elements(): - """reorders amplitudes of the conditioned states""" + """Reorders amplitudes of the conditioned states.""" for wire in reversed(range(1, N + 1)): control_wires = [0] + list(range(wire + 1, N + 1)) qml.MultiControlledX(wires=(*control_wires, wire)) @@ -156,13 +157,13 @@ def permute_elements(): # In the above code, we use ``reversed`` to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. # After the permutation is another CNOT ladder, which we already have a function for. # -# The last part is a phase adjustment of the ancilla qubit: a phase shift of :math:`-\pi/2`, followed by a rotation in :math:`Y` by :math:`\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. +# The last part is a phase adjustment of the ancilla qubit: a rotation in :math:`Y` by :math:`\pi/2`, a phase shift of :math:`-\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. # All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. # We can implement the multicontrolled :math:`X` rotation by using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX`` and the control wires as the second argument of ``qml.ctrl``. def adjust_phases(): - """adjusts the phase of the ancilla qubit""" + """Adjusts the phase of the ancilla qubit.""" qml.RY(-pi / 2, wires=0) qml.PhaseShift(-pi / 2, wires=0) # first Pauli Xs @@ -218,31 +219,67 @@ def circuit_to_draw(): ############################################# # Testing the QChT # ---------------- -# With our QChT circuit, let's see if the orthonormality described earlier holds. -# To do this, we'll use the computational state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. -# Then, we will compute the overlap at the nodes with all other :math:`|\tau(x_j^\mathrm{Ch})\rangle`. +# With our QChT circuit, let's first check if the ancilla ends in the state :math:`|0\rangle`, and the output state amplitudes are real valued. +# To do this, we'll input the computational basis state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. +# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with ancilla in |1\rangle). j = 7 # initial state in computational basis -# compute state after transform -total_state = circuit(state=j) +total_state = circuit(state=j) # state with ancilla +# round very small values to zero +total_state = np.where(np.abs(total_state)<1e-12, 0, total_state) print(total_state) +############################################# +# Indeed, we see the second half of the amplitude vector is zero. Furthermore, the first :math:`2^N` entries are real valued, but let's check if the amplitudes of the state in the computational basis agree with our definition. + # reduce state size, effectively removing the ancilla -state = total_state[: 2**N] +state = np.real(total_state[: 2**N]) # discard small imaginary components + +# computational basis indices +x = range(2**N) -print(state) # compute nodes -def ch_node(j, N): +def ch_node(j): return np.cos(pi * (2 * j + 1) / 2 ** (N + 1)) -js = list(range(int(len(state)))) -nodes = [ch_node(i, N) for i in js] +def tau_amplitudes(x, k): + """Computes the expected amplitudes of tau.""" + if k == 0: + prefactor = 1 / 2 ** (N / 2) + else: + prefactor = 1 / 2 ** ((N - 1) / 2) + return prefactor * np.cos(k * np.arccos(x)) + + +import matplotlib.pyplot as plt + +plt.style.use("pennylane.drawer.plot") + +fig = plt.figure(figsize=(6.4, 4.8)) +ax = fig.add_axes((0.15, 0.23, 0.80, 0.72)) # make room for caption +ax.plot(x, state, "o", label="circuit") +ax.plot(x, [tau_amplitudes(ch_node(j), xs) for xs in x], label="expectation") +ax.set(xlabel=r"$|k\rangle$", ylabel="Amplitude") +ax.legend() +fig.text(0.5, 0.05, + r"Figure 3. Amplitudes of $|\tau(x_7^\mathrm{Ch})\rangle$ in computational basis.", + horizontalalignment="center", + size="small", + weight="normal", +) +plt.show() + +############################################# +# The output state from the circuit is exactly what we want. +# +# Next, let's see if the orthonormality described earlier holds by computing the overlap at the nodes with all other :math:`|\tau(x_j^\mathrm{Ch})\rangle`. # compute overlap with other basis states using np.vdot() +js = list(range(int(len(state)))) overlaps = [np.vdot(state, circuit(state=i)[: 2**N]) for i in js] ############################################# @@ -254,36 +291,34 @@ def ch_node(j, N): # # where :math:`\tau(x)` is a generalization of one of Chebyshev basis states defined earlier, where :math:`x` can be any value in :math:`[-1,1]` rather than just one of the nodes. -import matplotlib.pyplot as plt - def T_n(x, n): - """Chebyshev polynomial of order n""" + """Chebyshev polynomial of order n.""" return np.cos(n * np.arccos(x)) -def overlap_sq(x, xp, N): - """computes the squared overlap""" +def overlap_sq(x, xp): + """Computes the squared overlap between Chebyshev states.""" numerator = T_n(xp, 2**N + 1) * T_n(x, 2**N) - T_n(xp, 2**N) * T_n(x, 2**N + 1) return numerator**2 / (2 ** (2 * N)) / (xp - x) ** 2 -plt.style.use("pennylane.drawer.plot") +nodes = [ch_node(i) for i in js] fig = plt.figure(figsize=(6.4, 2.4)) ax = fig.add_axes((0.15, 0.3, 0.8, 0.65)) # make room for caption -ax.set(xlabel=r"x", ylabel="Square Overlap") +ax.set(xlabel=r"x", ylabel="Squared Overlap") -# plot squared overlaps computed in circuit +# plot squared overlaps computed in the circuit ax.plot(nodes, np.abs(overlaps) ** 2, marker="o", label="circuit") # plot expected squared overlaps xs = np.linspace(-1, 1, 1000) -ax.plot(xs, [overlap_sq(x, nodes[j], N) for x in xs], label="expectation") +ax.plot(xs, [overlap_sq(x, nodes[j]) for x in xs], label="expectation") ax.legend() fig.text(0.5, 0.05, - "Figure 3. Squared overlap of Chebyshev basis states.", + "Figure 4. Squared overlap of Chebyshev states.", horizontalalignment="center", size="small", weight="normal", @@ -294,55 +329,9 @@ def overlap_sq(x, xp, N): ############################################# # We can see that the squared overlap between the basis states and the :math:`j=7` state :math:`|\tau(x_7^\mathrm{Ch})\rangle` is 0, unless :math:`x=x_7^\mathrm{Ch}\approx 0.1`, then the overlap is 1. # -# Let's also see if the amplitudes of the state in the computational basis agree with expectation. -# To do this, we just modify our ``circuit`` function to return the probabilities of each of the computational basis states (ignoring the ancilla). - - -@qml.qnode(dev) -def circuit(state=None): - qml.BasisState(state=state, wires=range(1, N + 1)) - QChT() - return qml.probs(wires=range(1, N + 1)) - - -probs = circuit(state=j) - -# computational basis indices -x = range(2**N) - - -def tau_amplitudes(x, k, N): - """computes the expected amplitud es of tau""" - if k == 0: - prefactor = 1 / 2 ** (N / 2) - else: - prefactor = 1 / 2 ** ((N - 1) / 2) - return prefactor * np.cos(k * np.arccos(x)) - - -fig = plt.figure(figsize=(6.4, 4.8)) -ax = fig.add_axes((0.15, 0.23, 0.80, 0.72)) # make room for caption -ax.plot(x, probs, "o", label="circuit") -ax.plot(x, [tau_amplitudes(nodes[j], xs, N) ** 2 for xs in x], label="expectation") -ax.set(xlabel=r"$|k\rangle$", ylabel="Probability") -ax.legend() -fig.text(0.5, 0.05, - "Figure 4. Squared overlap of Chebyshev basis state with computational basis states.", - horizontalalignment="center", - size="small", - weight="normal", -) -plt.show() - -############################################# -# The circuit output probabilities are exactly what we want. -# -# # Conclusion # ---------- -# In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. -# Further work could test the phase of the output to make sure it matches what we expect the QChT to output. -# One could also implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. +# In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. To build a generative model in the Chebyshev basis, one could implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. # # # References From c6676a29fd7672bda58ff1855a0d34ce8145e866 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 16 May 2025 14:20:01 -0400 Subject: [PATCH 42/64] small math mode correction --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 2b2569dd6a..997de71c90 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -221,7 +221,7 @@ def circuit_to_draw(): # ---------------- # With our QChT circuit, let's first check if the ancilla ends in the state :math:`|0\rangle`, and the output state amplitudes are real valued. # To do this, we'll input the computational basis state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. -# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with ancilla in |1\rangle). +# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with ancilla in :math:`|1\rangle``). j = 7 # initial state in computational basis From 7b33e3015fddf5438b2c2884efa81098c26047bf Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Fri, 16 May 2025 15:07:27 -0400 Subject: [PATCH 43/64] typo --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 997de71c90..5629f739a3 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -221,7 +221,7 @@ def circuit_to_draw(): # ---------------- # With our QChT circuit, let's first check if the ancilla ends in the state :math:`|0\rangle`, and the output state amplitudes are real valued. # To do this, we'll input the computational basis state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. -# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with ancilla in :math:`|1\rangle``). +# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with ancilla in :math:`|1\rangle`). j = 7 # initial state in computational basis From 68dc3ec1a0260318111e8d1aabb814bae40793d4 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 20 May 2025 12:40:57 -0400 Subject: [PATCH 44/64] Expanded on conclusion. Fixed some typos. --- .../tutorial_quantum_chebyshev_transform.py | 15 +++++++++------ 1 file changed, 9 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 5629f739a3..e93d91aa3e 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -2,7 +2,7 @@ ============================================================= This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. -They demonstrate the use of the Chebyshev basis space in generative modeling for probability distributions. A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. +They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. Crucial to the implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. @@ -69,7 +69,7 @@ Quantum Chebyshev basis --------------------------------------- The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. -The jth Chebyshev basis state using :math:`N` qubits is +The :math:`j`th Chebyshev basis state using :math:`N` qubits is .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, @@ -214,7 +214,7 @@ def circuit_to_draw(): fig.show() ############################################# -# Note we define a new function for the circuit to simplify the drawing, removing the returned ``qml.state``. +# Note we defined a new function for the circuit to simplify the drawing, removing the returned ``qml.state``. ############################################# # Testing the QChT @@ -232,10 +232,11 @@ def circuit_to_draw(): print(total_state) ############################################# -# Indeed, we see the second half of the amplitude vector is zero. Furthermore, the first :math:`2^N` entries are real valued, but let's check if the amplitudes of the state in the computational basis agree with our definition. +# Indeed, we see the second half of the amplitude vector is zero. +# Furthermore, the first :math:`2^N` entries are real valued, but let's check if the amplitudes of the state components in the computational basis agree with our definition. # reduce state size, effectively removing the ancilla -state = np.real(total_state[: 2**N]) # discard small imaginary components +state = np.real(total_state[: 2**N]) # discard the small imaginary components # computational basis indices x = range(2**N) @@ -331,7 +332,9 @@ def overlap_sq(x, xp): # # Conclusion # ---------- -# In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. To build a generative model in the Chebyshev basis, one could implement the quantum Chebyshev feature map from the same paper, which prepares a state in the Chevyshev space via a parameter :math:`x``. +# In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. +# The properties of Chebyshev polynomimals and the speed at which the QChT can be implemented make the Chebyshev basis an interesting choice of state space for quantum algorithms, such as generative modelling of probability distributions. +# To build a generative model in the Chebyshev basis, one could implement the quantum Chebyshev feature map from the same paper [#williams2023]_, which prepares a state in the Chevyshev space via a parameter :math:`x``. # # # References From 121622367f5538ca2c4ac99f2dda34b0b0f88e42 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 20 May 2025 12:42:31 -0400 Subject: [PATCH 45/64] More typos --- demonstrations/tutorial_quantum_chebyshev_transform.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index e93d91aa3e..371502d094 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -75,7 +75,7 @@ |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, where :math:`|k\rangle` are the computational basis states. -These states are orthonormal due to the orthgonality of the Chebyshev polynomials and the normalization convention defined above, that is +These states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization convention defined above, that is .. math:: \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \delta_{j, j'}\,. @@ -333,7 +333,7 @@ def overlap_sq(x, xp): # Conclusion # ---------- # In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. -# The properties of Chebyshev polynomimals and the speed at which the QChT can be implemented make the Chebyshev basis an interesting choice of state space for quantum algorithms, such as generative modelling of probability distributions. +# The properties of Chebyshev polynomials and the speed at which the QChT can be implemented make the Chebyshev basis an interesting choice of state space for quantum algorithms, such as generative modelling of probability distributions. # To build a generative model in the Chebyshev basis, one could implement the quantum Chebyshev feature map from the same paper [#williams2023]_, which prepares a state in the Chevyshev space via a parameter :math:`x``. # # From 05878e868c367eddd989c133da479e649e916c54 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 20 May 2025 13:23:54 -0400 Subject: [PATCH 46/64] Added thumbnail figure to intro. --- .../tutorial_quantum_chebyshev_transform.py | 18 +++++++++++++----- 1 file changed, 13 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 371502d094..34e92b1ec9 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -2,11 +2,19 @@ ============================================================= This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. -They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. +They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. +A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. Crucial to the implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. +.. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/quantum_chebyshev_transform/thumbnail_quantum_chebyshev_transform.png + :align: center + :width: 60% + :target: javascript:void(0) + :alt: Quantum Chebyshev Transform mapping between the uniform computational basis and the non-uniform Chebyshev basis. + + Figure 1: QChT -- a map between the uniform computational basis and the non-uniform Chebyshev basis. What are Chebyshev polynomials? --------------------------------------- @@ -42,7 +50,7 @@ :target: javascript:void(0) :alt: Plot of first six Chebyshev polynomials, with nodes denser near the boundary. - Figure 1. The first six Chebyshev polynomials, along with their corresponding nodes. + Figure 2. The first six Chebyshev polynomials, along with their corresponding nodes. The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of :math:`T_N(x)` in the following way for :math:`k, \ell`__ applied on all qubits. @@ -267,7 +275,7 @@ def tau_amplitudes(x, k): ax.set(xlabel=r"$|k\rangle$", ylabel="Amplitude") ax.legend() fig.text(0.5, 0.05, - r"Figure 3. Amplitudes of $|\tau(x_7^\mathrm{Ch})\rangle$ in computational basis.", + r"Figure 4. Amplitudes of $|\tau(x_7^\mathrm{Ch})\rangle$ in computational basis.", horizontalalignment="center", size="small", weight="normal", @@ -319,7 +327,7 @@ def overlap_sq(x, xp): ax.legend() fig.text(0.5, 0.05, - "Figure 4. Squared overlap of Chebyshev states.", + "Figure 5. Squared overlap of Chebyshev states.", horizontalalignment="center", size="small", weight="normal", From 400338136ebac1b47b45bd99576023037d5657ac Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 20 May 2025 13:27:44 -0400 Subject: [PATCH 47/64] Added qubitization to related content --- .../tutorial_quantum_chebyshev_transform.metadata.json | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json index 8ae9c851eb..cf7675f557 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json +++ b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json @@ -51,6 +51,11 @@ "type": "demonstration", "id": "tutorial_qcbm", "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_qubitization", + "weight": 1.0 } ] } From ea7d9d62178f33b4a3d08288511a18c8e738f0c1 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 20 May 2025 18:06:23 -0400 Subject: [PATCH 48/64] Fixed figure reference issue. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 34e92b1ec9..bf8e9cf099 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -8,7 +8,7 @@ We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. -.. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/quantum_chebyshev_transform/thumbnail_quantum_chebyshev_transform.png +.. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center :width: 60% :target: javascript:void(0) @@ -77,7 +77,7 @@ Quantum Chebyshev basis --------------------------------------- The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. -The :math:`j`th Chebyshev basis state using :math:`N` qubits is +The :math:`j` -th Chebyshev basis state using :math:`N` qubits is .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, From 33c54732bcaf963dc420a5e6fa36bed2c60af3fe Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Thu, 22 May 2025 13:24:15 -0400 Subject: [PATCH 49/64] Edits round 1 from Alvaro review --- .../tutorial_quantum_chebyshev_transform.py | 74 ++++++++++--------- 1 file changed, 39 insertions(+), 35 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index bf8e9cf099..d552358d00 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -4,27 +4,27 @@ This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. -Crucial to the implementation of learning models in Chebyshev space is the quantum Chebyshev transform (QChT), which is used to swap between the computational basis and the Chebyshev basis. +Crucial to the implementation of learning models in Chebyshev space is the quantum Chebyshev transform, which is used to swap between the computational basis and the Chebyshev basis. -We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the QChT can be implemented in PennyLane. +We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the quantum Chebyshev transform can be implemented in PennyLane. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center :width: 60% :target: javascript:void(0) - :alt: Quantum Chebyshev Transform mapping between the uniform computational basis and the non-uniform Chebyshev basis. + :alt: Quantum Chebyshev Transform mapping between the computational basis and the non-uniform Chebyshev basis. - Figure 1: QChT -- a map between the uniform computational basis and the non-uniform Chebyshev basis. + Figure 1: Quantum Chebyshev transform -- a map between the computational basis and the non-uniform Chebyshev basis. What are Chebyshev polynomials? --------------------------------------- -`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. They can be defined as +`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. The :math:`n` -th order Chebyshev polynomial is defined as .. math:: - T_n(x) \equiv \cos(n \arccos(x))\,, + T_n(x) \equiv \cos(n \arccos(x))\,. -where :math:`n` is the order of the polynomial. We can write out the first few orders explicitly. +We can write out the first few orders explicitly. .. math:: T_0(x) &= 1 \\ @@ -62,11 +62,11 @@ N/2 & k = \ell \neq 0\,. \end{cases} -The Chebyshev polynomials have a lot of *nice* properties. Because they are complete, any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. -To do this process numerically on a classical computer for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. -However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation (DChT) `__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. +The Chebyshev polynomials have a lot of *nice* properties. They are complete on the interval :math:`[-1,1]`, meaning any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. +To compute the Chebyshev-polynomial expansion of a function on a classical computer for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, the `discrete Chebyshev transformation `__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. -The DChT is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. +The discrete Chebyshev transformation is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. This non-uniform sampling has more resolution at the boundary, but less in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. @@ -76,7 +76,7 @@ Quantum Chebyshev basis --------------------------------------- -The quantum Chebyshev transform (QChT) circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. +The quantum Chebyshev transform circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. The :math:`j` -th Chebyshev basis state using :math:`N` qubits is .. math:: @@ -86,7 +86,11 @@ These states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization convention defined above, that is .. math:: - \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \delta_{j, j'}\,. + \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = + \begin{cases} + 0 & j \neq j'\,,\\ + 1 & j = j' \neq 0\,. + \end{cases} The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} |\tau(x_j^\mathrm{Ch})\rangle\langle x_j|`. @@ -94,30 +98,30 @@ Designing the transform circuit --------------------------------------- Let's start from the end and look at the circuit diagram generated from the code we want to write. -An ancilla qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`|x\rangle` which starts in the computational basis, shown below as :math:`|\psi\rangle`. We demonstrate for :math:`N=4` non-ancilla qubits. +An auxiliary qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`|x\rangle` which starts in the computational basis, shown below as :math:`|\psi\rangle`. We demonstrate for :math:`N=4` non-auxiliary qubits. -.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/qcht_circuit_diagram.png +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/QChT_circuit_diagram.png :align: center :width: 100% :target: javascript:void(0) - :alt: Quantum Chebyshev Transform circuit diagram drawn using PennyLane + :alt: Quantum Chebyshev transform circuit diagram drawn using PennyLane - Figure 3. Quantum Chebyshev Transform circuit. + Figure 3. Quantum Chebyshev transform circuit. The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. Notice the use of the `quantum Fourier transform (QFT) `__ applied on all qubits. -The QChT is an extended QFT circuit with some added interference and mixing of the elements. -Note the ancilla starts and ends in the state :math:`|0\rangle`, and the amplitudes of the transformed state are all real valued. +The quantum Chebyshev transform is an extended QFT circuit with some added interference and mixing of the elements. +Note the auxiliary qubit starts and ends in the state :math:`|0\rangle`, and the amplitudes of the transformed state are all real valued. Let's break down the circuit above into pieces that we will use inside our circuit function. -First, a Hadamard gate is applied to the ancilla, and then a CNOT ladder is applied, controlled on the ancilla. +First, a Hadamard gate is applied to the auxiliary qubit, and then a CNOT ladder is applied, controlled on the auxiliary qubit. To start, we will define a function for the CNOT ladder. """ import pennylane as qml -# number of qubits (non-ancilla) +# number of qubits (non-auxiliary qubit) N = 4 @@ -130,7 +134,7 @@ def CNOT_ladder(): # After the initial CNOT ladder comes an :math:`N+1` QFT circuit, which can be implemented using ``qml.QFT``. # # Next are phase rotations and shifts. -# For the ancilla, there is a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}` followed by a phase shift of :math:`-\pi/2^{(N+1)}` . +# For the auxiliary qubit, there is a :math:`Z` rotation of :math:`-\pi(2^N - 1)/2^{N+1}` followed by a phase shift of :math:`-\pi/2^{(N+1)}` . # The other qubits are rotated in :math:`Z` by :math:`\pi/2^{(j+1)}`, where :math:`j` is the index of the qubit as labelled in the circuit diagram. import numpy as np @@ -139,7 +143,7 @@ def CNOT_ladder(): def rotate_phases(): - """Rotates and shifts the phase of the ancilla and rotates the jth qubit by + """Rotates and shifts the phase of the auxiliary qubit and rotates the jth qubit by pi/2^(j+1) in Z.""" qml.RZ(-pi * (2**N - 1) / 2 ** (N + 1), wires=0) qml.PhaseShift(-pi / 2 ** (N + 1), wires=0) @@ -149,7 +153,7 @@ def rotate_phases(): ############################################# # Now a permutation of the qubits is used to reorder them. -# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the ancilla and all qubits with larger index than the target. +# This is built using a multicontrolled NOT gate applied to each qubit from the initial state, which is controlled on the auxiliary qubit and all qubits with larger index than the target. # The multicontrolled NOT gate can be implemented using a multicontrolled Pauli X gate. # Let's see what that looks like. @@ -165,13 +169,13 @@ def permute_elements(): # In the above code, we use ``reversed`` to loop over the qubits in reverse order, to apply the controlled gate to the last qubit first. # After the permutation is another CNOT ladder, which we already have a function for. # -# The last part is a phase adjustment of the ancilla qubit: a rotation in :math:`Y` by :math:`\pi/2`, a phase shift of :math:`-\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. +# The last part is a phase adjustment of the auxiliary qubit: a rotation in :math:`Y` by :math:`\pi/2`, a phase shift of :math:`-\pi/2` and a multicontrolled :math:`X` rotation by :math:`\pi/2`. # All of the other qubits control the :math:`X` rotation, but the control is sandwiched by Pauli :math:`X` operators. # We can implement the multicontrolled :math:`X` rotation by using the function ``qml.ctrl`` on ``qml.RX``, specifying the target wire in ``qml.RX`` and the control wires as the second argument of ``qml.ctrl``. def adjust_phases(): - """Adjusts the phase of the ancilla qubit.""" + """Adjusts the phase of the auxiliary qubit.""" qml.RY(-pi / 2, wires=0) qml.PhaseShift(-pi / 2, wires=0) # first Pauli Xs @@ -225,15 +229,15 @@ def circuit_to_draw(): # Note we defined a new function for the circuit to simplify the drawing, removing the returned ``qml.state``. ############################################# -# Testing the QChT +# Testing the quantum Chebyshev transform # ---------------- -# With our QChT circuit, let's first check if the ancilla ends in the state :math:`|0\rangle`, and the output state amplitudes are real valued. +# With our quantum Chebyshev transform circuit, let's first check if the auxiliary qubit ends in the state :math:`|0\rangle` and the output state amplitudes are real valued. # To do this, we'll input the computational basis state :math:`|7\rangle`, which will transform into :math:`|\tau(x_7^\mathrm{Ch})\rangle`. -# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with ancilla in :math:`|1\rangle`). +# We expect the full output state to be :math:`|0\rangle|\tau(x_7^\mathrm{Ch})\rangle`, which means the second half of the amplitude vector should be zero (corresponding to states with the auxiliary qubit in :math:`|1\rangle`). j = 7 # initial state in computational basis -total_state = circuit(state=j) # state with ancilla +total_state = circuit(state=j) # state with auxiliary qubit # round very small values to zero total_state = np.where(np.abs(total_state)<1e-12, 0, total_state) @@ -243,7 +247,7 @@ def circuit_to_draw(): # Indeed, we see the second half of the amplitude vector is zero. # Furthermore, the first :math:`2^N` entries are real valued, but let's check if the amplitudes of the state components in the computational basis agree with our definition. -# reduce state size, effectively removing the ancilla +# reduce state size, effectively removing the auxiliary qubit state = np.real(total_state[: 2**N]) # discard the small imaginary components # computational basis indices @@ -292,13 +296,13 @@ def tau_amplitudes(x, k): overlaps = [np.vdot(state, circuit(state=i)[: 2**N]) for i in js] ############################################# -# We compare these circuit calculated overlaps to the definition, for which we plot the squared overlaps at all values of :math:`x`. +# We compare these circuit-calculated overlaps to the definition, for which we plot the squared overlaps at all values of :math:`x`. # This continuous overlap function can be derived analytically as # # .. math:: # |\langle\tau(x_j^\mathrm{Ch})|\tau(x)\rangle|^2 = \frac{\left(T_{2^N+1}(x_j^\mathrm{Ch})T_{2^N}(x)-T_{2^N}(x_j^\mathrm{Ch})T_{2^N+1}(x)\right)^2}{2^{2N}(x_j^\mathrm{Ch}-x)^2}\,, # -# where :math:`\tau(x)` is a generalization of one of Chebyshev basis states defined earlier, where :math:`x` can be any value in :math:`[-1,1]` rather than just one of the nodes. +# where :math:`\tau(x)` is a generalization of one of Chebyshev basis states defined earlier and :math:`x` can be any value in :math:`[-1,1]` rather than just one of the nodes. def T_n(x, n): @@ -340,8 +344,8 @@ def overlap_sq(x, xp): # # Conclusion # ---------- -# In this tutorial, we've gone through how to implement the QChT from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. -# The properties of Chebyshev polynomials and the speed at which the QChT can be implemented make the Chebyshev basis an interesting choice of state space for quantum algorithms, such as generative modelling of probability distributions. +# In this tutorial, we've gone through how to implement the quantum Chebyshev transform from the paper by Williams *et al.*, and tested the circuit output by looking at the state amplitudes and the orthonormality. +# The properties of Chebyshev polynomials and the speed at which the quantum Chebyshev transform can be implemented make the Chebyshev basis an interesting choice of state space for quantum algorithms, such as generative modelling of probability distributions. # To build a generative model in the Chebyshev basis, one could implement the quantum Chebyshev feature map from the same paper [#williams2023]_, which prepares a state in the Chevyshev space via a parameter :math:`x``. # # From 7f9a50f56fa8f9b6a132b7ee3b2dedbd4127c435 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Thu, 22 May 2025 14:58:04 -0400 Subject: [PATCH 50/64] New introduction --- demonstrations/tutorial_quantum_chebyshev_transform.py | 7 ++----- 1 file changed, 2 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index d552358d00..9bd31e61d6 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -1,12 +1,9 @@ r"""Quantum Chebyshev Transform ============================================================= -This demo is inspired by the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_ wherein the authors describe a workflow for quantum Chebyshev-based model building. -They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. -A proposed protocol for learning and sampling multivariate probability distributions that arise in high-energy physics also makes use of the Chebyshev basis [#delejarza2025]_. -Crucial to the implementation of learning models in Chebyshev space is the quantum Chebyshev transform, which is used to swap between the computational basis and the Chebyshev basis. +Looking for ways to leverage the speed of the quantum Fourier transform is a common way to design quantum algorithms with exponential speed ups over classical algorithms. Working in the Fourier basis can be a better choice for some computations rather than the standard basis. In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe the Chebyshev basis and its associated transformation, the quantum Chebyshev transform. They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions and further work proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform, allowing for fast transformations between the standard and the Chebyshev basis. -We will start by discussing Chebyshev polynomials and why you may want to work in Chebyshev space. Then we will show how the quantum Chebyshev transform can be implemented in PennyLane. +In this demo, we will show how the quantum Chebyshev transform can be implemented in Pennylane. We will start by describing what Chebyshev polynomials are, why you may want to work with Chebyshev states, and what the Chebyshev basis in a quantum algorithm looks like. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center From ec34b759c039a358f79402a7d03d9b02302884dc Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Thu, 22 May 2025 15:29:44 -0400 Subject: [PATCH 51/64] Major revisions to text in background section --- .../tutorial_quantum_chebyshev_transform.py | 33 ++++++++++--------- 1 file changed, 18 insertions(+), 15 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 9bd31e61d6..8727290daa 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -1,9 +1,13 @@ r"""Quantum Chebyshev Transform ============================================================= -Looking for ways to leverage the speed of the quantum Fourier transform is a common way to design quantum algorithms with exponential speed ups over classical algorithms. Working in the Fourier basis can be a better choice for some computations rather than the standard basis. In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe the Chebyshev basis and its associated transformation, the quantum Chebyshev transform. They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions and further work proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform, allowing for fast transformations between the standard and the Chebyshev basis. +Looking for ways to leverage the speed of the `quantum Fourier transform `__ is a common way to design quantum algorithms with exponential speed ups over classical algorithms. Working in the Fourier basis can be a better choice for some computations rather than the standard basis. +In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe the Chebyshev basis and its associated transformation, the quantum Chebyshev transform. +They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions and further work proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. +Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform, allowing for fast transformations between the standard and the Chebyshev basis. -In this demo, we will show how the quantum Chebyshev transform can be implemented in Pennylane. We will start by describing what Chebyshev polynomials are, why you may want to work with Chebyshev states, and what the Chebyshev basis in a quantum algorithm looks like. +In this demo, we will show how the quantum Chebyshev transform can be implemented in Pennylane. +We will start by describing what Chebyshev polynomials are, why you may want to use Chebyshev states, and what the Chebyshev basis in a quantum algorithm looks like. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center @@ -16,7 +20,8 @@ What are Chebyshev polynomials? --------------------------------------- -`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. The :math:`n` -th order Chebyshev polynomial is defined as +`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. +The :math:`n` -th order Chebyshev polynomial is defined as .. math:: T_n(x) \equiv \cos(n \arccos(x))\,. @@ -49,7 +54,8 @@ Figure 2. The first six Chebyshev polynomials, along with their corresponding nodes. -The nodes are plotted above along with the corresponding polynomials. Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of :math:`T_N(x)` in the following way for :math:`k, \ell`__ can be related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. +The Chebyshev polynomials have a lot of *nice* properties. +They are complete on the interval :math:`[-1,1]`, meaning any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. +To compute the Chebyshev expansion of a function on a classical computer for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, one can define a `discrete Chebyshev transformation `__ that is related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. +This discrete Chebyshev transformation is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. +This non-uniform sampling has more resolution at the boundary than in the middle, which can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary. -The discrete Chebyshev transformation is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. -This non-uniform sampling has more resolution at the boundary, but less in the middle. -This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary, so the extra resolution there is useful. -In general, working in the Chebyshev basis can have advantages over the Fourier basis for polynomial decomposition. - -Next, we will describe the quantum analogue of this transformation. +The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing it to be designed effiency by utilizing the `quantum Fourier transform `__ . Quantum Chebyshev basis @@ -79,8 +83,7 @@ .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, -where :math:`|k\rangle` are the computational basis states. -These states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization convention defined above, that is +where :math:`|k\rangle` are the computational basis states and :math:`x_j^\mathrm{Ch}` is the :math:`j` -th node of the Chebyshev polynomial of order :math:`2^N-1`. Notice the amplitude of the :math:`k` -th basis state component is the :math:`k` -th order Chebyshev polynomial evaluated at the :math`j` -th Chebyshev node. This construction guarantees the states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization factors used, that is .. math:: \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = From 72149e122f83242067b40bbc1a9a6dee6b8e0cb7 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Thu, 22 May 2025 16:48:42 -0400 Subject: [PATCH 52/64] More intro edits. --- demonstrations/tutorial_quantum_chebyshev_transform.py | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 8727290daa..9dc60a0f24 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -1,13 +1,14 @@ r"""Quantum Chebyshev Transform ============================================================= -Looking for ways to leverage the speed of the `quantum Fourier transform `__ is a common way to design quantum algorithms with exponential speed ups over classical algorithms. Working in the Fourier basis can be a better choice for some computations rather than the standard basis. -In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe the Chebyshev basis and its associated transformation, the quantum Chebyshev transform. -They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions and further work proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. +Looking for ways to leverage the speed of the `quantum Fourier transform `__ is a common way to design quantum algorithms with exponential speed ups over classical algorithms. +Working in the Fourier basis can be a better choice for some computations rather than the standard basis. +In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe a different basis, the *Chebyshev* basis, and its associated transformation, the *quantum Chebyshev transform*. +They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. Further work also proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform, allowing for fast transformations between the standard and the Chebyshev basis. In this demo, we will show how the quantum Chebyshev transform can be implemented in Pennylane. -We will start by describing what Chebyshev polynomials are, why you may want to use Chebyshev states, and what the Chebyshev basis in a quantum algorithm looks like. +To start we'll describe what Chebyshev polynomials are, why you may want to use Chebyshev states, and what the quantum Chebyshev basis looks like. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center From 3124650b0d5c36ae9d60708d4c1d0b395ab6d5da Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Thu, 22 May 2025 17:03:24 -0400 Subject: [PATCH 53/64] Update tutorial_quantum_chebyshev_transform.py --- .../tutorial_quantum_chebyshev_transform.py | 18 ++++++++++-------- 1 file changed, 10 insertions(+), 8 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 9dc60a0f24..ad815e94e6 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -5,10 +5,10 @@ Working in the Fourier basis can be a better choice for some computations rather than the standard basis. In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe a different basis, the *Chebyshev* basis, and its associated transformation, the *quantum Chebyshev transform*. They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. Further work also proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. -Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform, allowing for fast transformations between the standard and the Chebyshev basis. +Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform to allow for fast transformations between the standard and the Chebyshev basis. -In this demo, we will show how the quantum Chebyshev transform can be implemented in Pennylane. -To start we'll describe what Chebyshev polynomials are, why you may want to use Chebyshev states, and what the quantum Chebyshev basis looks like. +In this demo we will show how the quantum Chebyshev transform can be implemented in Pennylane. +To start, we'll describe what Chebyshev polynomials are, why you may want to use Chebyshev states, and what the quantum Chebyshev basis looks like. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center @@ -69,11 +69,11 @@ The Chebyshev polynomials have a lot of *nice* properties. They are complete on the interval :math:`[-1,1]`, meaning any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. To compute the Chebyshev expansion of a function on a classical computer for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. -However, because of the way the Chebyshev polynomials are defined in terms of cosine, one can define a `discrete Chebyshev transformation `__ that is related to the `discrete cosine transform (DCT) `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, one can define a `discrete Chebyshev transformation `__ that is related to the `discrete cosine transform `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. This discrete Chebyshev transformation is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. This non-uniform sampling has more resolution at the boundary than in the middle, which can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary. -The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing it to be designed effiency by utilizing the `quantum Fourier transform `__ . +The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing it to be designed efficiently by utilizing the `quantum Fourier transform `__ . Quantum Chebyshev basis @@ -84,13 +84,15 @@ .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, -where :math:`|k\rangle` are the computational basis states and :math:`x_j^\mathrm{Ch}` is the :math:`j` -th node of the Chebyshev polynomial of order :math:`2^N-1`. Notice the amplitude of the :math:`k` -th basis state component is the :math:`k` -th order Chebyshev polynomial evaluated at the :math`j` -th Chebyshev node. This construction guarantees the states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization factors used, that is +where :math:`|k\rangle` are the computational basis states and :math:`x_j^\mathrm{Ch}` is the :math:`j` -th node of the Chebyshev polynomial of order :math:`2^N-1`. +Notice how the amplitudes of the basis state components the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node. +This construction guarantees the states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization factors used, that is .. math:: \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = \begin{cases} 0 & j \neq j'\,,\\ - 1 & j = j' \neq 0\,. + 1 & j = j'\,. \end{cases} The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} |\tau(x_j^\mathrm{Ch})\rangle\langle x_j|`. @@ -109,7 +111,7 @@ Figure 3. Quantum Chebyshev transform circuit. -The intuition for the structure of the above circuit comes from the link between the DChT and the DCT. +The intuition for the structure of the above circuit comes from the link between the discrete Chebyshev transform and the discrete cosine transform. Notice the use of the `quantum Fourier transform (QFT) `__ applied on all qubits. The quantum Chebyshev transform is an extended QFT circuit with some added interference and mixing of the elements. Note the auxiliary qubit starts and ends in the state :math:`|0\rangle`, and the amplitudes of the transformed state are all real valued. From c051f249d7ab1d04ed9d571b30a4e91e78445798 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 26 May 2025 10:46:48 -0400 Subject: [PATCH 54/64] Some more edits. --- .../tutorial_quantum_chebyshev_transform.py | 12 +++++++----- 1 file changed, 7 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index ad815e94e6..d41b79dee0 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -2,9 +2,11 @@ ============================================================= Looking for ways to leverage the speed of the `quantum Fourier transform `__ is a common way to design quantum algorithms with exponential speed ups over classical algorithms. -Working in the Fourier basis can be a better choice for some computations rather than the standard basis. +Working in the Fourier basis can be a more natural choice than the standard basis for some computations. +Swapping bases is feasible due to the efficiency of the quantum Fourier transform. In the paper `"Quantum Chebyshev transform: mapping, embedding, learning and sampling distributions" `__ [#williams2023]_, the authors describe a different basis, the *Chebyshev* basis, and its associated transformation, the *quantum Chebyshev transform*. -They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. Further work also proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. +They demonstrate the use of the Chebyshev basis space in generative modelling of probability distributions. +Further work also proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform to allow for fast transformations between the standard and the Chebyshev basis. In this demo we will show how the quantum Chebyshev transform can be implemented in Pennylane. @@ -68,8 +70,8 @@ The Chebyshev polynomials have a lot of *nice* properties. They are complete on the interval :math:`[-1,1]`, meaning any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. -To compute the Chebyshev expansion of a function on a classical computer for a discrete set of sampling points would take :math:`\mathcal{O}(N^2)` operations for a general set of complete functions 🐌. -However, because of the way the Chebyshev polynomials are defined in terms of cosine, one can define a `discrete Chebyshev transformation `__ that is related to the `discrete cosine transform `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms, which take :math:`\mathcal{O}(N \log N)` operations 🚀. +For a general set of complete functions, to compute the expansion of a function in that set on a classical computer for a discrete number of sampling points would take :math:`\mathcal{O}(N^2)` operations 🐌. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, one can define a `discrete Chebyshev transformation `__ that is related to the `discrete cosine transform `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms for expansion, which take :math:`\mathcal{O}(N \log N)` operations 🚀. This discrete Chebyshev transformation is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. This non-uniform sampling has more resolution at the boundary than in the middle, which can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary. @@ -85,7 +87,7 @@ |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, where :math:`|k\rangle` are the computational basis states and :math:`x_j^\mathrm{Ch}` is the :math:`j` -th node of the Chebyshev polynomial of order :math:`2^N-1`. -Notice how the amplitudes of the basis state components the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node. +Notice how the amplitudes of the basis state components are the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node. This construction guarantees the states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization factors used, that is .. math:: From ced758230c34c9f05b6193a067b30506c1f28648 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 27 May 2025 10:11:30 -0400 Subject: [PATCH 55/64] Added DCT section. --- .../tutorial_quantum_chebyshev_transform.py | 57 +++++++++++++++---- 1 file changed, 46 insertions(+), 11 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index d41b79dee0..8c0bf1e0e7 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -10,7 +10,7 @@ Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform to allow for fast transformations between the standard and the Chebyshev basis. In this demo we will show how the quantum Chebyshev transform can be implemented in Pennylane. -To start, we'll describe what Chebyshev polynomials are, why you may want to use Chebyshev states, and what the quantum Chebyshev basis looks like. +To start, we'll describe what Chebyshev polynomials are, including the classical discrete Chebyshev transform. After that, we'll look at the quantum Chebyshev basis and its transform. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center @@ -23,13 +23,14 @@ What are Chebyshev polynomials? --------------------------------------- -`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. -The :math:`n` -th order Chebyshev polynomial is defined as +`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. *Completeness* here means any function :math:`f(x)` on that interval can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. + +The :math:`n` -th order Chebyshev polynomial of the first kind is defined as .. math:: T_n(x) \equiv \cos(n \arccos(x))\,. -We can write out the first few orders explicitly. +Note there are more types of Chebyshev polynomials, but in this demo, we will only discuss those of the first kind. We can write out the first few orders explicitly. .. math:: T_0(x) &= 1 \\ @@ -68,14 +69,48 @@ N/2 & k = \ell \neq 0\,. \end{cases} -The Chebyshev polynomials have a lot of *nice* properties. -They are complete on the interval :math:`[-1,1]`, meaning any function :math:`f(x)` on the interval :math:`x\in [-1,1]` can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. -For a general set of complete functions, to compute the expansion of a function in that set on a classical computer for a discrete number of sampling points would take :math:`\mathcal{O}(N^2)` operations 🐌. -However, because of the way the Chebyshev polynomials are defined in terms of cosine, one can define a `discrete Chebyshev transformation `__ that is related to the `discrete cosine transform `__ to leverage the efficiency of the `fast-Fourier-transform `__-style algorithms for expansion, which take :math:`\mathcal{O}(N \log N)` operations 🚀. -This discrete Chebyshev transformation is sampled on the nodes of :math:`T_N(x)`, which are non-equidistant on the :math:`[-1,1]` interval. -This non-uniform sampling has more resolution at the boundary than in the middle, which can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary. +The Chebyshev polynomials have a lot of *nice* properties that make them well suited as a basis for series expansion of a function on a real interval. They are often utilized in numerical integration and interpolation methods, where the `discrete Chebyshev transformation `__ is needed to map between the function evaluated on a grid of points and the coefficients of the Chebyshev expansion of that function. Let's take a look at the discrete transform. + +Discrete Chebyshev transform +--------------------------------------- +A common definition of the discrete Chebyshev transform uses the grid of Chebyshev nodes, also known as the roots grid, as the sampling points. For a given function :math:`f(x)` sampled at :math:`N` nodes on the interval :math:`[-1,1]`, the transform computes the coefficents of that function expanded in Chebyshev polynomials evaluated on the grid. That is, the :math:`j` -th coefficient is, for :math:`j>0` + +.. math:: + a_j = \frac{2}{N}\sum_{k=0}^{N-1} f(x_k) T_j(x_k)\,, + +and for :math:`j=0`, there is a slightly different normalization factor + +.. math:: + a_0 = \frac{1}{N}\sum_{k=0}^{N-1} f(x_k)\,. + +The inverse of the transform is then just the series expansion evaluated on the grid + +.. math:: + f(x_k) = \sum_{j=0}^{N-1} a_j T_j(x_k)\,. + +Since a function expanded in Chebyshev polynomials in this way will be sampled on the non-uniformly spaced Chebyshev nodes, it will have more resolution at the boundary than in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary. + +For a general set of complete functions, computing the expansion of a function in that set on a classical computer for a discrete number of sampling points would take :math:`\mathcal{O}(N^2)` operations 🐌. +However, because of the way the Chebyshev polynomials are defined in terms of cosine, the discrete Chebyshev transformation is related to the `discrete cosine transform `__. This allows the discrete Chebyshev transform to be implemented in a way that leverages the efficiency of the `fast-Fourier-transform `__-style algorithms for expansion, which take :math:`\mathcal{O}(N \log N)` operations 🚀. + +We can see this relation by plugging in the definition of the Chebyshev polynomials into the inverse transform + +.. math:: + f(x_k) = \sum_{j=0}^{N-1} a_j \cos(j \cos^{-1}(x_k))\,, + +then if we also use the definition of the nodes we obtain + +.. math:: + f(x_k) = \sum_{j=0}^{N-1} a_j \cos(\frac{j\pi}{N}(N + k + 1/2))\,. + +Finally, we can use the cyclical property of cosine to convert an :math:`j \pi` term in the argument to a :math:`(-1)^{j}` factor in the coefficient + +.. math:: + f(x_k) = \sum_{j=0}^{N-1} a_j (-1)^{j}\cos(\frac{j\pi}{N}(k + 1/2))\,, + +Which looks just like a discrete cosine transform. -The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing it to be designed efficiently by utilizing the `quantum Fourier transform `__ . +The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing it to be designed efficiently by utilizing the `quantum Fourier transform `__ . Next we will show how Chebyshev polynomials could be represented in a quantum state, before we describe the transform. Quantum Chebyshev basis From 8c222c7bc7b6d531acb7632620e1a6d831cd6dfd Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 27 May 2025 10:55:53 -0400 Subject: [PATCH 56/64] Small edits --- demonstrations/tutorial_quantum_chebyshev_transform.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 8c0bf1e0e7..de5f77a0f8 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -54,7 +54,7 @@ :align: center :width: 100% :target: javascript:void(0) - :alt: Plot of first six Chebyshev polynomials, with nodes denser near the boundary. + :alt: Plot of first six Chebyshev polynomials, with nodes densely packed near the boundary. Figure 2. The first six Chebyshev polynomials, along with their corresponding nodes. @@ -140,7 +140,7 @@ Let's start from the end and look at the circuit diagram generated from the code we want to write. An auxiliary qubit is required, which will be the :math:`0` indexed qubit, and the rest compose the state :math:`|x\rangle` which starts in the computational basis, shown below as :math:`|\psi\rangle`. We demonstrate for :math:`N=4` non-auxiliary qubits. -.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/QChT_circuit_diagram.png +.. figure:: ../_static/demonstration_assets/quantum_chebyshev_transform/qcht_circuit_diagram.png :align: center :width: 100% :target: javascript:void(0) From 0787e3c9d3c160c8c8593ae66f87223922faef70 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 27 May 2025 11:34:30 -0400 Subject: [PATCH 57/64] Edited text. --- .../tutorial_quantum_chebyshev_transform.py | 29 +++++++++---------- 1 file changed, 14 insertions(+), 15 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index de5f77a0f8..e0082e8f8a 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -69,11 +69,11 @@ N/2 & k = \ell \neq 0\,. \end{cases} -The Chebyshev polynomials have a lot of *nice* properties that make them well suited as a basis for series expansion of a function on a real interval. They are often utilized in numerical integration and interpolation methods, where the `discrete Chebyshev transformation `__ is needed to map between the function evaluated on a grid of points and the coefficients of the Chebyshev expansion of that function. Let's take a look at the discrete transform. +The Chebyshev polynomials have a lot of *nice* properties that make them a good choice to series expansion a function on a real interval. They are often utilized in numerical integration and interpolation methods, where the `discrete Chebyshev transformation `__ is needed to map between the function evaluated on a grid of points and the coefficients of the Chebyshev expansion of that function. Let's take a look at the discrete transform. Discrete Chebyshev transform --------------------------------------- -A common definition of the discrete Chebyshev transform uses the grid of Chebyshev nodes, also known as the roots grid, as the sampling points. For a given function :math:`f(x)` sampled at :math:`N` nodes on the interval :math:`[-1,1]`, the transform computes the coefficents of that function expanded in Chebyshev polynomials evaluated on the grid. That is, the :math:`j` -th coefficient is, for :math:`j>0` +A common definition of the discrete Chebyshev transform uses the grid of Chebyshev nodes, also known as the roots grid, as the sampling points. For a given function :math:`f(x)` sampled at :math:`N` nodes on the interval :math:`[-1,1]`, the transform computes the coefficents of that function expanded in Chebyshev polynomials evaluated on the grid. That is, the :math:`j` -th coefficient, for :math:`j>0`, is .. math:: a_j = \frac{2}{N}\sum_{k=0}^{N-1} f(x_k) T_j(x_k)\,, @@ -88,41 +88,40 @@ .. math:: f(x_k) = \sum_{j=0}^{N-1} a_j T_j(x_k)\,. -Since a function expanded in Chebyshev polynomials in this way will be sampled on the non-uniformly spaced Chebyshev nodes, it will have more resolution at the boundary than in the middle. This can be a benefit if you are, for example, solving a differential equation and expect more interesting features at the boundary. +Since a function expanded in Chebyshev polynomials in this way will be sampled on the non-uniformly spaced Chebyshev nodes, it will have more resolution at the boundary than in the middle. This can be beneficial if you are, for example, solving a differential equation and expect more interesting features at the boundary. -For a general set of complete functions, computing the expansion of a function in that set on a classical computer for a discrete number of sampling points would take :math:`\mathcal{O}(N^2)` operations 🐌. +In general, computing the expansion of a function in a complete set on a classical computer for a discrete number of sampling points would take :math:`\mathcal{O}(N^2)` operations 🐌. However, because of the way the Chebyshev polynomials are defined in terms of cosine, the discrete Chebyshev transformation is related to the `discrete cosine transform `__. This allows the discrete Chebyshev transform to be implemented in a way that leverages the efficiency of the `fast-Fourier-transform `__-style algorithms for expansion, which take :math:`\mathcal{O}(N \log N)` operations 🚀. -We can see this relation by plugging in the definition of the Chebyshev polynomials into the inverse transform +We can see the relation to the cosine transform by plugging in the definition of the Chebyshev polynomials and the nodes into the inverse transform and simplifying. Starting with the polynomials .. math:: - f(x_k) = \sum_{j=0}^{N-1} a_j \cos(j \cos^{-1}(x_k))\,, + f(x_k) = \sum_{j=0}^{N-1} a_j \cos\left(j \cos^{-1}(x_k)\right)\,, -then if we also use the definition of the nodes we obtain +and using the definition of the nodes we obtain .. math:: - f(x_k) = \sum_{j=0}^{N-1} a_j \cos(\frac{j\pi}{N}(N + k + 1/2))\,. + f(x_k) = \sum_{j=0}^{N-1} a_j \cos\left(\frac{j\pi}{N}(N + k + 1/2)\right)\,. -Finally, we can use the cyclical property of cosine to convert an :math:`j \pi` term in the argument to a :math:`(-1)^{j}` factor in the coefficient +Finally, we can use the cyclical property of cosine to convert a :math:`j \pi` term in the argument to a :math:`(-1)^{j}` factor in the coefficient .. math:: - f(x_k) = \sum_{j=0}^{N-1} a_j (-1)^{j}\cos(\frac{j\pi}{N}(k + 1/2))\,, + f(x_k) = \sum_{j=0}^{N-1} a_j (-1)^{j}\cos\left(\frac{j\pi}{N}(k + 1/2)\right)\,, Which looks just like a discrete cosine transform. -The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing it to be designed efficiently by utilizing the `quantum Fourier transform `__ . Next we will show how Chebyshev polynomials could be represented in a quantum state, before we describe the transform. +The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing the transform to be designed efficiently by utilizing the `quantum Fourier transform `__. Next we will show how Chebyshev polynomials can be embedded in the amplitudes of a quantum state, to construct the quantum Chebyshev basis, allowing us to define the quantum transform. Quantum Chebyshev basis --------------------------------------- -The quantum Chebyshev transform circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. -The :math:`j` -th Chebyshev basis state using :math:`N` qubits is +We can define the :math:`j` -th Chebyshev basis state using :math:`N` qubits is .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, where :math:`|k\rangle` are the computational basis states and :math:`x_j^\mathrm{Ch}` is the :math:`j` -th node of the Chebyshev polynomial of order :math:`2^N-1`. -Notice how the amplitudes of the basis state components are the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node. +Notice how the amplitudes of the basis state components are the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node, and that the normalization of the components is different for the :math:`0` term, like in the classical transform. This construction guarantees the states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization factors used, that is .. math:: @@ -132,7 +131,7 @@ 1 & j = j'\,. \end{cases} -The goal is to design a circuit that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} |\tau(x_j^\mathrm{Ch})\rangle\langle x_j|`. +The quantum Chebyshev transform circuit described in Ref. [#williams2023]_ maps computational basis states :math:`\{|x_j\rangle\}_{j=0}^{2^N-1}` to Chebyshev basis states :math:`\{|\tau(x_j^\mathrm{Ch})\rangle\}_{j=0}^{2^N-1}`. Our goal is to design a circuit in PennyLane that applies the operation :math:`\mathcal{U}_\mathrm{QChT} = \sum_{j=0}^{2^N-1} |\tau(x_j^\mathrm{Ch})\rangle\langle x_j|`. Designing the transform circuit From 14db24a6d67c2704d568747b8e6d2d4b9ed82d0c Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 27 May 2025 16:46:11 -0400 Subject: [PATCH 58/64] More edits. --- .../tutorial_quantum_chebyshev_transform.py | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index e0082e8f8a..7edf8f5dd3 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -98,31 +98,31 @@ .. math:: f(x_k) = \sum_{j=0}^{N-1} a_j \cos\left(j \cos^{-1}(x_k)\right)\,, -and using the definition of the nodes we obtain +then, using the definition of the nodes we obtain .. math:: f(x_k) = \sum_{j=0}^{N-1} a_j \cos\left(\frac{j\pi}{N}(N + k + 1/2)\right)\,. -Finally, we can use the cyclical property of cosine to convert a :math:`j \pi` term in the argument to a :math:`(-1)^{j}` factor in the coefficient +Finally, we can use the cyclical property of cosine to convert a :math:`j \pi` term in the argument to a :math:`(-1)^{j}` factor in the coefficient, resulting in .. math:: f(x_k) = \sum_{j=0}^{N-1} a_j (-1)^{j}\cos\left(\frac{j\pi}{N}(k + 1/2)\right)\,, -Which looks just like a discrete cosine transform. +which looks just like a discrete cosine transform. -The quantum analogy of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing the transform to be designed efficiently by utilizing the `quantum Fourier transform `__. Next we will show how Chebyshev polynomials can be embedded in the amplitudes of a quantum state, to construct the quantum Chebyshev basis, allowing us to define the quantum transform. +The quantum analogue of the discrete Chebyshev transform, the quantum Chebyshev transform, inherits the relation to the Fourier transform, allowing the transform to be designed efficiently by utilizing the `quantum Fourier transform `__. Next we will discuss the quantum Chebyshev basis, where the Chebyshev polynomials appear in the state amplitudes. Quantum Chebyshev basis --------------------------------------- -We can define the :math:`j` -th Chebyshev basis state using :math:`N` qubits is +We can define the :math:`j` -th Chebyshev basis state using :math:`N` qubits as .. math:: |\tau(x_j^\mathrm{Ch})\rangle = \frac1{2^{N/2}}T_0(x_j^\mathrm{Ch})|0\rangle + \frac1{2^{(N-1)/2}}\sum_{k=1}^{2^N-1}T_k(x_j^\mathrm{Ch})|k\rangle\,, where :math:`|k\rangle` are the computational basis states and :math:`x_j^\mathrm{Ch}` is the :math:`j` -th node of the Chebyshev polynomial of order :math:`2^N-1`. -Notice how the amplitudes of the basis state components are the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node, and that the normalization of the components is different for the :math:`0` term, like in the classical transform. -This construction guarantees the states are orthonormal due to the orthogonality of the Chebyshev polynomials and the normalization factors used, that is +Notice how the amplitudes of the basis state components are the Chebyshev polynomials evaluated at the :math:`j` -th Chebyshev node, and that the normalization of the :math:`|0\rangle` component is different from the rest, like in the classical transform. +Due to the orthogonality of the Chebyshev polynomials and the normalization factors used, this construction guarantees the states are orthonormal, that is .. math:: \langle\tau(x_j^\mathrm{Ch})|\tau(x_{j'}^\mathrm{Ch})\rangle = From 17914a18cca2178c11f77cfb3bd4a6e370a2abd2 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 2 Jun 2025 15:17:50 -0400 Subject: [PATCH 59/64] Updated gitignore to ignore my ubuntu venv --- .gitignore | 1 + 1 file changed, 1 insertion(+) diff --git a/.gitignore b/.gitignore index bc97193548..311df1ffb0 100644 --- a/.gitignore +++ b/.gitignore @@ -11,6 +11,7 @@ backreferences .DS_Store demos_community.rst .venv/ +.venv_ubuntu/ .venv-build/ **/*.egg-info/ From f224efb0d4faedf1f061369204aab84fbfebc0a6 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 2 Jun 2025 15:24:52 -0400 Subject: [PATCH 60/64] Typo --- demonstrations/tutorial_quantum_chebyshev_transform.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 7edf8f5dd3..8be65b368a 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -9,8 +9,8 @@ Further work also proposes a protocol for learning and sampling multivariate probability distributions that arise in high-energy physics [#delejarza2025]_. Crucial to their implementation of the learning models is the quantum Chebyshev transform which utilizes the quantum Fourier transform to allow for fast transformations between the standard and the Chebyshev basis. -In this demo we will show how the quantum Chebyshev transform can be implemented in Pennylane. -To start, we'll describe what Chebyshev polynomials are, including the classical discrete Chebyshev transform. After that, we'll look at the quantum Chebyshev basis and its transform. +In this demo we will show how the quantum Chebyshev transform can be implemented in PennyLane. +To start, we'll describe what Chebyshev polynomials are, and what the classical discrete Chebyshev transform is. After that, we'll look at the quantum Chebyshev basis and its transform. .. figure:: ../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_quantum_chebyshev_transform.png :align: center From b9ba43fc0f61a78077ed19f45c919779b8cb9c40 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 3 Jun 2025 14:06:09 -0400 Subject: [PATCH 61/64] Update demonstrations/tutorial_quantum_chebyshev_transform.py Co-authored-by: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com> --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 8be65b368a..7ff6047ee6 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -59,7 +59,7 @@ Figure 2. The first six Chebyshev polynomials, along with their corresponding nodes. The nodes are plotted above along with the corresponding polynomials. -Note that the polynomials are normalized such that $T_n(1)=1$, and they satisfy a discrete orthogonality condition on the nodes of :math:`T_N(x)` in the following way for :math:`k, \ell Date: Tue, 3 Jun 2025 14:06:31 -0400 Subject: [PATCH 62/64] Update demonstrations/tutorial_quantum_chebyshev_transform.py Co-authored-by: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com> --- demonstrations/tutorial_quantum_chebyshev_transform.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index 7ff6047ee6..b76d6448c4 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -69,7 +69,7 @@ N/2 & k = \ell \neq 0\,. \end{cases} -The Chebyshev polynomials have a lot of *nice* properties that make them a good choice to series expansion a function on a real interval. They are often utilized in numerical integration and interpolation methods, where the `discrete Chebyshev transformation `__ is needed to map between the function evaluated on a grid of points and the coefficients of the Chebyshev expansion of that function. Let's take a look at the discrete transform. +The Chebyshev polynomials have a lot of *nice* properties that make them a good choice for expanding a function into a series on a real interval. They are often utilized in numerical integration and interpolation methods, where the `discrete Chebyshev transformation `__ is needed to map between the function evaluated on a grid of points and the coefficients of the Chebyshev expansion of that function. Let's take a look at the discrete transform. Discrete Chebyshev transform --------------------------------------- From 91affb0a15701ed69c1119df2e5a4513f083e5ba Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Tue, 3 Jun 2025 14:48:28 -0400 Subject: [PATCH 63/64] Alvaro changes implemented --- ...torial_quantum_chebyshev_transform.metadata.json | 2 +- .../tutorial_quantum_chebyshev_transform.py | 13 ++++++++----- 2 files changed, 9 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json index cf7675f557..a233880272 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json +++ b/demonstrations/tutorial_quantum_chebyshev_transform.metadata.json @@ -54,7 +54,7 @@ }, { "type": "demonstration", - "id": "tutorial_qubitization", + "id": "tutorial_period_finding", "weight": 1.0 } ] diff --git a/demonstrations/tutorial_quantum_chebyshev_transform.py b/demonstrations/tutorial_quantum_chebyshev_transform.py index b76d6448c4..79cd7de87d 100644 --- a/demonstrations/tutorial_quantum_chebyshev_transform.py +++ b/demonstrations/tutorial_quantum_chebyshev_transform.py @@ -22,15 +22,18 @@ What are Chebyshev polynomials? --------------------------------------- - -`Chebyshev polynomials `__ of the first kind :math:`T_n(x)` are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. *Completeness* here means any function :math:`f(x)` on that interval can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. - The :math:`n` -th order Chebyshev polynomial of the first kind is defined as .. math:: T_n(x) \equiv \cos(n \arccos(x))\,. -Note there are more types of Chebyshev polynomials, but in this demo, we will only discuss those of the first kind. We can write out the first few orders explicitly. +`Chebyshev polynomials `__ of the first kind are a set of orthogonal polynomials that are complete on the interval :math:`[-1,1]`. +*Completeness* here means any function :math:`f(x)` on that interval can be expanded as a series in :math:`T_n(x)` up to order :math:`N` as :math:`f(x) = \sum_{j=0}^N a_j T_j(x)`. + +.. note:: + There are more types of Chebyshev polynomials, but in this demo, we will only discuss those of the first kind. + +We can write out the first few orders explicitly. .. math:: T_0(x) &= 1 \\ @@ -280,7 +283,7 @@ def circuit_to_draw(): # round very small values to zero total_state = np.where(np.abs(total_state)<1e-12, 0, total_state) -print(total_state) +print(np.round(total_state, 3)) ############################################# # Indeed, we see the second half of the amplitude vector is zero. From 4474ba8301dd6d82c4af114aadffc3c59a624204 Mon Sep 17 00:00:00 2001 From: "Colin J. Dale" <79223094+colinjdale@users.noreply.github.com> Date: Mon, 14 Jul 2025 15:51:03 -0400 Subject: [PATCH 64/64] Tarik's thumbnails --- .gitignore | 3 +++ ...nail_large_quantum_chebyshev_transform.png | Bin 50430 -> 109394 bytes ...Gthumbnail_quantum_chebyshev_transform.png | Bin 0 -> 101495 bytes .../thumbnail_quantum_chebyshev_transform.png | Bin 26901 -> 26327 bytes 4 files changed, 3 insertions(+) create mode 100644 _static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_quantum_chebyshev_transform.png diff --git a/.gitignore b/.gitignore index 311df1ffb0..3633fc8e98 100644 --- a/.gitignore +++ b/.gitignore @@ -14,6 +14,9 @@ demos_community.rst .venv_ubuntu/ .venv-build/ **/*.egg-info/ +*.ipynb +*.deb +.test_figures/* # Generated after installation .github/workflows/qml_pipeline_utils/build/ diff --git a/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_quantum_chebyshev_transform.png 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