From a6e4463ae8be23a8568fc4aa5147841384c20460 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 7 May 2025 11:20:44 -0400 Subject: [PATCH 01/37] Added error correction demo --- .../tutorial_stabilizer_codes.metadata.json | 172 +++++++++++ demonstrations/tutorial_stabilizer_codes.py | 275 ++++++++++++++++++ 2 files changed, 447 insertions(+) create mode 100644 demonstrations/tutorial_stabilizer_codes.metadata.json create mode 100644 demonstrations/tutorial_stabilizer_codes.py diff --git a/demonstrations/tutorial_stabilizer_codes.metadata.json b/demonstrations/tutorial_stabilizer_codes.metadata.json new file mode 100644 index 0000000000..09a784afc3 --- /dev/null +++ b/demonstrations/tutorial_stabilizer_codes.metadata.json @@ -0,0 +1,172 @@ +{ + "title": "Stabilizer codes for quantum error correction", + "authors": [ + { + "username": "alvaro" + } + ], + "dateOfPublication": "2025-01-23T09:00:00+00:00", + "dateOfLastModification": "2025-01-23T09:00:00+00:00", + "categories": [ + "Getting Started", + "Quantum Computing", + "Algorithms" + ], + "tags": [], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_tensor_network_basics.png" + }, + { + "type": "large_thumbnail", + "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_tensor_network_basics.png" + } + ], + "seoDescription": "Learn all about the stabilizer formalism for quantum error correction with concrete coding examples.", + "doi": "", + "references": [ + { + "id": "Selinger2010", + "type": "book", + "title": "A Survey of Graphical Languages for Monoidal Categories", + "authors": "P. Selinger", + "year": "2010", + "publisher": "Springer Berlin Heidelberg", + "journal": "New Structures for Physics", + "doi": "10.1007/978-3-642-12821-9_4", + "url": "http://dx.doi.org/10.1007/978-3-642-12821-9_4" + }, + { + "id": "Arad2010", + "type": "preprint", + "title": "Quantum computation and the evaluation of tensor networks", + "authors": "I. Arad, Z. Landau", + "year": "2010", + "publisher": "", + "journal": "", + "doi": "10.48550/arXiv.0805.0040", + "url": "https://arxiv.org/abs/0805.0040" + }, + { + "id": "Bridgeman2017", + "type": "article", + "title": "Hand-waving and interpretive dance: an introductory course on tensor networks", + "authors": "J. C. Bridgeman, C. T. Chubb", + "year": "2017", + "publisher": "", + "journal": "Journal of Physics A: Mathematical and Theoretical", + "doi": "10.1088/1751-8121/aa6dc3", + "url": "http://dx.doi.org/10.1088/1751-8121/aa6dc3" + }, + { + "id": "Lam1997", + "type": "article", + "title": "On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution", + "authors": "C.-C. Lam, P. Sadayappan, R. Wenger", + "year": "1997", + "publisher": "", + "journal": "Parallel Processing Letters", + "doi": "10.1142/S0129626497000176", + "url": "https://doi.org/10.1142/S0129626497000176" + }, + { + "id": "Gray2021", + "type": "article", + "title": "Hyper-optimized tensor network contraction", + "authors": "J. Gray, S. Kourtis, S. Choi", + "year": "2021", + "publisher": "", + "journal": "Quantum", + "doi": "10.22331/q-2021-03-15-410", + "url": "https://quantum-journal.org/papers/q-2021-03-15-410/" + }, + { + "id": "Bienstock1990", + "type": "article", + "title": "On embedding graphs in trees", + "authors": "Dan Bienstock", + "year": "1990", + "journal": "Journal of Combinatorial Theory, Series B", + "volume": "49", + "number": "1", + "pages": "103–136", + "doi": "10.1016/0095-8956(90)90066-9", + "url": "https://www.sciencedirect.com/science/article/pii/0095895690900669" + }, + { + "id": "Pfeifer2014", + "type": "article", + "title": "Faster identification of optimal contraction sequences for tensor networks", + "authors": "R. N. C. Pfeifer, J. Haegeman, F. Verstraete", + "year": "2014", + "publisher": "", + "journal": "Physical Review E", + "doi": "10.1103/PhysRevE.90.033315", + "url": "http://dx.doi.org/10.1103/PhysRevE.90.033315" + }, + { + "id": "Pirvu2010", + "type": "article", + "title": "Matrix product operator representations", + "authors": "B. Pirvu, V. Murg, J. I. Cirac, F. Verstraete", + "year": "2010", + "publisher": "", + "journal": "New Journal of Physics", + "doi": "10.1088/1367-2630/12/2/025012", + "url": "http://dx.doi.org/10.1088/1367-2630/12/2/025012" + }, + { + "id": "Ferris2012", + "type": "article", + "title": "Perfect sampling with unitary tensor networks", + "authors": "A. J. Ferris, G. Vidal", + "year": "2012", + "publisher": "", + "journal": "Physical Review B", + "doi": "10.1103/PhysRevB.85.165146", + "url": "http://dx.doi.org/10.1103/PhysRevB.85.165146" + }, + { + "id": "Schuch2008", + "type": "article", + "title": "Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions", + "authors": "N. Schuch, M. M. Wolf, F. Verstraete, J. I. Cirac", + "year": "2008", + "publisher": "", + "journal": "Physical Review Letters", + "doi": "10.1103/PhysRevLett.100.040501", + "url": "https://doi.org/10.1103/PhysRevLett.100.040501" + }, + { + "id": "Schollwoeck2011", + "type": "article", + "title": "The density-matrix renormalization group in the age of matrix product states", + "authors": "U. Schollwöck", + "year": "2011", + "publisher": "", + "journal": "Annals of Physics", + "doi": "10.1016/j.aop.2010.09.012", + "url": "https://doi.org/10.1016/j.aop.2010.09.012" + } + ], + "basedOnPapers": [], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_tn_circuits", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_mps", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_How_to_simulate_quantum_circuits_with_tensor_networks", + "weight": 1.0 + } + ] +} \ No newline at end of file diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py new file mode 100644 index 0000000000..ec6585d0a2 --- /dev/null +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -0,0 +1,275 @@ +r"""Stabilizer codes for Quantum Error correction +================================================= + + State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than + a thousand qubits. Does this qubit count suffice for valuable quantum algorithms with clear speedups? + The reality is that there is more to the story than the sheer number of qubits. As we currently stand, quantum + devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing-- +has been gaining traction. + +Quantum error correction is done via schemes known as **error correction codes.** +These are quantum algorithms that come in many varieties and address different error types . +Is there a unified way to understand these codes? Indeed, The **stabilizer formalism** provides such a framework for +a large class of quantum error correction codes. The so-called +**stabilizer **, such as the repetition, Shor, Steane, and surface codes, +fall under this formalism. Other codes, like GKP codes used by Xanadu, lie outside of it. + +In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct and +some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which +the code can be reconstructed. This enables the construction of a wide range of error correction codes +directly from their stabilizer generators. + +A toy example: the repetition code +---------------------------------- + +To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** +We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. +The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: + +- Qubit encoding +- Error detection +- Error correction + +Qubit encoding +~~~~~~~~~~~~~~ + +The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** +The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea +of what the original logical qubit was by looking at the rest of the qubits. For example in the three-qubit repetition code, the logical basis +state qubits :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits as + +.. math:: + + \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \range \mapsto \vert 111 \rangle + +A general qubit :math:` \vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle` + \beta \vert \bar{1}\rangle` is then encoded as + +.. math:: + + \alpha \vert \bar{0}\rangle` + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}. + +This encoding can be done via the following quantum circuit. + +**INSERT PICTURE** + +Let's code this below and verify the output +""" + +def encode(alpha, beta): + + #subcircuit + + qml.StatePrep([alpha, beta], wires = 0) + qml.CNOT(wires = [0, 1]) + qml.CNOT(wires = [0, 2]) + +@qml.qnode(qml.device("default.qubit")) +def encoded_state(alpha, beta): + + encode(alpha, beta) + return qml.state() + +alpha = 1 / np.sqrt(2) +beta = 1 / np.sqrt(2) + + +print("|000> component: ", encoded_state(alpha, beta)[0]) +print("|111> component: ", encoded_state(alpha, beta)[7]) + + +########################################################################################## + +# Now, suppose that a **bit-flip** error occurs on the second qubit, meaning that the qubit is randomly flipped. This can be modelled as +# an unwanted Pauli-$X$ operator being applied on :math:`\vert \bar{\psi}\rangle:` +# +# .. math:: +# +# X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \range + \beta \vert 101 +# +# If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and +# :math:`\alpha \vert 111 \rangle` are allowed, be can fix this error by flipping the qubit back. The problem here is that, to know that this happened, +# we have to measure the state. This collapses the wave function, rendering it useless for future calculations. Let us see how to get around this. +# +# .. note:: +# Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, +# but it is not resource efficient: errors compound too quickly on single qubits. If the quantum state is not known, the no-cloning +# theorem states it is impossible to make a copy of the state. +# +# Error detection +# ~~~~~~~~~~~~~~~ +# +# To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **syndrome measurement.** A syndrome measurement +# acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure in the computational +# on two auxiliary qubits after after applying some :math:`\textrm{CNOT}` gates, as shown below +# +# **NEEDS PICTURE** +# +# The results of the measurements will will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, +# they can tell us which qubit was flipped. The following table shows how to interpret the results of the syndrome measurements. +# +# **INSERT TABLE** +# +# Let us verify this by implementing the syndrome measurement in PennyLane +# +def error_detection(error_wire): + + # encode + + qml.StatePrep([alpha, beta], wires = 0) + qml.CNOT(wires = [0, 1]) + qml.CNOT(wires = [0, 2]) + + # error + + qml.PauliX(wires = error_wire) + + # CNOTs and syndrome measurement + + qml.CNOT(wires = [0, 3]) + qml.CNOT(wires = [1, 3]) + qml.CNOT(wires = [1, 4]) + qml.CNOT(wires = [2, 4]) + + return qml.sample(wires = [3,4]) + +dev = qml.device("default.qubit", wires = 5, shots = 1) # A single sample will flag the error +error_detection_qnode = qml.QNode(error_detection, dev) +print("Measurement if error on wire 0: ", error_detection_qnode(0)) +print("Measurement if error on wire 1: ", error_detection_qnode(1)) +print("Measurement if error on wire 2: ", error_detection_qnode(2)) + +################################################################################################ +# +# Error Correction +# ~~~~~~~~~~~~~~~~ +# +# Once a single bit-flip error is detected, correction is straightforward. Since the Pauli-X operator is its own inverse +# (i.e., :math:`X^2 = \mathbb{I}`), applying the :mathbb:`X` operator to the erroneous qubit restores the original state, for example, +# if the syndrome measurement shows the error occurred on the second qubit, we apply +# +# ..math:: +# +# X_2 (X_2 \vert \bar{\psi}\rangle) = \vert \bar{\psi} \rangle. +# +# By applying the appropriate corrective operation, the repetition code effectively protects and repairs the encoded quantum information. +# The full workflow is shown in the circuit below. +# +# We can use PennyLane's mid-circuit measurement features to implement the full three-qubit repetition code. + +def error_correction(error_wire): + + qml.StatePrep([alpha, beta], wires = 0) + qml.CNOT(wires = [0, 1]) + qml.CNOT(wires = [0, 2]) + + qml.PauliX(wires = error_wire) + + qml.CNOT(wires = [0, 3]) + qml.CNOT(wires = [1, 3]) + qml.CNOT(wires = [1, 4]) + qml.CNOT(wires = [2, 4]) + + m3 = qml.measure(3) + m4 = qml.measure(4) + + qml.cond(m3 & ~m4 , qml.PauliX)(wires = 0) + qml.cond(m3 & m4, qml.PauliX)(wires = 1) + qml.cond(~m3 & m4, qml.PauliX)(wires = 2) + + return qml.density_matrix(wires = [0, 1, 2]) +############################################################################# +# +# Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so return the density matrix instead. +# With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction +# as follows + +dev = qml.device("default.qubit", wires = 5) +error_correction_qnode = qml.QNode(error_correction, dev) +encoded_state = qml.math.dm_from_state_vector(encode_qnode(alpha, beta)) + +# Compute fidelity of final corrected state with initial encoded state + +print("Fidelity if error on wire 0: ", qml.math.fidelity(encoded_state, error_correction_qnode(0)).round(2)) +print("Fidelity if error on wire 1: ", qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2)) +print("Fidelity if error on wire 2: ", qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2)) + +################################################################################ +# +# The error is corrected no matter which qubit was flipped! +# +# The operator picture +# -------------------- +# +# We have worked with a simple example, but it is quite limited. The three-qubit code, for example, only works for +# bit flip errors, but more powerful codes need more resources. For example, Shor's code, involving 9 qubits, +# can correct more types of errors on a single logical qubit. Moreover, to avoid errors at an acceptable level, +# the industry standard is 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly +# complex! To deal with these situations, we resort to a different +# representation of error correction codes, known as the operator picture. +# +############################################################################## +# References +# ---------- +# +# .. [#Selinger2010] +# P. Selinger. +# "A Survey of Graphical Languages for Monoidal Categories", +# ``__, in *New Structures for Physics*, Springer Berlin Heidelberg, pp. 289–355, 2010. +# +# .. [#Arad2010] +# I. Arad and Z. Landau. +# "Quantum computation and the evaluation of tensor networks", +# ``__, 2010. +# +# .. [#Bridgeman2017] +# J. C. Bridgeman and C. T. Chubb. +# "Hand-waving and interpretive dance: an introductory course on tensor networks," +# ``__, Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 22, 2017. +# +# .. [#Lam1997] +# C.-C. Lam, P. Sadayappan, and R. Wenger. +# "On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution", +# ``__, Parallel Processing Letters, Vol. 7, No. 2, pp. 157-168, 1997. +# +# .. [#Gray2021] +# J. Gray, S. Kourtis, and S. Choi. +# "Hyper-optimized tensor network contraction", +# ``__, Quantum, Vol. 5, pp. 410, 2021. +# +# .. [#Bienstock1990] +# D. Bienstock. +# "On embedding graphs in trees," +# ``__, Journal of Combinatorial Theory, Series B, vol. 49, no. 1, pp. 103–136, 1990. +# +# .. [#Pfeifer2014] +# R. N. C. Pfeifer, J. Haegeman, and F. Verstraete. +# "Faster identification of optimal contraction sequences for tensor networks", +# ``__, Physical Review +# +# .. [#Pirvu2010] +# B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete. +# "Matrix product operator representations," +# ``__, New Journal of Physics, vol. 12, no. 2, p. 025012, 2010. +# +# .. [#Ferris2012] +# A. J. Ferris and G. Vidal. +# "Perfect sampling with unitary tensor networks," +# ``__, Physical Review B, vol. 85, no. 16, 2012. +# +# .. [#Schuch2008] +# N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac. +# "Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions," +# ``__, Physical Review Letters, vol. 100, no. 4, Jan 2008. +# +# .. [#Schollwoeck2011] +# U. Schollwöck. +# "The density-matrix renormalization group in the age of matrix product states," +# ``__, Annals of Physics, vol. 326, no. 1, pp. 96–192, Jan 2011. + + + +############################################################################## +# About the author +# ---------------- +# +# From 55121b4b782fc4db73b5992d801091c16b7fd44c Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 7 May 2025 11:33:56 -0400 Subject: [PATCH 02/37] added imports --- demonstrations/tutorial_stabilizer_codes.py | 3 +++ 1 file changed, 3 insertions(+) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index ec6585d0a2..b1c92a7a17 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -55,6 +55,9 @@ Let's code this below and verify the output """ +import pennylane as qml +from pennylane import numpy as np + def encode(alpha, beta): #subcircuit From 0cffde17f79c73c1763dbf424008e3c70d7bce83 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 7 May 2025 11:47:12 -0400 Subject: [PATCH 03/37] fixed code --- demonstrations/tutorial_stabilizer_codes.py | 83 ++++++++++----------- 1 file changed, 40 insertions(+), 43 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index b1c92a7a17..0a31ecd92d 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -59,26 +59,26 @@ from pennylane import numpy as np def encode(alpha, beta): - - #subcircuit - qml.StatePrep([alpha, beta], wires = 0) - qml.CNOT(wires = [0, 1]) - qml.CNOT(wires = [0, 2]) + qml.StatePrep([alpha, beta], wires = 0) + qml.CNOT(wires = [0, 1]) + qml.CNOT(wires = [0, 2]) + -@qml.qnode(qml.device("default.qubit")) def encoded_state(alpha, beta): - encode(alpha, beta) - return qml.state() + encode(alpha, beta) + return qml.state() + +encode_qnode = qml.QNode(encoded_state, qml.device("default.qubit")) alpha = 1 / np.sqrt(2) beta = 1 / np.sqrt(2) +encode_qnode = qml.QNode(encoded_state, qml.device("default.qubit")) -print("|000> component: ", encoded_state(alpha, beta)[0]) -print("|111> component: ", encoded_state(alpha, beta)[7]) - +print("|000> component: ", encode_qnode(alpha, beta)[0]) +print("|111> component: ", encode_qnode(alpha, beta)[7]) ########################################################################################## @@ -114,32 +114,29 @@ def encoded_state(alpha, beta): # # Let us verify this by implementing the syndrome measurement in PennyLane # -def error_detection(error_wire): +def error_detection(): - # encode + qml.CNOT(wires = [0, 3]) + qml.CNOT(wires = [1, 3]) + qml.CNOT(wires = [1, 4]) + qml.CNOT(wires = [2, 4]) - qml.StatePrep([alpha, beta], wires = 0) - qml.CNOT(wires = [0, 1]) - qml.CNOT(wires = [0, 2]) - # error +@qml.qnode(qml.device("default.qubit", wires = 5, shots = 1)) # A single sample flags error +def syndrome_measurement(error_wire): - qml.PauliX(wires = error_wire) + encode(alpha, beta) - # CNOTs and syndrome measurement + qml.PauliX(wires = error_wire) # Unwanted Pauli Operator - qml.CNOT(wires = [0, 3]) - qml.CNOT(wires = [1, 3]) - qml.CNOT(wires = [1, 4]) - qml.CNOT(wires = [2, 4]) + error_detection() - return qml.sample(wires = [3,4]) + return qml.sample(wires = [3,4]) -dev = qml.device("default.qubit", wires = 5, shots = 1) # A single sample will flag the error -error_detection_qnode = qml.QNode(error_detection, dev) -print("Measurement if error on wire 0: ", error_detection_qnode(0)) -print("Measurement if error on wire 1: ", error_detection_qnode(1)) -print("Measurement if error on wire 2: ", error_detection_qnode(2)) + +print("Syndrome if error on wire 0: ", syndrome_measurement(0)) +print("Syndrome if error on wire 1: ", syndrome_measurement(1)) +print("Syndrome if error on wire 2: ", syndrome_measurement(2)) ################################################################################################ # @@ -159,27 +156,27 @@ def error_detection(error_wire): # # We can use PennyLane's mid-circuit measurement features to implement the full three-qubit repetition code. +@qml.qnode(qml.device("default.qubit", wires = 5)) def error_correction(error_wire): - qml.StatePrep([alpha, beta], wires = 0) - qml.CNOT(wires = [0, 1]) - qml.CNOT(wires = [0, 2]) + encode(alpha, beta) + + qml.PauliX(wires = error_wire) + + error_detection() - qml.PauliX(wires = error_wire) + # Mid circuit measurements - qml.CNOT(wires = [0, 3]) - qml.CNOT(wires = [1, 3]) - qml.CNOT(wires = [1, 4]) - qml.CNOT(wires = [2, 4]) + m3 = qml.measure(3) + m4 = qml.measure(4) - m3 = qml.measure(3) - m4 = qml.measure(4) + # Operations conditional on measurements - qml.cond(m3 & ~m4 , qml.PauliX)(wires = 0) - qml.cond(m3 & m4, qml.PauliX)(wires = 1) - qml.cond(~m3 & m4, qml.PauliX)(wires = 2) + qml.cond(m3 & ~m4 , qml.PauliX)(wires = 0) + qml.cond(m3 & m4, qml.PauliX)(wires = 1) + qml.cond(~m3 & m4, qml.PauliX)(wires = 2) - return qml.density_matrix(wires = [0, 1, 2]) + return qml.density_matrix(wires = [0, 1, 2]) # qml.state not supported, but density matrices are ############################################################################# # # Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so return the density matrix instead. From ee6b8365ac0a960fc25fc1e8c45587c132c4b0ae Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 7 May 2025 12:30:46 -0400 Subject: [PATCH 04/37] fixed some rendering --- demonstrations/tutorial_stabilizer_codes.py | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 0a31ecd92d..b15e8dc2ec 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -1,17 +1,17 @@ r"""Stabilizer codes for Quantum Error correction ================================================= - State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than - a thousand qubits. Does this qubit count suffice for valuable quantum algorithms with clear speedups? - The reality is that there is more to the story than the sheer number of qubits. As we currently stand, quantum - devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing-- +State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than +a thousand qubits. Does this qubit count suffice for valuable quantum algorithms with clear speedups? +The reality is that there is more to the story than the sheer number of qubits. As we currently stand, quantum +devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing-- has been gaining traction. Quantum error correction is done via schemes known as **error correction codes.** These are quantum algorithms that come in many varieties and address different error types . Is there a unified way to understand these codes? Indeed, The **stabilizer formalism** provides such a framework for a large class of quantum error correction codes. The so-called -**stabilizer **, such as the repetition, Shor, Steane, and surface codes, +**stabilizer codes**, such as the repetition, Shor, Steane, and surface codes, fall under this formalism. Other codes, like GKP codes used by Xanadu, lie outside of it. In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct and @@ -40,13 +40,13 @@ .. math:: - \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \range \mapsto \vert 111 \rangle + \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle -A general qubit :math:` \vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle` + \beta \vert \bar{1}\rangle` is then encoded as +A general qubit :math:`\vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle` is then encoded as .. math:: - \alpha \vert \bar{0}\rangle` + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}. + \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}\rangle. This encoding can be done via the following quantum circuit. @@ -81,7 +81,7 @@ def encoded_state(alpha, beta): print("|111> component: ", encode_qnode(alpha, beta)[7]) ########################################################################################## - +# # Now, suppose that a **bit-flip** error occurs on the second qubit, meaning that the qubit is randomly flipped. This can be modelled as # an unwanted Pauli-$X$ operator being applied on :math:`\vert \bar{\psi}\rangle:` # @@ -147,7 +147,7 @@ def syndrome_measurement(error_wire): # (i.e., :math:`X^2 = \mathbb{I}`), applying the :mathbb:`X` operator to the erroneous qubit restores the original state, for example, # if the syndrome measurement shows the error occurred on the second qubit, we apply # -# ..math:: +# .. math:: # # X_2 (X_2 \vert \bar{\psi}\rangle) = \vert \bar{\psi} \rangle. # From a8baac81a49ec98ac124f40900313b0084933958 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 7 May 2025 13:05:53 -0400 Subject: [PATCH 05/37] more rendering errors --- demonstrations/tutorial_stabilizer_codes.py | 14 ++++++-------- 1 file changed, 6 insertions(+), 8 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index b15e8dc2ec..c8ac1940f2 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -87,7 +87,7 @@ def encoded_state(alpha, beta): # # .. math:: # -# X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \range + \beta \vert 101 +# X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 # # If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and # :math:`\alpha \vert 111 \rangle` are allowed, be can fix this error by flipping the qubit back. The problem here is that, to know that this happened, @@ -144,7 +144,7 @@ def syndrome_measurement(error_wire): # ~~~~~~~~~~~~~~~~ # # Once a single bit-flip error is detected, correction is straightforward. Since the Pauli-X operator is its own inverse -# (i.e., :math:`X^2 = \mathbb{I}`), applying the :mathbb:`X` operator to the erroneous qubit restores the original state, for example, +# (i.e., :math:`X^2 = \mathbb{I}`), applying the :math:`X` operator to the erroneous qubit restores the original state, for example, # if the syndrome measurement shows the error occurred on the second qubit, we apply # # .. math:: @@ -177,7 +177,8 @@ def error_correction(error_wire): qml.cond(~m3 & m4, qml.PauliX)(wires = 2) return qml.density_matrix(wires = [0, 1, 2]) # qml.state not supported, but density matrices are -############################################################################# + +#################################################################################################### # # Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so return the density matrix instead. # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction @@ -193,7 +194,7 @@ def error_correction(error_wire): print("Fidelity if error on wire 1: ", qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2)) print("Fidelity if error on wire 2: ", qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2)) -################################################################################ +############################################################################################################# # # The error is corrected no matter which qubit was flipped! # @@ -207,7 +208,6 @@ def error_correction(error_wire): # complex! To deal with these situations, we resort to a different # representation of error correction codes, known as the operator picture. # -############################################################################## # References # ---------- # @@ -265,9 +265,7 @@ def error_correction(error_wire): # U. Schollwöck. # "The density-matrix renormalization group in the age of matrix product states," # ``__, Annals of Physics, vol. 326, no. 1, pp. 96–192, Jan 2011. - - - +# ############################################################################## # About the author # ---------------- From 8fc39f2608ba8c7f2b5506d1ac769eee84045f53 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 8 May 2025 10:07:26 -0400 Subject: [PATCH 06/37] fixes --- demonstrations/tutorial_stabilizer_codes.py | 59 +-------------------- 1 file changed, 2 insertions(+), 57 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index c8ac1940f2..53a600235a 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -87,7 +87,7 @@ def encoded_state(alpha, beta): # # .. math:: # -# X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 +# X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # # If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and # :math:`\alpha \vert 111 \rangle` are allowed, be can fix this error by flipping the qubit back. The problem here is that, to know that this happened, @@ -113,7 +113,7 @@ def encoded_state(alpha, beta): # **INSERT TABLE** # # Let us verify this by implementing the syndrome measurement in PennyLane -# + def error_detection(): qml.CNOT(wires = [0, 3]) @@ -210,61 +210,6 @@ def error_correction(error_wire): # # References # ---------- -# -# .. [#Selinger2010] -# P. Selinger. -# "A Survey of Graphical Languages for Monoidal Categories", -# ``__, in *New Structures for Physics*, Springer Berlin Heidelberg, pp. 289–355, 2010. -# -# .. [#Arad2010] -# I. Arad and Z. Landau. -# "Quantum computation and the evaluation of tensor networks", -# ``__, 2010. -# -# .. [#Bridgeman2017] -# J. C. Bridgeman and C. T. Chubb. -# "Hand-waving and interpretive dance: an introductory course on tensor networks," -# ``__, Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 22, 2017. -# -# .. [#Lam1997] -# C.-C. Lam, P. Sadayappan, and R. Wenger. -# "On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution", -# ``__, Parallel Processing Letters, Vol. 7, No. 2, pp. 157-168, 1997. -# -# .. [#Gray2021] -# J. Gray, S. Kourtis, and S. Choi. -# "Hyper-optimized tensor network contraction", -# ``__, Quantum, Vol. 5, pp. 410, 2021. -# -# .. [#Bienstock1990] -# D. Bienstock. -# "On embedding graphs in trees," -# ``__, Journal of Combinatorial Theory, Series B, vol. 49, no. 1, pp. 103–136, 1990. -# -# .. [#Pfeifer2014] -# R. N. C. Pfeifer, J. Haegeman, and F. Verstraete. -# "Faster identification of optimal contraction sequences for tensor networks", -# ``__, Physical Review -# -# .. [#Pirvu2010] -# B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete. -# "Matrix product operator representations," -# ``__, New Journal of Physics, vol. 12, no. 2, p. 025012, 2010. -# -# .. [#Ferris2012] -# A. J. Ferris and G. Vidal. -# "Perfect sampling with unitary tensor networks," -# ``__, Physical Review B, vol. 85, no. 16, 2012. -# -# .. [#Schuch2008] -# N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac. -# "Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions," -# ``__, Physical Review Letters, vol. 100, no. 4, Jan 2008. -# -# .. [#Schollwoeck2011] -# U. Schollwöck. -# "The density-matrix renormalization group in the age of matrix product states," -# ``__, Annals of Physics, vol. 326, no. 1, pp. 96–192, Jan 2011. # ############################################################################## # About the author From ca0ed07197d9d8c8196d7a497936020ada3bdfba Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 8 May 2025 10:33:58 -0400 Subject: [PATCH 07/37] header text --- demonstrations/tutorial_stabilizer_codes.py | 73 +++++++++++---------- 1 file changed, 37 insertions(+), 36 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 53a600235a..914bb557f6 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -1,4 +1,5 @@ -r"""Stabilizer codes for Quantum Error correction +r""" +Stabilizer codes for Quantum Error correction ================================================= State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than @@ -18,43 +19,43 @@ some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which the code can be reconstructed. This enables the construction of a wide range of error correction codes directly from their stabilizer generators. - -A toy example: the repetition code ----------------------------------- - -To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** -We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. -The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: - -- Qubit encoding -- Error detection -- Error correction - -Qubit encoding -~~~~~~~~~~~~~~ - -The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** -The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea -of what the original logical qubit was by looking at the rest of the qubits. For example in the three-qubit repetition code, the logical basis -state qubits :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits as - -.. math:: - - \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle - -A general qubit :math:`\vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle` is then encoded as - -.. math:: - - \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}\rangle. - -This encoding can be done via the following quantum circuit. - -**INSERT PICTURE** - -Let's code this below and verify the output """ +# A toy example: the repetition code +# ---------------------------------- +# +# To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** +# We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. +# The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: +# +# - Qubit encoding +# - Error detection +# - Error correction +# +# Qubit encoding +# ~~~~~~~~~~~~~~ +# +# The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** +#The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea +# of what the original logical qubit was by looking at the rest of the qubits. For example in the three-qubit repetition code, the logical basis +# state qubits :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits as +# +# .. math:: +# +# \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle +# +# A general qubit :math:`\vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle` is then encoded as +# +# .. math:: +# +# \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}\rangle. +# +# This encoding can be done via the following quantum circuit. +# +# **INSERT PICTURE** +# +# Let's code this below and verify the output + import pennylane as qml from pennylane import numpy as np From 38d20ca0967a38d59432e9a8124718825396bc88 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 8 May 2025 10:49:47 -0400 Subject: [PATCH 08/37] fixes --- demonstrations/tutorial_stabilizer_codes.py | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 914bb557f6..0f9ca5872c 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -19,8 +19,12 @@ some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which the code can be reconstructed. This enables the construction of a wide range of error correction codes directly from their stabilizer generators. + """ + +############################################################################## +# # A toy example: the repetition code # ---------------------------------- # @@ -81,7 +85,7 @@ def encoded_state(alpha, beta): print("|000> component: ", encode_qnode(alpha, beta)[0]) print("|111> component: ", encode_qnode(alpha, beta)[7]) -########################################################################################## +############################################################################## # # Now, suppose that a **bit-flip** error occurs on the second qubit, meaning that the qubit is randomly flipped. This can be modelled as # an unwanted Pauli-$X$ operator being applied on :math:`\vert \bar{\psi}\rangle:` @@ -139,7 +143,7 @@ def syndrome_measurement(error_wire): print("Syndrome if error on wire 1: ", syndrome_measurement(1)) print("Syndrome if error on wire 2: ", syndrome_measurement(2)) -################################################################################################ +############################################################################## # # Error Correction # ~~~~~~~~~~~~~~~~ @@ -179,7 +183,7 @@ def error_correction(error_wire): return qml.density_matrix(wires = [0, 1, 2]) # qml.state not supported, but density matrices are -#################################################################################################### +############################################################################## # # Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so return the density matrix instead. # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction @@ -195,7 +199,7 @@ def error_correction(error_wire): print("Fidelity if error on wire 1: ", qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2)) print("Fidelity if error on wire 2: ", qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2)) -############################################################################################################# +############################################################################## # # The error is corrected no matter which qubit was flipped! # From f5045afc51470f9d179916583f97c82841dce8b9 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 8 May 2025 12:04:48 -0400 Subject: [PATCH 09/37] sections --- demonstrations/tutorial_stabilizer_codes.py | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 0f9ca5872c..cd197580ba 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -26,7 +26,7 @@ ############################################################################## # # A toy example: the repetition code -# ---------------------------------- +# ----------------------------------- # # To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** # We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. @@ -204,7 +204,7 @@ def error_correction(error_wire): # The error is corrected no matter which qubit was flipped! # # The operator picture -# -------------------- +# --------------------- # # We have worked with a simple example, but it is quite limited. The three-qubit code, for example, only works for # bit flip errors, but more powerful codes need more resources. For example, Shor's code, involving 9 qubits, @@ -216,6 +216,7 @@ def error_correction(error_wire): # References # ---------- # + ############################################################################## # About the author # ---------------- From 2e4dedc10e57d05f14850b96eba78a94730269ee Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 8 May 2025 12:57:27 -0400 Subject: [PATCH 10/37] whatsgoingon --- demonstrations/tutorial_stabilizer_codes.py | 12 +++++------- 1 file changed, 5 insertions(+), 7 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index cd197580ba..48b8bb0d3c 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -37,7 +37,7 @@ # - Error correction # # Qubit encoding -# ~~~~~~~~~~~~~~ +# ~~~~~~~~~~~~~~~ # # The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** #The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea @@ -104,7 +104,7 @@ def encoded_state(alpha, beta): # theorem states it is impossible to make a copy of the state. # # Error detection -# ~~~~~~~~~~~~~~~ +# ~~~~~~~~~~~~~~~~ # # To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **syndrome measurement.** A syndrome measurement # acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure in the computational @@ -146,7 +146,7 @@ def syndrome_measurement(error_wire): ############################################################################## # # Error Correction -# ~~~~~~~~~~~~~~~~ +# ~~~~~~~~~~~~~~~~~ # # Once a single bit-flip error is detected, correction is straightforward. Since the Pauli-X operator is its own inverse # (i.e., :math:`X^2 = \mathbb{I}`), applying the :math:`X` operator to the erroneous qubit restores the original state, for example, @@ -214,11 +214,9 @@ def error_correction(error_wire): # representation of error correction codes, known as the operator picture. # # References -# ---------- +# ----------- # - -############################################################################## # About the author -# ---------------- +# ----------------- # # From caddf2aa74b70c83486ec836941882eba98f2716 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 8 May 2025 13:13:31 -0400 Subject: [PATCH 11/37] try again --- demonstrations/tutorial_stabilizer_codes.py | 71 ++++++++++----------- 1 file changed, 34 insertions(+), 37 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 48b8bb0d3c..ed8e8277f1 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -20,45 +20,42 @@ the code can be reconstructed. This enables the construction of a wide range of error correction codes directly from their stabilizer generators. -""" +A toy example: the repetition code +----------------------------------- +To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** +We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. +The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: -############################################################################## -# -# A toy example: the repetition code -# ----------------------------------- -# -# To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** -# We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. -# The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: -# -# - Qubit encoding -# - Error detection -# - Error correction -# -# Qubit encoding -# ~~~~~~~~~~~~~~~ -# -# The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** -#The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea -# of what the original logical qubit was by looking at the rest of the qubits. For example in the three-qubit repetition code, the logical basis -# state qubits :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits as -# -# .. math:: -# -# \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle -# -# A general qubit :math:`\vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle` is then encoded as -# -# .. math:: -# -# \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}\rangle. -# -# This encoding can be done via the following quantum circuit. -# -# **INSERT PICTURE** -# -# Let's code this below and verify the output +- Qubit encoding +- Error detection +- Error correction + +Qubit encoding +~~~~~~~~~~~~~~~ + +The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** +The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea +of what the original logical qubit was by looking at the rest of the qubits. For example in the three-qubit repetition code, the logical basis +state qubits :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits as + +.. math:: + + \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle + +A general qubit :math:`\vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle` is then encoded as + +.. math:: + + \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}\rangle. + +This encoding can be done via the following quantum circuit. + +**INSERT PICTURE** + +Let's code this below and verify the output + +""" import pennylane as qml from pennylane import numpy as np From 399f2597296126aa071892f55ec7ca27a0951acd Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Fri, 9 May 2025 10:20:56 -0400 Subject: [PATCH 12/37] more on operator picture --- demonstrations/tutorial_stabilizer_codes.py | 58 ++++++++++++++++----- 1 file changed, 45 insertions(+), 13 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index ed8e8277f1..117b8a851b 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -5,15 +5,15 @@ State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than a thousand qubits. Does this qubit count suffice for valuable quantum algorithms with clear speedups? The reality is that there is more to the story than the sheer number of qubits. As we currently stand, quantum -devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing-- -has been gaining traction. +devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing--has +been gaining traction. Quantum error correction is done via schemes known as **error correction codes.** -These are quantum algorithms that come in many varieties and address different error types . +These are quantum algorithms that come in many varieties that address different error types. Is there a unified way to understand these codes? Indeed, The **stabilizer formalism** provides such a framework for a large class of quantum error correction codes. The so-called **stabilizer codes**, such as the repetition, Shor, Steane, and surface codes, -fall under this formalism. Other codes, like GKP codes used by Xanadu, lie outside of it. +all fall under this formalism. Other codes, like the GKP codes used by Xanadu, lie outside of it. In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct and some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which @@ -34,14 +34,14 @@ Qubit encoding ~~~~~~~~~~~~~~~ -The first step in an error correction code is **encoding** one abstract or **logical qubit** is into a set of many on-device **physical qubits.** -The rationale for doing this is that, if some external factor changes the state of one of the qubits, we will still have an idea -of what the original logical qubit was by looking at the rest of the qubits. For example in the three-qubit repetition code, the logical basis -state qubits :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits as +The first step in an error correction code is **encoding** one abstract or **logical qubit** into a set of many on-device **physical qubits.** +The rationale is that, if some external factor changes the state of one of the qubits, we will still have an idea +of what the original logical qubit was by looking at the rest of the qubits. For example, in the three-qubit repetition code, the logical basis-state +qubits, or **logical codewords**, :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits via .. math:: - \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle + \vert \bar{0} \rangle \mapsto \vert 000 \rangle, \quad \vert \bar{1} \rangle \mapsto \vert 111 \rangle. A general qubit :math:`\vert \bar{\psi}\rangle = \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle` is then encoded as @@ -97,7 +97,7 @@ def encoded_state(alpha, beta): # # .. note:: # Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, -# but it is not resource efficient: errors compound too quickly on single qubits. If the quantum state is not known, the no-cloning +# but even state preparation is prone to errors! If the quantum state is not known, the no-cloning # theorem states it is impossible to make a copy of the state. # # Error detection @@ -200,16 +200,48 @@ def error_correction(error_wire): # # The error is corrected no matter which qubit was flipped! # -# The operator picture -# --------------------- +# Operator picture and stabilizers +# --------------------------------- # -# We have worked with a simple example, but it is quite limited. The three-qubit code, for example, only works for +# We have worked with a simple example, but it is quite limited. Indeed, the three-qubit code only works for # bit flip errors, but more powerful codes need more resources. For example, Shor's code, involving 9 qubits, # can correct more types of errors on a single logical qubit. Moreover, to avoid errors at an acceptable level, # the industry standard is 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly # complex! To deal with these situations, we resort to a different # representation of error correction codes, known as the operator picture. # +# To gain some intuition about the operator picture, let us express the three-qubit repetition code in a different way. Using the +# identity +# +# **Hadamard and reversed CNOT identities** +# +# and the fact that :math:`HXH = Z,`, the error correction code can be expressed in the following way: +# +# **Three-qubit circuit in the Stabilizer formalism** +# +# This is the same circuit, but the controls are all now in the auxiliary qubits, while the physical qubits act as target qubits. +# This does not seem desirable--we do not want to change the state of the physical qubits! However, let us observe that the operators +# that act on the logical qubits are :math:`Z_0 Z_1 I_2` and :math:`I_0 Z_1 Z_2,` which leave the logical codewords invariant: +# +# .. math:: +# +# Z_0 Z_1 I_2 \vert 000 \rangle = \vert 000 \rangle, \quad Z_0 Z_1 I_2 \vert 111 \rangle = \vert 111 \rangle. +# +# .. math:: +# +# I_0 Z_1 Z_2 \vert 000 \rangle = \vert 000 \rangle, \quad I_0 Z_1 Z_2 \vert 111 \rangle = \vert 111 \rangle. +# +# This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be +# some operations done on the state, but we'll still be able to fix them via an error correction scheme. Notably, the invariance property +# **only holds true for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigen +# value :math:`-1`, as shown in the table below. +# +# **INSERT TABLE** +# +# +# +# +# # References # ----------- # From ae42540a970e9fb7a6b7c53c2b333d5cffd9b3fc Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Fri, 9 May 2025 12:22:09 -0400 Subject: [PATCH 13/37] stabilizer formalism and table --- demonstrations/tutorial_stabilizer_codes.py | 42 ++++++++++++++++----- 1 file changed, 32 insertions(+), 10 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 117b8a851b..c165fda7eb 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -10,7 +10,7 @@ Quantum error correction is done via schemes known as **error correction codes.** These are quantum algorithms that come in many varieties that address different error types. -Is there a unified way to understand these codes? Indeed, The **stabilizer formalism** provides such a framework for +Is there a unified way to understand these codes? The **stabilizer formalism** provides such a framework for a large class of quantum error correction codes. The so-called **stabilizer codes**, such as the repetition, Shor, Steane, and surface codes, all fall under this formalism. Other codes, like the GKP codes used by Xanadu, lie outside of it. @@ -109,10 +109,18 @@ def encoded_state(alpha, beta): # # **NEEDS PICTURE** # -# The results of the measurements will will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, +# The results of the measurements will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, # they can tell us which qubit was flipped. The following table shows how to interpret the results of the syndrome measurements. # -# **INSERT TABLE** +# +------------+------------+------------+ +# | Error | Syndrome 0 | Syndrome 1 | +# +============+============+============+ +# | :math:`X_0`| :math:`1` | :math:`0` | +# +------------+------------+------------+ +# | :math:`X_1`| :math:`1` | :math:`1` | +# +------------+------------+------------+ +# | :math:`X_2`| :math:`0` | :math:`1` | +# +------------+------------+------------+ # # Let us verify this by implementing the syndrome measurement in PennyLane @@ -200,15 +208,15 @@ def error_correction(error_wire): # # The error is corrected no matter which qubit was flipped! # -# Operator picture and stabilizers -# --------------------------------- +# Operator picture and stabilizer generators +# ------------------------------------------ # # We have worked with a simple example, but it is quite limited. Indeed, the three-qubit code only works for # bit flip errors, but more powerful codes need more resources. For example, Shor's code, involving 9 qubits, # can correct more types of errors on a single logical qubit. Moreover, to avoid errors at an acceptable level, # the industry standard is 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly # complex! To deal with these situations, we resort to a different -# representation of error correction codes, known as the operator picture. +# representation of error correction codes, known as the **operator picture.** # # To gain some intuition about the operator picture, let us express the three-qubit repetition code in a different way. Using the # identity @@ -232,16 +240,30 @@ def error_correction(error_wire): # I_0 Z_1 Z_2 \vert 000 \rangle = \vert 000 \rangle, \quad I_0 Z_1 Z_2 \vert 111 \rangle = \vert 111 \rangle. # # This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be -# some operations done on the state, but we'll still be able to fix them via an error correction scheme. Notably, the invariance property -# **only holds true for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigen -# value :math:`-1`, as shown in the table below. +# some operations applied on the state, but we will be able to fix them via an error correction scheme. Notably, the invariance property +# **only holds true for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue +# :math:`-1`, as shown in the table below. # # **INSERT TABLE** # -# +# This gives us a new option for characterizing error correction codes. What if instead of building codewords and trying to find +# the syndrome measurement operators from them, we went the the other way round? Namely, we could start by specifying a set of operators, find the states +# that remain invariant under their action, and make these our codewords. These operators are known as **stabilizers generators.** # +# The stabilizer formalism +# ------------------------- # +# The stabilizer formalism takes the operator picture representation seriously and uses it to find error correction codes starting from a +# set of **Pauli words**--tensor products of Pauli operators. We start by specifying the stabilizer set. A stabilizer set on :math:`n` +# qubits satisfies the following properties. # +# 1. It contains the identity operator :math:`-I_0\otimes I_1 \times \ldots I_{n-1},` but does not contain the negative identity. +# 2. All elements of the set commute with each other. +# 3. The matrix product of two elements in the set yields an element that is also in the set. +# +# ..note :: +# Codes generated with this approach do have a limitation: they can only correct for Pauli errors. +# # References # ----------- # From 1f19cdf32225c48606215e85c797a5c170c050fc Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Mon, 12 May 2025 11:28:05 -0400 Subject: [PATCH 14/37] Added section on stabilizer generators --- demonstrations/tutorial_stabilizer_codes.py | 71 ++++++++++++++++++--- 1 file changed, 61 insertions(+), 10 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index c165fda7eb..44dea57a88 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -103,14 +103,14 @@ def encoded_state(alpha, beta): # Error detection # ~~~~~~~~~~~~~~~~ # -# To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **syndrome measurement.** A syndrome measurement +# To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **parity measurement.** A parity measurement # acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure in the computational # on two auxiliary qubits after after applying some :math:`\textrm{CNOT}` gates, as shown below # # **NEEDS PICTURE** # -# The results of the measurements will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, -# they can tell us which qubit was flipped. The following table shows how to interpret the results of the syndrome measurements. +# The result of the measurements is known as the **syndrome**. It will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, +# it can tell us which qubit was flipped. The following table shows how to interpret the syndromes. # # +------------+------------+------------+ # | Error | Syndrome 0 | Syndrome 1 | @@ -190,7 +190,7 @@ def error_correction(error_wire): ############################################################################## # -# Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so return the density matrix instead. +# Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so we return the density matrix instead. # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction # as follows @@ -241,28 +241,79 @@ def error_correction(error_wire): # # This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be # some operations applied on the state, but we will be able to fix them via an error correction scheme. Notably, the invariance property -# **only holds true for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue +# **holds true only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue # :math:`-1`, as shown in the table below. # # **INSERT TABLE** # # This gives us a new option for characterizing error correction codes. What if instead of building codewords and trying to find -# the syndrome measurement operators from them, we went the the other way round? Namely, we could start by specifying a set of operators, find the states -# that remain invariant under their action, and make these our codewords. These operators are known as **stabilizers generators.** +# the syndrome measurement operators from them, we went the the other way round? Namely, we could start by specifying a set of operators, find the states +# that remain invariant under their action, and make these our codewords. These operators are known as **stabilizer generators.** # # The stabilizer formalism # ------------------------- # +# Stabilizer generators +# ~~~~~~~~~~~~~~~~~~~~~~ +# # The stabilizer formalism takes the operator picture representation seriously and uses it to find error correction codes starting from a # set of **Pauli words**--tensor products of Pauli operators. We start by specifying the stabilizer set. A stabilizer set on :math:`n` # qubits satisfies the following properties. # -# 1. It contains the identity operator :math:`-I_0\otimes I_1 \times \ldots I_{n-1},` but does not contain the negative identity. +# 1. It contains the identity operator :math:`I_0\otimes I_1 \times \ldots I_{n-1},` but does not contain the negative identity. # 2. All elements of the set commute with each other. # 3. The matrix product of two elements in the set yields an element that is also in the set. # -# ..note :: -# Codes generated with this approach do have a limitation: they can only correct for Pauli errors. +# The set can be more succinctly specified set of generators: a minimal set of operators in the stabilizer set that can produce all +# the other elements through pairwise multiplication. As a simple example, consider the stabilizer set +# +# .. math:: +# +# S = \left\lbrace I_0 \otimes I_1 \otimes I_2, \ Z_0 \otimes Z_1 \otimes I_2, \ Z_0 \otimes I_1 \otimes Z_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rbrace. +# +# We can check that it satisfies the defining properties 1. to 3. The most cumbersome to check is property 3, where we have to take all +# possible products of the elements and check whether the result is in :math:`S.` For example +# +# .. math:: +# +# (Z_0 \otimes Z_1 \otimes I_2)\cdot (I_0 \otimes Z_1 \otimes Z_2) = Z_0 \otimes I_1 \otimes Z_2, \\ +# (Z_0 \otimes Z_1 \otimes I_2 )^2 = I_0 \otimes I_1 \otimes I_2, +# +# and so on. Note that we can obtain all the elements in :math:`S` just from :math:`Z_0 \otimes Z_1 \otimes I_2` and :math:`I_0 \otimes Z_1 \otimes Z_2.` Because +# of this property these elements are **stabilizer generators** for :math:`S`. We write this fact as +# +# .. math:: +# +# S = \left\langle Z_0 \otimes Z_1 \otimes I_2 \otimes I_0 \otimes Z_1 \otimes Z_2 \right\rangle. +# +# It turns out that specifying these generators is enough to completely define an error correcting code. +# +# Now that we know how stabilizer generators work, let us create a tool that creates the full stabilizer set from its generators. +# +def generate_stabilizer_group(gens, num_wires): + group = [] + init_op =I(0) + for i in range(1,num_wires): + init_op = init_op @ I(i) + for bits in itertools.product([0, 1], repeat=len(gens)): + op = init_op + for i, bit in enumerate(bits): + if bit: + op = qml.prod(op, gens[i]).simplify() + group.append(op) + return set(group) + +generators = [Z(0)@Z(1)@I(2), I(0)@Z(1)@Z(2)] +generate_stabilizer_group(generators, 3) + +############################################################################## +# +# Stabilizer codes +# ~~~~~~~~~~~~~~~~~ +# +# +# +# # # References # ----------- From 7acf0e8b9f92657c58469cb2fb53c1b33cb887ef Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Mon, 12 May 2025 11:44:45 -0400 Subject: [PATCH 15/37] imports --- demonstrations/tutorial_stabilizer_codes.py | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 44dea57a88..e4e3a4f054 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -290,6 +290,10 @@ def error_correction(error_wire): # # Now that we know how stabilizer generators work, let us create a tool that creates the full stabilizer set from its generators. # +import itertools +from pennylane import X, Y, Z +from pennylane import Identity as I + def generate_stabilizer_group(gens, num_wires): group = [] init_op =I(0) From 8e050e1eb46b361f5e278e8675cc23225e224ddf Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Tue, 13 May 2025 11:28:54 -0400 Subject: [PATCH 16/37] Added logical operators --- demonstrations/tutorial_stabilizer_codes.py | 53 ++++++++++++++++++--- 1 file changed, 47 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index e4e3a4f054..736e72dbba 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -31,6 +31,8 @@ - Error detection - Error correction +Let us describe each of these steps in detail. + Qubit encoding ~~~~~~~~~~~~~~~ @@ -92,8 +94,9 @@ def encoded_state(alpha, beta): # X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # # If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and -# :math:`\alpha \vert 111 \rangle` are allowed, be can fix this error by flipping the qubit back. The problem here is that, to know that this happened, -# we have to measure the state. This collapses the wave function, rendering it useless for future calculations. Let us see how to get around this. +# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occured on the second qubit and we can fix this error by flipping it back. The problem here is that, +# in a quantum circuit, we do not have access to the state. To know that a flip did occur, +# we have to measure the state. But this collapses the wave function, rendering it useless for future calculations. Let us see how to get around this. # # .. note:: # Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, @@ -280,11 +283,11 @@ def error_correction(error_wire): # (Z_0 \otimes Z_1 \otimes I_2 )^2 = I_0 \otimes I_1 \otimes I_2, # # and so on. Note that we can obtain all the elements in :math:`S` just from :math:`Z_0 \otimes Z_1 \otimes I_2` and :math:`I_0 \otimes Z_1 \otimes Z_2.` Because -# of this property these elements are **stabilizer generators** for :math:`S`. We write this fact as +# of this property, these elements are **stabilizer generators** for :math:`S`. We write this fact as # # .. math:: # -# S = \left\langle Z_0 \otimes Z_1 \otimes I_2 \otimes I_0 \otimes Z_1 \otimes Z_2 \right\rangle. +# S = \left\langle Z_0 \otimes Z_1 \otimes I_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rangle. # # It turns out that specifying these generators is enough to completely define an error correcting code. # @@ -312,10 +315,48 @@ def generate_stabilizer_group(gens, num_wires): ############################################################################## # -# Stabilizer codes -# ~~~~~~~~~~~~~~~~~ +# Defining the codespace +# ~~~~~~~~~~~~~~~~~~~~~~~ +# +# At this point, the pressing question is given a set of stabilizer operators, how do we recover the error correction circuit? That is, +# we need to find a way to go from the operator picture to the state picture. Let us recall that the property that inspired using stabilizer generators +# is that they must leave the codewords invariant. That is, for any stabilizer element :math:`S` and codeword :math:`\vert \psi \rangle`, we myst +# have +# +# .. math:: +# +# S\vert \psi rangle = \vert \psi \rangle. +# +# The **codespace** is defined as the set made up of all states such that :math`S_i\vert\psi\rangle = \vert \psi\rangle` for all stabilizer +# generators :math:`S_i.` The **codewords** can then be recovered by choosing an orthogonal basis of the codespace. +# +# With this in mind, the error correcting code can be recovered for the set :math:`\left\lbrace S_i \right\rbrace`, as shown below +# +# **FIGURE: GENERAL ERROR CORRECTION CIRCUIT FROM STABILIZERS** +# +# The stabilizer generators act as the controlled operators in the codewords and yield unique syndromes for each Pauli Error. +# +# Logical operators +# ~~~~~~~~~~~~~~~~~~ +# +# Thus far, we have defined the stabilizer generators, which correspond to the operators that implement syndrome measurements, and the +# codewords, which are states invariant under the stabilizers and hence usable as logical qubits. One missing ingredient are gates we can implement +# on logical qubits without leaving the codespace. Such operators would act non-trivially on the codewords, so they cannot be in the +# stabilizer set, but have to preserve the codespace, meaning they **must commute** with all the stabilizer generators. In particular +# we are interested in the logical Pauli $\bar{X}$ and $\bar{Z}$ operators, defined by +# +# .. math:: +# +# \bar{X}\vert \bar{0} \rangle = \vert {1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert {0} \rangle +# \bar{Z}\vert \bar{0} \rangle = \vert {0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert {1} \rangle # +# For example, in the three qubit bit flip error correcting code, the logical operators are :math:`\bar{X} = X_0 X_1 X_2` and +# :math:`\bar{Z} = Z_0 Z_1 Z_2,` but they will not always be this simple. In general, given a stabilizer set $S$, the logical +# operators for the code satisfy the following properties: # +# 1. They commute with all elements in :math:`S,` so they leave the codespace invariant, +# 2. They are not in the stabilizer group, +# 3. They anticommute with each other, which means they act in a non-trivial on the codewords. # # # From 846164f44a613439b7d5cff5d5638869a0357033 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Tue, 13 May 2025 15:21:58 -0400 Subject: [PATCH 17/37] 5 qubit code --- demonstrations/tutorial_stabilizer_codes.py | 64 +++++++++++++++++---- 1 file changed, 52 insertions(+), 12 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 736e72dbba..c2b2fa8b18 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -23,7 +23,7 @@ A toy example: the repetition code ----------------------------------- -To start with, we will explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** +To start with, let us explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: @@ -31,14 +31,14 @@ - Error detection - Error correction -Let us describe each of these steps in detail. +Let us describe each element in detail. Qubit encoding ~~~~~~~~~~~~~~~ The first step in an error correction code is **encoding** one abstract or **logical qubit** into a set of many on-device **physical qubits.** -The rationale is that, if some external factor changes the state of one of the qubits, we will still have an idea -of what the original logical qubit was by looking at the rest of the qubits. For example, in the three-qubit repetition code, the logical basis-state +The rationale is that, if some external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. +For example, in the three-qubit repetition code, the logical basis-state qubits, or **logical codewords**, :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits via .. math:: @@ -94,9 +94,9 @@ def encoded_state(alpha, beta): # X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # # If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and -# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occured on the second qubit and we can fix this error by flipping it back. The problem here is that, +# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occured on the second qubit and we can fix this error by flipping it back. But there is a problem with this reasoning: # in a quantum circuit, we do not have access to the state. To know that a flip did occur, -# we have to measure the state. But this collapses the wave function, rendering it useless for future calculations. Let us see how to get around this. +# we have to measure the state. But this collapses the state of the qubit, rendering it useless for future calculations. Let us see how to get around this. # # .. note:: # Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, @@ -260,7 +260,7 @@ def error_correction(error_wire): # ~~~~~~~~~~~~~~~~~~~~~~ # # The stabilizer formalism takes the operator picture representation seriously and uses it to find error correction codes starting from a -# set of **Pauli words**--tensor products of Pauli operators. We start by specifying the stabilizer set. A stabilizer set on :math:`n` +# **stabilizer set** of **Pauli words**--tensor products of Pauli operators. A stabilizer set on :math:`n` # qubits satisfies the following properties. # # 1. It contains the identity operator :math:`I_0\otimes I_1 \times \ldots I_{n-1},` but does not contain the negative identity. @@ -287,11 +287,12 @@ def error_correction(error_wire): # # .. math:: # -# S = \left\langle Z_0 \otimes Z_1 \otimes I_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rangle. +# S = \left\langle Z_0 \otimes Z_1 \otimes I_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rangle, # +# which reads :math:`S` *is the stabilizer set generated by the elements* :math:`Z_0 \otimes Z_1 \otimes I_2` *and* :math:`I_0 \otimes Z_1 \otimes Z_2.` # It turns out that specifying these generators is enough to completely define an error correcting code. # -# Now that we know how stabilizer generators work, let us create a tool that creates the full stabilizer set from its generators. +# Now that we know how stabilizer generators work, let us create a tool for later use that creates the full stabilizer set from its generators. # import itertools from pennylane import X, Y, Z @@ -315,12 +316,14 @@ def generate_stabilizer_group(gens, num_wires): ############################################################################## # +# Indeed, obtain all the elements of the set by inputting the generators only. Feel free to try out this code with different generators! +# # Defining the codespace # ~~~~~~~~~~~~~~~~~~~~~~~ # -# At this point, the pressing question is given a set of stabilizer operators, how do we recover the error correction circuit? That is, +# At this point, the pressing question is, given a set of stabilizer operators, how do we recover the error correction circuit? That is, # we need to find a way to go from the operator picture to the state picture. Let us recall that the property that inspired using stabilizer generators -# is that they must leave the codewords invariant. That is, for any stabilizer element :math:`S` and codeword :math:`\vert \psi \rangle`, we myst +# is that they must leave the codewords invariant. For any stabilizer element :math:`S` and codeword :math:`\vert \psi \rangle`, we must # have # # .. math:: @@ -347,7 +350,7 @@ def generate_stabilizer_group(gens, num_wires): # # .. math:: # -# \bar{X}\vert \bar{0} \rangle = \vert {1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert {0} \rangle +# \bar{X}\vert \bar{0} \rangle = \vert {1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert {0} \rangle \\ # \bar{Z}\vert \bar{0} \rangle = \vert {0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert {1} \rangle # # For example, in the three qubit bit flip error correcting code, the logical operators are :math:`\bar{X} = X_0 X_1 X_2` and @@ -358,7 +361,44 @@ def generate_stabilizer_group(gens, num_wires): # 2. They are not in the stabilizer group, # 3. They anticommute with each other, which means they act in a non-trivial on the codewords. # +# The codespace, the logical operators, and the syndrome measurements are all we need to define an error correcting code in the state +# picture. We are now ready to look at a more advanced example, the famous... +# +# Five-qubit stabilizer code +# --------------------------- +# +# Unlike other error correcting codes, the 5-qubit code does not have a special name, but it holds a special place as the smallest +# error correcting protocol capable of correcting arbitrary Pauli Errors--unwanted applications of :math:`X`, :math:`Y`, or :math:`Z` gates. +# In this section, we will build it starting from its stabilizer generators: +# +# .. math:: +# +# S = \langle S_0, \ S_1, \ S_2, \ S_3 \rangle, +# +# with +# +# .. math:: +# +# S_0 = X_0 Z_1 Z_2 X_3 I_4,\\ +# S_1 = I_0 X_1 Z_2 Z_3 X_4,\\ +# S_2 = X_0 I_1 X_2 Z_3 Z_4,\\ +# S_3 = Z_0 X_1 I_2 Z_3 X_4. +# +# The calculations are a bit cumbersome, but with some patience we can find the common :math:`+1`-eigenspace of the stabilizer generators, +# spanned by the codewords +# +# .. math:: +# +# \begin{align*} +# \vert \bar{0}\rangle = &\frac{1}{4}\left(\vert 00000 \rangle \vert 10010 \rangle + \vert 01001 \rangle + \vert 10100 \rangle + \vert 01010 \rangle - \vert 11011 \rangle - \vert 00110 \rangle \right.\\ +# &\left. - \vert 11101 \rangle - \vert 00011\rangle - \vert 11110 \rangle - \vert 01111 \rangle - \vert 10001 \rangle - \vert 01100 \rangle - \vert 10111 \rangle + \vert 00101 \rangle \right) +# \end{align*} +# +# .. math:: # +# \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0}. +# +# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`Z^{\otimes 5}.` # # References # ----------- From 9a4baed65b12d9d749ebb30fed62df147f32d86d Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Tue, 13 May 2025 15:25:39 -0400 Subject: [PATCH 18/37] fix latex typos --- demonstrations/tutorial_stabilizer_codes.py | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index c2b2fa8b18..06e84379e9 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -49,7 +49,7 @@ .. math:: - \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert \bar{111}\rangle. + \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert 111\rangle. This encoding can be done via the following quantum circuit. @@ -328,9 +328,9 @@ def generate_stabilizer_group(gens, num_wires): # # .. math:: # -# S\vert \psi rangle = \vert \psi \rangle. +# S\vert \psi \rangle = \vert \psi \rangle. # -# The **codespace** is defined as the set made up of all states such that :math`S_i\vert\psi\rangle = \vert \psi\rangle` for all stabilizer +# The **codespace** is defined as the set made up of all states such that :math:`S_i \vert\psi\rangle = \vert \psi\rangle` for all stabilizer # generators :math:`S_i.` The **codewords** can then be recovered by choosing an orthogonal basis of the codespace. # # With this in mind, the error correcting code can be recovered for the set :math:`\left\lbrace S_i \right\rbrace`, as shown below @@ -346,7 +346,7 @@ def generate_stabilizer_group(gens, num_wires): # codewords, which are states invariant under the stabilizers and hence usable as logical qubits. One missing ingredient are gates we can implement # on logical qubits without leaving the codespace. Such operators would act non-trivially on the codewords, so they cannot be in the # stabilizer set, but have to preserve the codespace, meaning they **must commute** with all the stabilizer generators. In particular -# we are interested in the logical Pauli $\bar{X}$ and $\bar{Z}$ operators, defined by +# we are interested in the logical Pauli :math:`\bar{X}` and :math:`\bar{Z}` operators, defined by # # .. math:: # From da0ef98af0ae8f6cbcb387c3f30830fd9081187b Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Fri, 16 May 2025 14:20:46 -0400 Subject: [PATCH 19/37] Added code and figure --- .../stabilizer_codes/syndrome_table3.png | Bin 0 -> 62293 bytes demonstrations/tutorial_stabilizer_codes.py | 141 ++++++++++++++++-- 2 files changed, 125 insertions(+), 16 deletions(-) create mode 100644 _static/demonstration_assets/stabilizer_codes/syndrome_table3.png diff --git a/_static/demonstration_assets/stabilizer_codes/syndrome_table3.png b/_static/demonstration_assets/stabilizer_codes/syndrome_table3.png new file mode 100644 index 0000000000000000000000000000000000000000..69ea5bf4a7d9d7a1063a13ce8ac1d5a63578b4ce GIT binary patch literal 62293 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z_kw=K>XG&v6ApqMqqvFi-Y@_H86%>_{qncK2SE*j8V7geT2A#=CpqxQwR{URaEn1c z{Ukk^A|(_|84l|}a}d-ZsDTt6bbMg!Xao82$d3o~MC6Ld6-BPNt7$+NP~>V%IEY-) zAsKiVC~}<=lTD6D5V;xv)G;_gk?UkVnIa_=xuVDwMXpAJGj}SAT#X6`eStz>prCdR z<=7bW%n`XFaz*57ls^!;8s!gkNM`ID2(T6%k{J^YIwUhj#mJ90O2sIX%;=FCf|@CZ oMUzzsK@EbM-~4$zOZ(2LJAB(k_3@>U%5Je{b=b!Re~LN&Z<*q0SpWb4 literal 0 HcmV?d00001 diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 06e84379e9..85ee532509 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -115,15 +115,11 @@ def encoded_state(alpha, beta): # The result of the measurements is known as the **syndrome**. It will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, # it can tell us which qubit was flipped. The following table shows how to interpret the syndromes. # -# +------------+------------+------------+ -# | Error | Syndrome 0 | Syndrome 1 | -# +============+============+============+ -# | :math:`X_0`| :math:`1` | :math:`0` | -# +------------+------------+------------+ -# | :math:`X_1`| :math:`1` | :math:`1` | -# +------------+------------+------------+ -# | :math:`X_2`| :math:`0` | :math:`1` | -# +------------+------------+------------+ +# .. figure:: ../_static/demonstration_assets/stabilizer_codes/syndrome_table3.png +# :align: center +# :width: 25% +# +# .. # # Let us verify this by implementing the syndrome measurement in PennyLane @@ -153,6 +149,8 @@ def syndrome_measurement(error_wire): ############################################################################## # +# The measurement outputs confirm the syndrome table. +# # Error Correction # ~~~~~~~~~~~~~~~~~ # @@ -167,6 +165,8 @@ def syndrome_measurement(error_wire): # By applying the appropriate corrective operation, the repetition code effectively protects and repairs the encoded quantum information. # The full workflow is shown in the circuit below. # +# **NEED FIGURE WITH FULL CIRCUIT** +# # We can use PennyLane's mid-circuit measurement features to implement the full three-qubit repetition code. @qml.qnode(qml.device("default.qubit", wires = 5)) @@ -193,7 +193,7 @@ def error_correction(error_wire): ############################################################################## # -# Unfortunately, circuits with mid-circuit measurements cannot return a quantum state, so we return the density matrix instead. +# At the time of writing, PennyLane circuits with mid-circuit measurements cannot return a quantum state, so we return the density matrix instead. # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction # as follows @@ -337,7 +337,8 @@ def generate_stabilizer_group(gens, num_wires): # # **FIGURE: GENERAL ERROR CORRECTION CIRCUIT FROM STABILIZERS** # -# The stabilizer generators act as the controlled operators in the codewords and yield unique syndromes for each Pauli Error. +# The stabilizer generators act as the controlled operators in the codewords and the measurement in the auxiliary wires yield unique syndromes for the +# Pauli error that the code deals with. # # Logical operators # ~~~~~~~~~~~~~~~~~~ @@ -359,10 +360,43 @@ def generate_stabilizer_group(gens, num_wires): # # 1. They commute with all elements in :math:`S,` so they leave the codespace invariant, # 2. They are not in the stabilizer group, -# 3. They anticommute with each other, which means they act in a non-trivial on the codewords. +# 3. They anticommute with each other, which means they act in a non-trivial way on the codewords. # -# The codespace, the logical operators, and the syndrome measurements are all we need to define an error correcting code in the state -# picture. We are now ready to look at a more advanced example, the famous... +# The codespace, the logical operators, and the syndrome measurements are all we need to define an error correcting code. We can even write +# a script to classify operators given a set of stabilizer generators. +# +def classify_pauli(operator, logical_ops, generators, n_wires): + + allowed_wires = set(range(n_wires)) + operator_wires = set(operator.wires) + + assert operator_wires.issubset(allowed_wires), "Operator has wires not allowed by the code" + + operator_names = set([op.name for op in operator.decomposition()]) + allowed_operators = set(['Identity', 'PauliX', 'PauliY', 'PauliZ', 'SProd']) + + assert operator_names.issubset(allowed_operators), "Operator contains an illegal operation" + + stabilizer_group = generate_stabilizer_group(generators, n_wires) + + if operator in stabilizer_group: + return f"{operator} is a Stabilizer." + + if all(qml.is_commuting(operator, g) for g in generators): + if operator in logical_operators: + return f"{operator} is a Logical Operator." + else: + return f"{operator} commutes with all stabilizers — it's a Logical Operator (or a multiple of one)." + + return f"{operator} is an Error Operator (Destabilizer)." + +generators = [Z(0)@Z(1)@I(2), I(0)@Z(1)@Z(2)] +logical_ops = [X(0)@X(1)@X(2), Z(0)@Z(1)@Z(2)] +print(classify_pauli(Z(0)@I(1)@Z(2), logical_ops, generators, 3)) +print(classify_pauli(Y(0)@Y(1)@Y(2), logical_ops, generators, 3)) +print(classify_pauli(X(0)@Y(1)@Z(2), logical_ops, generators, 3)) + +############################################################################## # # Five-qubit stabilizer code # --------------------------- @@ -398,8 +432,83 @@ def generate_stabilizer_group(gens, num_wires): # # \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0}. # -# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`Z^{\otimes 5}.` -# +# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`Z^{\otimes 5}.` With these +# defining features in mind, let us build the five-qubit stabilizer code. First, we need to prepare our logical qubit $\vert \bar{0} \rangle,$ +# which can be done using the circuit below. +# +# **INSERT CIRCUIT IMAGE** +# +# This is straightforward to implement in PennyLane. +# + +def five_qubit_encode(alpha, beta): + + qml.StatePrep([alpha, beta], wires = 4) + qml.Hadamard(wires = 0) + qml.S(wires = 0) + qml.CZ(wires = [0,1]) + qml.CZ(wires = [0,3]) + qml.CY(wires = [0,4]) + qml.Hadamard(wires = 1) + qml.CZ(wires = [1,2]) + qml.CZ(wires = [1,3]) + qml.CNOT(wires = [1,4]) + qml.Hadamard(wires = 2) + qml.CZ(wires = [2,0]) + qml.CZ(wires = [2,1]) + qml.CNOT(wires = [2,4]) + qml.Hadamard(wires = 3) + qml.S(wires = 3) + qml.CZ(wires = [3,0]) + qml.CZ(wires = [3,2]) + qml.CY(wires = [3,4]) + +############################################################################## +# +# Having encoded the logical state, we can use the stabilizers to measure obtain the syndrome table like we did with the three-qubit code. + +dev = qml.device("default.qubit", wires = 9, shots = 1) + +stabilizers = [X(0)@Z(1)@Z(2)@X(3)@I(4), I(0)@X(1)@Z(2)@Z(3)@X(4), + X(0)@I(1)@X(2)@Z(3)@Z(4), Z(0)@X(1)@I(2)@X(3)@Z(4)] + +@qml.qnode(dev) +def five_qubit_code(alpha, beta, error_type, error_wire): + + five_qubit_encode(alpha, beta) + + if error_type == 'X': + qml.PauliX(wires = error_wire) + + elif error_type == 'Y': + qml.PauliY(wires = error_wire) + + elif error_type == 'Z': + qml.PauliZ(wires = error_wire) + + for wire in range(5,9): + qml.Hadamard(wires = wire) + + for i in range(len(stabilizers)): + + qml.ctrl(stabilizers[i], control = [i + 5]) + + for wire in range(5,9): + qml.Hadamard(wires = wire) + + return qml.sample(wires = range(5,9)) + +for wire in (0, 1, 2, 3, 4): + + for error in ('X', 'Y', 'Z'): + + print(f"{error} {wire}", five_qubit_code(1/2, np.sqrt(3)/2, error, wire)) +############################################################################## +# +# The syndrome table is printed, and with we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward +# to generalize to any valid set of stabilizers. +# +# # References # ----------- # From 099afb449b9294c8e7e0d6d8a6f8e1dc8a111084 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Fri, 16 May 2025 14:28:16 -0400 Subject: [PATCH 20/37] indentation --- demonstrations/tutorial_stabilizer_codes.py | 70 ++++++++++----------- 1 file changed, 35 insertions(+), 35 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 85ee532509..5e7cbf9a74 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -443,25 +443,25 @@ def classify_pauli(operator, logical_ops, generators, n_wires): def five_qubit_encode(alpha, beta): - qml.StatePrep([alpha, beta], wires = 4) - qml.Hadamard(wires = 0) - qml.S(wires = 0) - qml.CZ(wires = [0,1]) - qml.CZ(wires = [0,3]) - qml.CY(wires = [0,4]) - qml.Hadamard(wires = 1) - qml.CZ(wires = [1,2]) - qml.CZ(wires = [1,3]) - qml.CNOT(wires = [1,4]) - qml.Hadamard(wires = 2) - qml.CZ(wires = [2,0]) - qml.CZ(wires = [2,1]) - qml.CNOT(wires = [2,4]) - qml.Hadamard(wires = 3) - qml.S(wires = 3) - qml.CZ(wires = [3,0]) - qml.CZ(wires = [3,2]) - qml.CY(wires = [3,4]) + qml.StatePrep([alpha, beta], wires = 4) + qml.Hadamard(wires = 0) + qml.S(wires = 0) + qml.CZ(wires = [0,1]) + qml.CZ(wires = [0,3]) + qml.CY(wires = [0,4]) + qml.Hadamard(wires = 1) + qml.CZ(wires = [1,2]) + qml.CZ(wires = [1,3]) + qml.CNOT(wires = [1,4]) + qml.Hadamard(wires = 2) + qml.CZ(wires = [2,0]) + qml.CZ(wires = [2,1]) + qml.CNOT(wires = [2,4]) + qml.Hadamard(wires = 3) + qml.S(wires = 3) + qml.CZ(wires = [3,0]) + qml.CZ(wires = [3,2]) + qml.CY(wires = [3,4]) ############################################################################## # @@ -475,34 +475,34 @@ def five_qubit_encode(alpha, beta): @qml.qnode(dev) def five_qubit_code(alpha, beta, error_type, error_wire): - five_qubit_encode(alpha, beta) + five_qubit_encode(alpha, beta) - if error_type == 'X': - qml.PauliX(wires = error_wire) + if error_type == 'X': + qml.PauliX(wires = error_wire) - elif error_type == 'Y': - qml.PauliY(wires = error_wire) + elif error_type == 'Y': + qml.PauliY(wires = error_wire) - elif error_type == 'Z': - qml.PauliZ(wires = error_wire) + elif error_type == 'Z': + qml.PauliZ(wires = error_wire) - for wire in range(5,9): - qml.Hadamard(wires = wire) + for wire in range(5,9): + qml.Hadamard(wires = wire) - for i in range(len(stabilizers)): + for i in range(len(stabilizers)): - qml.ctrl(stabilizers[i], control = [i + 5]) + qml.ctrl(stabilizers[i], control = [i + 5]) - for wire in range(5,9): - qml.Hadamard(wires = wire) + for wire in range(5,9): + qml.Hadamard(wires = wire) - return qml.sample(wires = range(5,9)) + return qml.sample(wires = range(5,9)) for wire in (0, 1, 2, 3, 4): - for error in ('X', 'Y', 'Z'): + for error in ('X', 'Y', 'Z'): - print(f"{error} {wire}", five_qubit_code(1/2, np.sqrt(3)/2, error, wire)) + print(f"{error} {wire}", five_qubit_code(1/2, np.sqrt(3)/2, error, wire)) ############################################################################## # # The syndrome table is printed, and with we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward From 833e7e204ecf970e20029be9b6326e3d08390660 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Fri, 16 May 2025 14:40:17 -0400 Subject: [PATCH 21/37] fix code --- demonstrations/tutorial_stabilizer_codes.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 5e7cbf9a74..f5dc77ca7c 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -379,11 +379,11 @@ def classify_pauli(operator, logical_ops, generators, n_wires): stabilizer_group = generate_stabilizer_group(generators, n_wires) - if operator in stabilizer_group: + if operator.simplify() in stabilizer_group: return f"{operator} is a Stabilizer." if all(qml.is_commuting(operator, g) for g in generators): - if operator in logical_operators: + if operator in logical_ops: return f"{operator} is a Logical Operator." else: return f"{operator} commutes with all stabilizers — it's a Logical Operator (or a multiple of one)." From e4418e0d187ec2be83f1da18457b7765390ef528 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Fri, 16 May 2025 16:45:51 -0400 Subject: [PATCH 22/37] more stuff --- demonstrations/tutorial_stabilizer_codes.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index f5dc77ca7c..9502da9fee 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -108,7 +108,7 @@ def encoded_state(alpha, beta): # # To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **parity measurement.** A parity measurement # acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure in the computational -# on two auxiliary qubits after after applying some :math:`\textrm{CNOT}` gates, as shown below +# on two auxiliary qubits after after applying some :math:`\textrm{CNOT}` gates, as shown below. # # **NEEDS PICTURE** # @@ -117,7 +117,7 @@ def encoded_state(alpha, beta): # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/syndrome_table3.png # :align: center -# :width: 25% +# :width: 100% # # .. # From a7817d4c07d4bdebc58efe6be14f254e7cc309d7 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 28 May 2025 16:02:35 -0400 Subject: [PATCH 23/37] Added table --- demonstrations/tutorial_stabilizer_codes.py | 16 +++++++++++----- 1 file changed, 11 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 9502da9fee..d6bd6f19a9 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -94,7 +94,7 @@ def encoded_state(alpha, beta): # X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # # If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and -# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occured on the second qubit and we can fix this error by flipping it back. But there is a problem with this reasoning: +# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occurred on the second qubit and we can fix this error by flipping it back. But there is a problem with this reasoning: # in a quantum circuit, we do not have access to the state. To know that a flip did occur, # we have to measure the state. But this collapses the state of the qubit, rendering it useless for future calculations. Let us see how to get around this. # @@ -115,11 +115,17 @@ def encoded_state(alpha, beta): # The result of the measurements is known as the **syndrome**. It will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, # it can tell us which qubit was flipped. The following table shows how to interpret the syndromes. # -# .. figure:: ../_static/demonstration_assets/stabilizer_codes/syndrome_table3.png -# :align: center -# :width: 100% +# .. math:: # -# .. +# \begin{table}[] +# \begin{tabular}{|c|c|c|} +# \hline +# \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline +# $X_0$ & $1$ & $0$ \\ \hline +# $X_1$ & $1$ & $1$ \\ \hline +# $X_2$ & $0$ & $1$ \\ \hline +# \end{tabular} +# \end{table} # # Let us verify this by implementing the syndrome measurement in PennyLane From ace050156ff3ad36afb57e361d139a64c3e17f36 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 28 May 2025 16:49:03 -0400 Subject: [PATCH 24/37] more tables --- demonstrations/tutorial_stabilizer_codes.py | 15 ++++++++++++++- 1 file changed, 14 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index d6bd6f19a9..6292b7fc14 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -253,7 +253,20 @@ def error_correction(error_wire): # **holds true only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue # :math:`-1`, as shown in the table below. # -# **INSERT TABLE** +# .. math:: +# +# \begin{align*} +# +# ZZI \vert 000 \rangle = + 1 \vert 000 \rangle \qquad IZZ \vert 000 \rangle + 1 \vert 000 \rangle \\ +# ZZI \vert 001 \rangle = + 1 \vert 001 \rangle \qquad IZZ \vert 001 \rangle - 1 \vert 001 \rangle \\ +# ZZI \vert 010 \rangle = - 1 \vert 010 \rangle \qquad IZZ \vert 010 \rangle - 1 \vert 010 \rangle \\ +# ZZI \vert 011 \rangle = - 1 \vert 011 \rangle \qquad IZZ \vert 011 \rangle + 1 \vert 011 \rangle \\ +# ZZI \vert 100 \rangle = - 1 \vert 100 \rangle \qquad IZZ \vert 100 \rangle + 1 \vert 100 \rangle \\ +# ZZI \vert 101 \rangle = - 1 \vert 101 \rangle \qquad IZZ \vert 101 \rangle - 1 \vert 101 \rangle \\ +# ZZI \vert 110 \rangle = + 1 \vert 110 \rangle \qquad IZZ \vert 110 \rangle - 1 \vert 110 \rangle \\ +# ZZI \vert 111 \rangle = + 1 \vert 111 \rangle \qquad IZZ \vert 111 \rangle + 1 \vert 111 \rangle \\ +# +# \end{align*} # # This gives us a new option for characterizing error correction codes. What if instead of building codewords and trying to find # the syndrome measurement operators from them, we went the the other way round? Namely, we could start by specifying a set of operators, find the states From a0cfeb43f0b634869545cab683d756c45cf3164d Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 28 May 2025 16:57:09 -0400 Subject: [PATCH 25/37] try and fix table --- demonstrations/tutorial_stabilizer_codes.py | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 6292b7fc14..1e8db97b14 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -117,15 +117,13 @@ def encoded_state(alpha, beta): # # .. math:: # -# \begin{table}[] -# \begin{tabular}{|c|c|c|} +# \begin{tabular}{|c|c|c|} # \hline # \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline # $X_0$ & $1$ & $0$ \\ \hline # $X_1$ & $1$ & $1$ \\ \hline # $X_2$ & $0$ & $1$ \\ \hline -# \end{tabular} -# \end{table} +# \end{tabular} # # Let us verify this by implementing the syndrome measurement in PennyLane From 47f8d77192cb197b98244255b147ad718e1c69ba Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 28 May 2025 17:22:07 -0400 Subject: [PATCH 26/37] formulas not rendering --- demonstrations/tutorial_stabilizer_codes.py | 26 ++++++++++----------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 1e8db97b14..2d59d6e831 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -117,13 +117,13 @@ def encoded_state(alpha, beta): # # .. math:: # -# \begin{tabular}{|c|c|c|} -# \hline -# \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline +# \begin{tabular}{|c|c|c|} +# \hline +# \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline # $X_0$ & $1$ & $0$ \\ \hline # $X_1$ & $1$ & $1$ \\ \hline # $X_2$ & $0$ & $1$ \\ \hline -# \end{tabular} +# \end{tabular} # # Let us verify this by implementing the syndrome measurement in PennyLane @@ -255,16 +255,16 @@ def error_correction(error_wire): # # \begin{align*} # -# ZZI \vert 000 \rangle = + 1 \vert 000 \rangle \qquad IZZ \vert 000 \rangle + 1 \vert 000 \rangle \\ -# ZZI \vert 001 \rangle = + 1 \vert 001 \rangle \qquad IZZ \vert 001 \rangle - 1 \vert 001 \rangle \\ -# ZZI \vert 010 \rangle = - 1 \vert 010 \rangle \qquad IZZ \vert 010 \rangle - 1 \vert 010 \rangle \\ -# ZZI \vert 011 \rangle = - 1 \vert 011 \rangle \qquad IZZ \vert 011 \rangle + 1 \vert 011 \rangle \\ -# ZZI \vert 100 \rangle = - 1 \vert 100 \rangle \qquad IZZ \vert 100 \rangle + 1 \vert 100 \rangle \\ -# ZZI \vert 101 \rangle = - 1 \vert 101 \rangle \qquad IZZ \vert 101 \rangle - 1 \vert 101 \rangle \\ -# ZZI \vert 110 \rangle = + 1 \vert 110 \rangle \qquad IZZ \vert 110 \rangle - 1 \vert 110 \rangle \\ -# ZZI \vert 111 \rangle = + 1 \vert 111 \rangle \qquad IZZ \vert 111 \rangle + 1 \vert 111 \rangle \\ +# ZZI \vert 000 \rangle &= + 1 \vert 000 \rangle \qquad IZZ \vert 000 \rangle + 1 \vert 000 \rangle \\ +# ZZI \vert 001 \rangle &= + 1 \vert 001 \rangle \qquad IZZ \vert 001 \rangle - 1 \vert 001 \rangle \\ +# ZZI \vert 010 \rangle &= - 1 \vert 010 \rangle \qquad IZZ \vert 010 \rangle - 1 \vert 010 \rangle \\ +# ZZI \vert 011 \rangle &= - 1 \vert 011 \rangle \qquad IZZ \vert 011 \rangle + 1 \vert 011 \rangle \\ +# ZZI \vert 100 \rangle &= - 1 \vert 100 \rangle \qquad IZZ \vert 100 \rangle + 1 \vert 100 \rangle \\ +# ZZI \vert 101 \rangle &= - 1 \vert 101 \rangle \qquad IZZ \vert 101 \rangle - 1 \vert 101 \rangle \\ +# ZZI \vert 110 \rangle &= + 1 \vert 110 \rangle \qquad IZZ \vert 110 \rangle - 1 \vert 110 \rangle \\ +# ZZI \vert 111 \rangle &= + 1 \vert 111 \rangle \qquad IZZ \vert 111 \rangle + 1 \vert 111 \rangle \\ # -# \end{align*} +# \end{align*} # # This gives us a new option for characterizing error correction codes. What if instead of building codewords and trying to find # the syndrome measurement operators from them, we went the the other way round? Namely, we could start by specifying a set of operators, find the states From 23a7d0af118bbacda32ecbf2dcfc7ee611aab5ab Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 29 May 2025 09:39:16 -0400 Subject: [PATCH 27/37] weird math --- demonstrations/tutorial_stabilizer_codes.py | 28 ++++++++++----------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 2d59d6e831..81d8199f1b 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -118,12 +118,12 @@ def encoded_state(alpha, beta): # .. math:: # # \begin{tabular}{|c|c|c|} -# \hline -# \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline -# $X_0$ & $1$ & $0$ \\ \hline -# $X_1$ & $1$ & $1$ \\ \hline -# $X_2$ & $0$ & $1$ \\ \hline -# \end{tabular} +# \hline +# \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline +# $X_0$ & $1$ & $0$ \\ \hline +# $X_1$ & $1$ & $1$ \\ \hline +# $X_2$ & $0$ & $1$ \\ \hline +# \end{tabular} # # Let us verify this by implementing the syndrome measurement in PennyLane @@ -255,14 +255,14 @@ def error_correction(error_wire): # # \begin{align*} # -# ZZI \vert 000 \rangle &= + 1 \vert 000 \rangle \qquad IZZ \vert 000 \rangle + 1 \vert 000 \rangle \\ -# ZZI \vert 001 \rangle &= + 1 \vert 001 \rangle \qquad IZZ \vert 001 \rangle - 1 \vert 001 \rangle \\ -# ZZI \vert 010 \rangle &= - 1 \vert 010 \rangle \qquad IZZ \vert 010 \rangle - 1 \vert 010 \rangle \\ -# ZZI \vert 011 \rangle &= - 1 \vert 011 \rangle \qquad IZZ \vert 011 \rangle + 1 \vert 011 \rangle \\ -# ZZI \vert 100 \rangle &= - 1 \vert 100 \rangle \qquad IZZ \vert 100 \rangle + 1 \vert 100 \rangle \\ -# ZZI \vert 101 \rangle &= - 1 \vert 101 \rangle \qquad IZZ \vert 101 \rangle - 1 \vert 101 \rangle \\ -# ZZI \vert 110 \rangle &= + 1 \vert 110 \rangle \qquad IZZ \vert 110 \rangle - 1 \vert 110 \rangle \\ -# ZZI \vert 111 \rangle &= + 1 \vert 111 \rangle \qquad IZZ \vert 111 \rangle + 1 \vert 111 \rangle \\ +# ZZI \vert 000 \rangle &= + 1 \vert 000 \rangle \qquad IZZ \vert 000 \rangle + 1 \vert 000 \rangle \\ +# ZZI \vert 001 \rangle &= + 1 \vert 001 \rangle \qquad IZZ \vert 001 \rangle - 1 \vert 001 \rangle \\ +# ZZI \vert 010 \rangle &= - 1 \vert 010 \rangle \qquad IZZ \vert 010 \rangle - 1 \vert 010 \rangle \\ +# ZZI \vert 011 \rangle &= - 1 \vert 011 \rangle \qquad IZZ \vert 011 \rangle + 1 \vert 011 \rangle \\ +# ZZI \vert 100 \rangle &= - 1 \vert 100 \rangle \qquad IZZ \vert 100 \rangle + 1 \vert 100 \rangle \\ +# ZZI \vert 101 \rangle &= - 1 \vert 101 \rangle \qquad IZZ \vert 101 \rangle - 1 \vert 101 \rangle \\ +# ZZI \vert 110 \rangle &= + 1 \vert 110 \rangle \qquad IZZ \vert 110 \rangle - 1 \vert 110 \rangle \\ +# ZZI \vert 111 \rangle &= + 1 \vert 111 \rangle \qquad IZZ \vert 111 \rangle + 1 \vert 111 \rangle \\ # # \end{align*} # From bcbc685853b8ec6a59a4798b54bbd3a377563ce5 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 29 May 2025 10:45:25 -0400 Subject: [PATCH 28/37] try table out --- demonstrations/tutorial_stabilizer_codes.py | 19 ++++++++++++++++--- 1 file changed, 16 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 81d8199f1b..cdff02921e 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -115,14 +115,27 @@ def encoded_state(alpha, beta): # The result of the measurements is known as the **syndrome**. It will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, # it can tell us which qubit was flipped. The following table shows how to interpret the syndromes. # +# .. list-table:: +# :header-rows: 1 +# :widths: auto 20 20 +# * - Column 1 +# - Column 2 +# - Column 3 +# * - Row 1 +# - :math:`\sqrt{16}` +# - 9 +# * - Row 2 +# - :math:`x^2` +# - 16 +# # .. math:: # # \begin{tabular}{|c|c|c|} # \hline # \textbf{Error} & \textbf{Syndrome 0} & \textbf{Syndrome 1} \\ \hline -# $X_0$ & $1$ & $0$ \\ \hline -# $X_1$ & $1$ & $1$ \\ \hline -# $X_2$ & $0$ & $1$ \\ \hline +# X_0 & 1 & 0 \\ \hline +# X_1 & 1 & 1 \\ \hline +# X_2 & 0 & 1 \\ \hline # \end{tabular} # # Let us verify this by implementing the syndrome measurement in PennyLane From 56d26d470c245086a2fa3a7dce580918291a71ab Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 29 May 2025 15:41:34 -0400 Subject: [PATCH 29/37] figures --- .../stabilizer_codes/3_qubit_code.png | Bin 0 -> 220765 bytes .../3_qubit_stabilizer_circ.png | Bin 0 -> 239077 bytes .../stabilizer_codes/cnot_identity.png | Bin 0 -> 103792 bytes .../stabilizer_codes/five_qubit_encode.png | Bin 0 -> 256488 bytes .../general_stabilizer_circuit.png | Bin 0 -> 261909 bytes .../stabilizer_codes/parity_measurements.png | Bin 0 -> 168161 bytes .../stabilizer_codes/three_qubit_encode.png | Bin 0 -> 116183 bytes demonstrations/tutorial_stabilizer_codes.py | 73 ++++++++++-------- 8 files changed, 42 insertions(+), 31 deletions(-) create mode 100644 _static/demonstration_assets/stabilizer_codes/3_qubit_code.png create mode 100644 _static/demonstration_assets/stabilizer_codes/3_qubit_stabilizer_circ.png create mode 100644 _static/demonstration_assets/stabilizer_codes/cnot_identity.png create mode 100644 _static/demonstration_assets/stabilizer_codes/five_qubit_encode.png create mode 100644 _static/demonstration_assets/stabilizer_codes/general_stabilizer_circuit.png create mode 100644 _static/demonstration_assets/stabilizer_codes/parity_measurements.png create mode 100644 _static/demonstration_assets/stabilizer_codes/three_qubit_encode.png 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zKKChGI~lZ+PxDcYyTb|;BAa#_OB-QZFM_&n8o+9&J|VdH4v2G&pAz(g}SvW=!z0DIuT z*z$-e{vR}j zf(O9&-SH0=a~KdqPeASJ>;=Yw?B)dXQ(HDcZ^?oLIb9CdZK(u!%c<#~%p#_yfASp< z0e{N+M?%2F2aGA_!`5w0IUlycV#@hUIUh_IpE~(q&N)9mZT`PK`7rViZj_leZJGst zMgGH0OaHDOE&)Bp&rcRJd&KEMcgFNhb?*){Rl+6?M&n%Z)R#|aOjFY`H7!${=m#5^ y@+}jd^Z$3J#ci>AA%u;Rty-J*{@eKWS^d-XZe5hq#KE0S+p4XvmAu*F Date: Thu, 29 May 2025 17:37:20 -0400 Subject: [PATCH 31/37] some corrections --- demonstrations/tutorial_stabilizer_codes.py | 61 +++++++++++---------- 1 file changed, 33 insertions(+), 28 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index eedc95ffb4..5a9e8100e0 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -99,20 +99,20 @@ def encoded_state(alpha, beta): # # If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and # :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occurred on the second qubit and we can fix this error by flipping it back. But there is a problem with this reasoning: -# in a quantum circuit, we do not have access to the state. To know that a flip did occur, +# To know that a flip did occur, # we have to measure the state. But this collapses the state of the qubit, rendering it useless for future calculations. Let us see how to get around this. # # .. note:: # Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, -# but even state preparation is prone to errors! If the quantum state is not known, the no-cloning +# but this only increases the amount of quantum resources we need! If the quantum state is not known, the no-cloning # theorem states it is impossible to make a copy of the state. # # Error detection # ~~~~~~~~~~~~~~~~ # # To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **parity measurement.** A parity measurement -# acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure in the computational -# on two auxiliary qubits after after applying some :math:`\textrm{CNOT}` gates, as shown below. +# acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure the two auxiliary qubits +# in the computational basis after applying some :math:`\textrm{CNOT}` gates, as shown below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/parity_measurements.png # :align: center @@ -120,8 +120,8 @@ def encoded_state(alpha, beta): # # .. # -# The result of the measurements is known as the **syndrome**. It will tell us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, -# it can tell us which qubit was flipped. The following table shows how to interpret the syndromes. +# The result of the measurements is known as the **syndrome**. It tells us whether one of the qubits in :math:`\vert \bar{\psi} \rangle` was flipped and moreover, +# it has information on which qubit was flipped. The following table shows how to interpret the syndromes. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/syndrome_table3.png # :align: center @@ -163,7 +163,7 @@ def syndrome_measurement(error_wire): # ~~~~~~~~~~~~~~~~~ # # Once a single bit-flip error is detected, correction is straightforward. Since the Pauli-X operator is its own inverse -# (i.e., :math:`X^2 = \mathbb{I}`), applying the :math:`X` operator to the erroneous qubit restores the original state, for example, +# (i.e., :math:`X^2 = \mathbb{I}`), reapplying the :math:`X` operator to the erroneous qubit restores the original state. For example, # if the syndrome measurement shows the error occurred on the second qubit, we apply # # .. math:: @@ -205,7 +205,7 @@ def error_correction(error_wire): ############################################################################## # -# At the time of writing, PennyLane circuits with mid-circuit measurements cannot return a quantum state, so we return the density matrix instead. +# At the time of writing, PennyLane circuits with mid-circuit measurements cannot return a quantum state vector, so we return the density matrix instead. # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction # as follows @@ -228,9 +228,9 @@ def error_correction(error_wire): # # We have worked with a simple example, but it is quite limited. Indeed, the three-qubit code only works for # bit flip errors, but more powerful codes need more resources. For example, Shor's code, involving 9 qubits, -# can correct more types of errors on a single logical qubit. Moreover, to avoid errors at an acceptable level, -# the industry standard is 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly -# complex! To deal with these situations, we resort to a different +# can correct more types of errors on a single logical qubit. To make matters worse, to avoid errors at an acceptable level, +# the industry standard is about 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly +# complex, so 1000 seems quite daunting! To deal with these situations, we resort to a different # representation of error correction codes, known as the **operator picture.** # # To gain some intuition about the operator picture, let us express the three-qubit repetition code in a different way. Using the @@ -242,7 +242,7 @@ def error_correction(error_wire): # # .. # -# the error correction code can be expressed in the following way: +# the three-qubit repetition code can be expressed in the following way: # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/3_qubit_stabilizer_circ.png # :align: center @@ -263,8 +263,8 @@ def error_correction(error_wire): # I_0 Z_1 Z_2 \vert 000 \rangle = \vert 000 \rangle, \quad I_0 Z_1 Z_2 \vert 111 \rangle = \vert 111 \rangle. # # This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be -# some operations applied on the state, but we will be able to fix them via an error correction scheme. Notably, the invariance property -# **holds true only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue +# some operations applied on the state, but we will be able to fix them via an error correction scheme. The invariance property +# **holds true for and only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue # :math:`-1`, as shown in the table below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/table_eigenvalues.png @@ -274,8 +274,8 @@ def error_correction(error_wire): # .. # # This gives us a new option for characterizing error correction codes. What if instead of building codewords and trying to find -# the syndrome measurement operators from them, we went the the other way round? Namely, we could start by specifying a set of operators, find the states -# that remain invariant under their action, and make these our codewords. These operators are known as **stabilizer generators.** +# the syndrome measurement operators from them, we went the other way round? Namely, we could start by specifying a set of operators, find the states +# that remain invariant under their action, and make these our codewords. The operators in this initial set are known as **stabilizer generators.** # # The stabilizer formalism # ------------------------- @@ -289,9 +289,9 @@ def error_correction(error_wire): # # 1. It contains the identity operator :math:`I_0\otimes I_1 \times \ldots I_{n-1},` but does not contain the negative identity. # 2. All elements of the set commute with each other. -# 3. The matrix product of two elements in the set yields an element that is also in the set. +# 3. The matrix product of any two elements in the set yields an element that is also in the set. # -# The set can be more succinctly specified set of generators: a minimal set of operators in the stabilizer set that can produce all +# The set can be more succinctly specified via a set of generators: a minimal set of operators in the stabilizer set that can produce all # the other elements through pairwise multiplication. As a simple example, consider the stabilizer set # # .. math:: @@ -299,7 +299,7 @@ def error_correction(error_wire): # S = \left\lbrace I_0 \otimes I_1 \otimes I_2, \ Z_0 \otimes Z_1 \otimes I_2, \ Z_0 \otimes I_1 \otimes Z_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rbrace. # # We can check that it satisfies the defining properties 1. to 3. The most cumbersome to check is property 3, where we have to take all -# possible products of the elements and check whether the result is in :math:`S.` For example +# possible products of the elements and check whether the result is in :math:`S.` For example, # # .. math:: # @@ -313,7 +313,7 @@ def error_correction(error_wire): # # S = \left\langle Z_0 \otimes Z_1 \otimes I_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rangle, # -# which reads :math:`S` *is the stabilizer set generated by the elements* :math:`Z_0 \otimes Z_1 \otimes I_2` *and* :math:`I_0 \otimes Z_1 \otimes Z_2.` +# which reads ":math:`S` *is the stabilizer set generated by the elements* :math:`Z_0 \otimes Z_1 \otimes I_2` *and* :math:`I_0 \otimes Z_1 \otimes Z_2.`" # It turns out that specifying these generators is enough to completely define an error correcting code. # # Now that we know how stabilizer generators work, let us create a tool for later use that creates the full stabilizer set from its generators. @@ -365,8 +365,8 @@ def generate_stabilizer_group(gens, num_wires): # # .. # -# The stabilizer generators act as the controlled operators in the codewords and the measurement in the auxiliary wires yield unique syndromes for the -# Pauli error that the code deals with. +# The stabilizer generators act as the controlled operators in the codewords and the measurements in the auxiliary wires yield unique syndromes for the +# Pauli errors that the code deals with. # # Logical operators # ~~~~~~~~~~~~~~~~~~ @@ -380,7 +380,7 @@ def generate_stabilizer_group(gens, num_wires): # .. math:: # # \bar{X}\vert \bar{0} \rangle = \vert {1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert {0} \rangle \\ -# \bar{Z}\vert \bar{0} \rangle = \vert {0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert {1} \rangle +# \ \bar{Z}\vert \bar{0} \rangle = \vert {0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert {1} \rangle # # For example, in the three qubit bit flip error correcting code, the logical operators are :math:`\bar{X} = X_0 X_1 X_2` and # :math:`\bar{Z} = Z_0 Z_1 Z_2,` but they will not always be this simple. In general, given a stabilizer set $S$, the logical @@ -430,7 +430,7 @@ def classify_pauli(operator, logical_ops, generators, n_wires): # --------------------------- # # Unlike other error correcting codes, the 5-qubit code does not have a special name, but it holds a special place as the smallest -# error correcting protocol capable of correcting arbitrary Pauli Errors--unwanted applications of :math:`X`, :math:`Y`, or :math:`Z` gates. +# error correcting protocol capable of correcting arbitrary Pauli Errors--unwanted applications of :math:`X,` :math:`Y,` or :math:`Z` gates. # In this section, we will build it starting from its stabilizer generators: # # .. math:: @@ -458,11 +458,16 @@ def classify_pauli(operator, logical_ops, generators, n_wires): # # .. math:: # -# \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0}. +# \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0} \rangle. # -# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`Z^{\otimes 5}.` With these -# defining features in mind, let us build the five-qubit stabilizer code. First, we need to prepare our logical qubit $\vert \bar{0} \rangle,$ -# which can be done using the circuit below. +# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`\bar{Z}=Z^{\otimes 5}.` With these +# defining features in mind, let us build the five-qubit stabilizer code. First, we need to prepare an arbitrary logical qubit +# +# .. math:: +# +# \vert \bar{\psi}\rangle = \alpha \vert \bar{0} \rangle + \beta \vert \bar{1} \rangle +# +# starting from the qubit :math:`\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle,` which can be done using the circuit below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/five_qubit_encode.png # :align: center From 1de298eb895d3b90cbf62cde1b6f4c3d6e044121 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Thu, 29 May 2025 18:04:00 -0400 Subject: [PATCH 32/37] added note --- demonstrations/tutorial_stabilizer_codes.py | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 5a9e8100e0..b6b09bfed7 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -426,6 +426,12 @@ def classify_pauli(operator, logical_ops, generators, n_wires): ############################################################################## # +# .. note:: +# In the literature, you may have come across an error correction code being called an ":math:`[n,k]`-stabilizer code." In this notation, the number :math:`n` represents +# the number of physical qubit used to encode the logical qubit, and :math:`k` is the number of logical qubits and it is +# equal to :math:`1` for all the codes in this demo. It is possible to show that the number of stabilizer generators :math:`m` +# is related to :math:`n` and :math:`k` via :math: `m = n - k.` +# # Five-qubit stabilizer code # --------------------------- # @@ -545,7 +551,6 @@ def five_qubit_code(alpha, beta, error_type, error_wire): # The syndrome table is printed, and with we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward # to generalize to any valid set of stabilizers. # -# # References # ----------- # From dd1538f0e9ee4144a9d275c755dc0785387f2e02 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Mon, 2 Jun 2025 15:23:55 -0400 Subject: [PATCH 33/37] changes to text and metadata --- .../tutorial_stabilizer_codes.metadata.json | 138 ++---------------- demonstrations/tutorial_stabilizer_codes.py | 41 +++--- 2 files changed, 35 insertions(+), 144 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.metadata.json b/demonstrations/tutorial_stabilizer_codes.metadata.json index 09a784afc3..1914ead2cf 100644 --- a/demonstrations/tutorial_stabilizer_codes.metadata.json +++ b/demonstrations/tutorial_stabilizer_codes.metadata.json @@ -1,6 +1,9 @@ { "title": "Stabilizer codes for quantum error correction", "authors": [ + { + "username": "chsid" + }, { "username": "alvaro" } @@ -8,9 +11,9 @@ "dateOfPublication": "2025-01-23T09:00:00+00:00", "dateOfLastModification": "2025-01-23T09:00:00+00:00", "categories": [ - "Getting Started", + "Algorithms", "Quantum Computing", - "Algorithms" + "Quantum Hardware" ], "tags": [], "previewImages": [ @@ -27,127 +30,12 @@ "doi": "", "references": [ { - "id": "Selinger2010", - "type": "book", - "title": "A Survey of Graphical Languages for Monoidal Categories", - "authors": "P. Selinger", - "year": "2010", - "publisher": "Springer Berlin Heidelberg", - "journal": "New Structures for Physics", - "doi": "10.1007/978-3-642-12821-9_4", - "url": "http://dx.doi.org/10.1007/978-3-642-12821-9_4" - }, - { - "id": "Arad2010", - "type": "preprint", - "title": "Quantum computation and the evaluation of tensor networks", - "authors": "I. Arad, Z. Landau", - "year": "2010", - "publisher": "", - "journal": "", - "doi": "10.48550/arXiv.0805.0040", - "url": "https://arxiv.org/abs/0805.0040" - }, - { - "id": "Bridgeman2017", - "type": "article", - "title": "Hand-waving and interpretive dance: an introductory course on tensor networks", - "authors": "J. C. Bridgeman, C. T. Chubb", - "year": "2017", - "publisher": "", - "journal": "Journal of Physics A: Mathematical and Theoretical", - "doi": "10.1088/1751-8121/aa6dc3", - "url": "http://dx.doi.org/10.1088/1751-8121/aa6dc3" - }, - { - "id": "Lam1997", - "type": "article", - "title": "On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution", - "authors": "C.-C. Lam, P. Sadayappan, R. Wenger", + "id": "gottesman1997stabilizercodesquantumerror", + "type": "thesis", + "title": "Stabilizer Codes and Quantum Error Correction", + "authors": "Daniel Gottesman", "year": "1997", - "publisher": "", - "journal": "Parallel Processing Letters", - "doi": "10.1142/S0129626497000176", - "url": "https://doi.org/10.1142/S0129626497000176" - }, - { - "id": "Gray2021", - "type": "article", - "title": "Hyper-optimized tensor network contraction", - "authors": "J. Gray, S. Kourtis, S. Choi", - "year": "2021", - "publisher": "", - "journal": "Quantum", - "doi": "10.22331/q-2021-03-15-410", - "url": "https://quantum-journal.org/papers/q-2021-03-15-410/" - }, - { - "id": "Bienstock1990", - "type": "article", - "title": "On embedding graphs in trees", - "authors": "Dan Bienstock", - "year": "1990", - "journal": "Journal of Combinatorial Theory, Series B", - "volume": "49", - "number": "1", - "pages": "103–136", - "doi": "10.1016/0095-8956(90)90066-9", - "url": "https://www.sciencedirect.com/science/article/pii/0095895690900669" - }, - { - "id": "Pfeifer2014", - "type": "article", - "title": "Faster identification of optimal contraction sequences for tensor networks", - "authors": "R. N. C. Pfeifer, J. Haegeman, F. Verstraete", - "year": "2014", - "publisher": "", - "journal": "Physical Review E", - "doi": "10.1103/PhysRevE.90.033315", - "url": "http://dx.doi.org/10.1103/PhysRevE.90.033315" - }, - { - "id": "Pirvu2010", - "type": "article", - "title": "Matrix product operator representations", - "authors": "B. Pirvu, V. Murg, J. I. Cirac, F. Verstraete", - "year": "2010", - "publisher": "", - "journal": "New Journal of Physics", - "doi": "10.1088/1367-2630/12/2/025012", - "url": "http://dx.doi.org/10.1088/1367-2630/12/2/025012" - }, - { - "id": "Ferris2012", - "type": "article", - "title": "Perfect sampling with unitary tensor networks", - "authors": "A. J. Ferris, G. Vidal", - "year": "2012", - "publisher": "", - "journal": "Physical Review B", - "doi": "10.1103/PhysRevB.85.165146", - "url": "http://dx.doi.org/10.1103/PhysRevB.85.165146" - }, - { - "id": "Schuch2008", - "type": "article", - "title": "Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions", - "authors": "N. Schuch, M. M. Wolf, F. Verstraete, J. I. Cirac", - "year": "2008", - "publisher": "", - "journal": "Physical Review Letters", - "doi": "10.1103/PhysRevLett.100.040501", - "url": "https://doi.org/10.1103/PhysRevLett.100.040501" - }, - { - "id": "Schollwoeck2011", - "type": "article", - "title": "The density-matrix renormalization group in the age of matrix product states", - "authors": "U. Schollwöck", - "year": "2011", - "publisher": "", - "journal": "Annals of Physics", - "doi": "10.1016/j.aop.2010.09.012", - "url": "https://doi.org/10.1016/j.aop.2010.09.012" + "url": "https://arxiv.org/abs/quant-ph/9705052" } ], "basedOnPapers": [], @@ -155,17 +43,17 @@ "relatedContent": [ { "type": "demonstration", - "id": "tutorial_tn_circuits", + "id": "tutorial_game_of_surface_codes", "weight": 1.0 }, { "type": "demonstration", - "id": "tutorial_mps", + "id": "tutorial_toric_code", "weight": 1.0 }, { "type": "demonstration", - "id": "tutorial_How_to_simulate_quantum_circuits_with_tensor_networks", + "id": "tutorial_mbqc", "weight": 1.0 } ] diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index b6b09bfed7..8b87bed44f 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -97,10 +97,10 @@ def encoded_state(alpha, beta): # # X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # -# If we are sure that **only one bit-flip error occurred**, and since only the superpositions of :math:`\alpha \vert 000 \rangle` and -# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occurred on the second qubit and we can fix this error by flipping it back. But there is a problem with this reasoning: -# To know that a flip did occur, -# we have to measure the state. But this collapses the state of the qubit, rendering it useless for future calculations. Let us see how to get around this. +# If we assume that **only one bit-flip error occurred**, and considering that only the superpositions of :math:`\alpha \vert 000 \rangle` and +# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occurred on the second qubit and we can fix it by flipping the qubit back. But there is a problem with this reasoning: +# to know that a flip did occur, +# we need to measure the state. But this collapses the state of the qubit, rendering it useless for future computations. Let us see how to get around this issue. # # .. note:: # Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, @@ -110,9 +110,9 @@ def encoded_state(alpha, beta): # Error detection # ~~~~~~~~~~~~~~~~ # -# To detect whether a bit-flip error has occurred on one of the physical qubits, we perform a **parity measurement.** A parity measurement -# acts on auxiliary qubits to avoid disturbing the encoded logical state. In the case of the three-qubit repetition code, we measure the two auxiliary qubits -# in the computational basis after applying some :math:`\textrm{CNOT}` gates, as shown below. +# To detect a bit-flip error on one of the physical qubits without disturbing the encoded logical state, we perform a **parity measurement.** A parity measurement +# acts on some auxiliary qubits to avoid disturbing the encoded logical state. IFor the three-qubit repetition code, this involves measuring two auxiliary qubits in the +# computational basis after applying a series of :math:`\textrm{CNOT}` gates, as illustrated in the circuit below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/parity_measurements.png # :align: center @@ -129,7 +129,7 @@ def encoded_state(alpha, beta): # # .. # -# Let us verify this by implementing the syndrome measurement in PennyLane +# Let us verify this by implementing the syndrome measurement in PennyLane. def error_detection(): @@ -201,13 +201,13 @@ def error_correction(error_wire): qml.cond(m3 & m4, qml.PauliX)(wires = 1) qml.cond(~m3 & m4, qml.PauliX)(wires = 2) - return qml.density_matrix(wires = [0, 1, 2]) # qml.state not supported, but density matrices are + return qml.density_matrix(wires = [0, 1, 2]) # qml.state not supported, but density matrices are. ############################################################################## # # At the time of writing, PennyLane circuits with mid-circuit measurements cannot return a quantum state vector, so we return the density matrix instead. # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction -# as follows +# as follows. dev = qml.device("default.qubit", wires = 5) error_correction_qnode = qml.QNode(error_correction, dev) @@ -215,9 +215,12 @@ def error_correction(error_wire): # Compute fidelity of final corrected state with initial encoded state -print("Fidelity if error on wire 0: ", qml.math.fidelity(encoded_state, error_correction_qnode(0)).round(2)) -print("Fidelity if error on wire 1: ", qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2)) -print("Fidelity if error on wire 2: ", qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2)) +print("Fidelity if error on wire 0: ", + qml.math.fidelity(encoded_state, error_correction_qnode(0)).round(2)) +print("Fidelity if error on wire 1: ", + qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2)) +print("Fidelity if error on wire 2: ", + qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2)) ############################################################################## # @@ -242,7 +245,7 @@ def error_correction(error_wire): # # .. # -# the three-qubit repetition code can be expressed in the following way: +# the three-qubit repetition code can be expressed in the following way. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/3_qubit_stabilizer_circ.png # :align: center @@ -252,7 +255,7 @@ def error_correction(error_wire): # # This is the same circuit, but the controls are all now in the auxiliary qubits, while the physical qubits act as target qubits. # This does not seem desirable--we do not want to change the state of the physical qubits! However, let us observe that the operators -# that act on the logical qubits are :math:`Z_0 Z_1 I_2` and :math:`I_0 Z_1 Z_2,` which leave the logical codewords invariant: +# that act on the logical qubits are :math:`Z_0 Z_1 I_2` and :math:`I_0 Z_1 Z_2,` which leave the logical codewords invariant. # # .. math:: # @@ -264,7 +267,7 @@ def error_correction(error_wire): # # This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be # some operations applied on the state, but we will be able to fix them via an error correction scheme. The invariance property -# **holds true for and only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue +# **holds true for and only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue # :math:`-1`, as shown in the table below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/table_eigenvalues.png @@ -428,9 +431,9 @@ def classify_pauli(operator, logical_ops, generators, n_wires): # # .. note:: # In the literature, you may have come across an error correction code being called an ":math:`[n,k]`-stabilizer code." In this notation, the number :math:`n` represents -# the number of physical qubit used to encode the logical qubit, and :math:`k` is the number of logical qubits and it is +# the number of physical qubit used to encode the logical qubit. The integer :math:`k` is the number of logical qubits and it is # equal to :math:`1` for all the codes in this demo. It is possible to show that the number of stabilizer generators :math:`m` -# is related to :math:`n` and :math:`k` via :math: `m = n - k.` +# is related to :math:`n` and :math:`k` via :math:`m = n - k.` # # Five-qubit stabilizer code # --------------------------- @@ -508,7 +511,7 @@ def five_qubit_encode(alpha, beta): ############################################################################## # -# Having encoded the logical state, we can use the stabilizers to measure obtain the syndrome table like we did with the three-qubit code. +# Having encoded the logical state, we can implement the general circuit built from the stabilizers to obtain the syndrome table like we did with the three-qubit code. dev = qml.device("default.qubit", wires = 9, shots = 1) From 11cafb9b6c04962bced80150738787ae54681488 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Mon, 2 Jun 2025 15:29:48 -0400 Subject: [PATCH 34/37] phdthesis --- demonstrations/tutorial_stabilizer_codes.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_stabilizer_codes.metadata.json b/demonstrations/tutorial_stabilizer_codes.metadata.json index 1914ead2cf..d4924e62f0 100644 --- a/demonstrations/tutorial_stabilizer_codes.metadata.json +++ b/demonstrations/tutorial_stabilizer_codes.metadata.json @@ -31,7 +31,7 @@ "references": [ { "id": "gottesman1997stabilizercodesquantumerror", - "type": "thesis", + "type": "phdthesis", "title": "Stabilizer Codes and Quantum Error Correction", "authors": "Daniel Gottesman", "year": "1997", From 54a91195c73142751c449c46d6d92a5112d16d44 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Mon, 30 Jun 2025 14:05:07 -0400 Subject: [PATCH 35/37] siddhu's version --- demonstrations/tutorial_stabilizer_codes.py | 513 ++++++++++++-------- 1 file changed, 313 insertions(+), 200 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index 8b87bed44f..f9adc585fa 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -2,30 +2,31 @@ Stabilizer codes for quantum error correction ================================================= -State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than +State-of-the-art quantum devices, such as IBM's Condor and Atom Computing's optical lattices, contain more than a thousand qubits. Does this qubit count suffice for valuable quantum algorithms with clear speedups? The reality is that there is more to the story than the sheer number of qubits. As we currently stand, quantum -devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing--has -been gaining traction. +devices are still prone to errors that increase with device size. For this reason, **quantum error correction**--one of the most important domain in the universe of quantum computing--has +been gaining traction. -Quantum error correction is done via schemes known as **error correction codes.** +Quantum error correction is implemented through **error correction codes.** These are quantum algorithms that come in many varieties that address different error types. -Is there a unified way to understand these codes? The **stabilizer formalism** provides such a framework for +Is there a unified way to understand all these codes? The **stabilizer formalism** provides such a framework for a large class of quantum error correction codes. The so-called -**stabilizer codes**, such as the repetition, Shor, Steane, and surface codes, +**stabilizer codes**, such as the repetition, Shor, Steane, and surface codes, all fall under this formalism. Other codes, like the GKP codes used by Xanadu, lie outside of it. -In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct and -some well-known codes using the quantum circuit formalism and then derive their **stabilizer generators,** from which -the code can be reconstructed. This enables the construction of a wide range of error correction codes -directly from their stabilizer generators. +In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct +some well-known codes using **stabilizer generators,** from which +the other elements of the code (codewords, syndrome measurements, etc.) can be reconstructed. This enables the construction of a wide range of error correction codes +directly from their stabilizer generators. We will use the quantum circuit formalism to construct the codes. + A toy example: the repetition code ----------------------------------- -To start with, let us explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** -We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. -The quantum circuit representation is known as the **state picture**. In this formalism, error correction codes follow a simple structure: +To start with, let us explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** +We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. +We represent the states of the qubits in the circuit using **state vectors** (the Schrödinger picture). In this formalism, error correction codes follow a simple structure: - Qubit encoding - Error detection @@ -36,9 +37,9 @@ Qubit encoding ~~~~~~~~~~~~~~~ -The first step in an error correction code is **encoding** one abstract or **logical qubit** into a set of many on-device **physical qubits.** -The rationale is that, if some external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. -For example, in the three-qubit repetition code, the logical basis-state +The first step in an error correction code is **encoding** one abstract or **logical qubit** into a set of many on-device **physical qubits.** +The rationale is that, if some external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. +For example, in the three-qubit repetition code, the logical basis-state qubits, or **logical codewords**, :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits via .. math:: @@ -51,8 +52,8 @@ \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert 111\rangle. -This encoding can be done via the following quantum circuit. - +This encoding can be done via the following quantum circuit. It should be explicitly stated that the 0 bar is logical 0 and 1 bar is logical 1. + .. figure:: ../_static/demonstration_assets/stabilizer_codes/three_qubit_encode.png :align: center :width: 70% @@ -66,18 +67,18 @@ import pennylane as qml from pennylane import numpy as np -def encode(alpha, beta): - qml.StatePrep([alpha, beta], wires = 0) - qml.CNOT(wires = [0, 1]) - qml.CNOT(wires = [0, 2]) +def encode(alpha, beta): + qml.StatePrep([alpha, beta], wires=0) + qml.CNOT(wires=[0, 1]) + qml.CNOT(wires=[0, 2]) def encoded_state(alpha, beta): - encode(alpha, beta) return qml.state() + encode_qnode = qml.QNode(encoded_state, qml.device("default.qubit")) alpha = 1 / np.sqrt(2) @@ -90,6 +91,14 @@ def encoded_state(alpha, beta): ############################################################################## # +# .. note:: +# Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, +# but this only increases the amount of quantum resources we need! If the quantum state is not known, the no-cloning +# theorem states it is impossible to make a copy of the state. +# +# Error detection +# ~~~~~~~~~~~~~~~~ +# # Now, suppose that a **bit-flip** error occurs on the second qubit, meaning that the qubit is randomly flipped. This can be modelled as # an unwanted Pauli-$X$ operator being applied on :math:`\vert \bar{\psi}\rangle:` # @@ -97,22 +106,15 @@ def encoded_state(alpha, beta): # # X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # -# If we assume that **only one bit-flip error occurred**, and considering that only the superpositions of :math:`\alpha \vert 000 \rangle` and -# :math:`\alpha \vert 111 \rangle` are allowed, we can deduce that error occurred on the second qubit and we can fix it by flipping the qubit back. But there is a problem with this reasoning: -# to know that a flip did occur, -# we need to measure the state. But this collapses the state of the qubit, rendering it useless for future computations. Let us see how to get around this issue. +# How do we detect this error? # -# .. note:: -# Why do we encode qubits in this way, instead of preparing many copies of the state? If the quantum state is known, we could do this, -# but this only increases the amount of quantum resources we need! If the quantum state is not known, the no-cloning -# theorem states it is impossible to make a copy of the state. +# As you already know that measuring the state collapses it, we cannot measure the state to detect the error. # -# Error detection -# ~~~~~~~~~~~~~~~~ # -# To detect a bit-flip error on one of the physical qubits without disturbing the encoded logical state, we perform a **parity measurement.** A parity measurement -# acts on some auxiliary qubits to avoid disturbing the encoded logical state. IFor the three-qubit repetition code, this involves measuring two auxiliary qubits in the -# computational basis after applying a series of :math:`\textrm{CNOT}` gates, as illustrated in the circuit below. +# To detect a bit-flip error on one of the physical qubits without disturbing the encoded logical state, we perform a **parity measurement.** +# This checks whether all physical qubits are in the same state by comparing them two at a time, without directly measuring them. +# Instead, auxiliary qubits are used and measured. For the three-qubit repetition code, this involves measuring two auxiliary qubits +# in the computational basis after applying a series of :math:\textrm{CNOT} gates, as illustrated in the circuit below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/parity_measurements.png # :align: center @@ -131,24 +133,23 @@ def encoded_state(alpha, beta): # # Let us verify this by implementing the syndrome measurement in PennyLane. -def error_detection(): - qml.CNOT(wires = [0, 3]) - qml.CNOT(wires = [1, 3]) - qml.CNOT(wires = [1, 4]) - qml.CNOT(wires = [2, 4]) +def error_detection(): + qml.CNOT(wires=[0, 3]) + qml.CNOT(wires=[1, 3]) + qml.CNOT(wires=[1, 4]) + qml.CNOT(wires=[2, 4]) -@qml.qnode(qml.device("default.qubit", wires = 5, shots = 1)) # A single sample flags error +@qml.qnode(qml.device("default.qubit", wires=5, shots=1)) # A single sample flags error def syndrome_measurement(error_wire): - encode(alpha, beta) - qml.PauliX(wires = error_wire) # Unwanted Pauli Operator + qml.PauliX(wires=error_wire) # Unwanted Pauli Operator error_detection() - return qml.sample(wires = [3,4]) + return qml.sample(wires=[3, 4]) print("Syndrome if error on wire 0: ", syndrome_measurement(0)) @@ -163,11 +164,11 @@ def syndrome_measurement(error_wire): # ~~~~~~~~~~~~~~~~~ # # Once a single bit-flip error is detected, correction is straightforward. Since the Pauli-X operator is its own inverse -# (i.e., :math:`X^2 = \mathbb{I}`), reapplying the :math:`X` operator to the erroneous qubit restores the original state. For example, -# if the syndrome measurement shows the error occurred on the second qubit, we apply +# (i.e., :math:`X^2 = \mathbb{I}`), re-applying the :math:`X` operator to the erroneous qubit restores the original state. For example, +# if the syndrome measurement shows the error occurred on the second qubit, we apply # # .. math:: -# +# # X_2 (X_2 \vert \bar{\psi}\rangle) = \vert \bar{\psi} \rangle. # # By applying the appropriate corrective operation, the repetition code effectively protects and repairs the encoded quantum information. @@ -181,12 +182,12 @@ def syndrome_measurement(error_wire): # # We can use PennyLane's mid-circuit measurement features to implement the full three-qubit repetition code. -@qml.qnode(qml.device("default.qubit", wires = 5)) -def error_correction(error_wire): +@qml.qnode(qml.device("default.qubit", wires=5)) +def error_correction(error_wire): encode(alpha, beta) - qml.PauliX(wires = error_wire) + qml.PauliX(wires=error_wire) error_detection() @@ -197,11 +198,14 @@ def error_correction(error_wire): # Operations conditional on measurements - qml.cond(m3 & ~m4 , qml.PauliX)(wires = 0) - qml.cond(m3 & m4, qml.PauliX)(wires = 1) - qml.cond(~m3 & m4, qml.PauliX)(wires = 2) + qml.cond(m3 & ~m4, qml.PauliX)(wires=0) + qml.cond(m3 & m4, qml.PauliX)(wires=1) + qml.cond(~m3 & m4, qml.PauliX)(wires=2) + + return qml.density_matrix( + wires=[0, 1, 2] + ) # qml.state not supported, but density matrices are. - return qml.density_matrix(wires = [0, 1, 2]) # qml.state not supported, but density matrices are. ############################################################################## # @@ -209,34 +213,42 @@ def error_correction(error_wire): # With this result, we can verify that the fidelity of the encoded state is the same as the final state after correction # as follows. -dev = qml.device("default.qubit", wires = 5) +dev = qml.device("default.qubit", wires=5) error_correction_qnode = qml.QNode(error_correction, dev) encoded_state = qml.math.dm_from_state_vector(encode_qnode(alpha, beta)) # Compute fidelity of final corrected state with initial encoded state -print("Fidelity if error on wire 0: ", - qml.math.fidelity(encoded_state, error_correction_qnode(0)).round(2)) -print("Fidelity if error on wire 1: ", - qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2)) -print("Fidelity if error on wire 2: ", - qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2)) +print( + "Fidelity if error on wire 0: ", + qml.math.fidelity(encoded_state, error_correction_qnode(0)).round(2), +) +print( + "Fidelity if error on wire 1: ", + qml.math.fidelity(encoded_state, error_correction_qnode(1)).round(2), +) +print( + "Fidelity if error on wire 2: ", + qml.math.fidelity(encoded_state, error_correction_qnode(2)).round(2), +) ############################################################################## # # The error is corrected no matter which qubit was flipped! # -# Operator picture and stabilizer generators -# ------------------------------------------ +# Revisiting the three-qubit repetition code with Pauli operators +# --------------------------------------------------------------- # -# We have worked with a simple example, but it is quite limited. Indeed, the three-qubit code only works for -# bit flip errors, but more powerful codes need more resources. For example, Shor's code, involving 9 qubits, -# can correct more types of errors on a single logical qubit. To make matters worse, to avoid errors at an acceptable level, +# We have worked with a simple example (repetition code), but it is quite limited. Indeed, the three-qubit code only works for +# a single bit flip error. There are more powerful codes but they also need more resources. For example, Shor's code +# can correct any error on a single logical qubit but it already needs 9 qubits. To make matters worse, to avoid errors at an acceptable level, # the industry standard is about 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly -# complex, so 1000 seems quite daunting! To deal with these situations, we resort to a different -# representation of error correction codes, known as the **operator picture.** +# complex, so writing a 1000 qubit state vector seems quite daunting! To deal with these situations, we resort to a different +# representation of error correction codes, using **Pauli operators** instead of state vectors. # -# To gain some intuition about the operator picture, let us express the three-qubit repetition code in a different way. Using the +# It is great time to look at Pauli operators and their properties now, if you are not familiar with them. +# +# To gain some intuition about the operator picture, let us express the three-qubit repetition code in a different way. Using the # identity below, # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/cnot_identity.png @@ -253,22 +265,22 @@ def error_correction(error_wire): # # .. # -# This is the same circuit, but the controls are all now in the auxiliary qubits, while the physical qubits act as target qubits. +# This is the same circuit, but the controls are all now in the auxiliary qubits, while the physical qubits act as target qubits. # This does not seem desirable--we do not want to change the state of the physical qubits! However, let us observe that the operators # that act on the logical qubits are :math:`Z_0 Z_1 I_2` and :math:`I_0 Z_1 Z_2,` which leave the logical codewords invariant. -# +# # .. math:: # # Z_0 Z_1 I_2 \vert 000 \rangle = \vert 000 \rangle, \quad Z_0 Z_1 I_2 \vert 111 \rangle = \vert 111 \rangle. -# +# # .. math:: # # I_0 Z_1 Z_2 \vert 000 \rangle = \vert 000 \rangle, \quad I_0 Z_1 Z_2 \vert 111 \rangle = \vert 111 \rangle. # # This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be -# some operations applied on the state, but we will be able to fix them via an error correction scheme. The invariance property -# **holds true for and only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue -# :math:`-1`, as shown in the table below. +# some operations applied on the state, but we will be able to fix them via an error correction scheme. The invariance property +# **holds true for and only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue +# :math:`-1`, as shown in the table below. Therefore measuring the eigenvalues of these operators will tell us if an error has occurred. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/table_eigenvalues.png # :align: center @@ -286,29 +298,39 @@ def error_correction(error_wire): # Stabilizer generators # ~~~~~~~~~~~~~~~~~~~~~~ # -# The stabilizer formalism takes the operator picture representation seriously and uses it to find error correction codes starting from a -# **stabilizer set** of **Pauli words**--tensor products of Pauli operators. A stabilizer set on :math:`n` -# qubits satisfies the following properties. +# Note: It is important to have a minimal understanding of group theory to understand the stabilizer formalism. # -# 1. It contains the identity operator :math:`I_0\otimes I_1 \times \ldots I_{n-1},` but does not contain the negative identity. -# 2. All elements of the set commute with each other. -# 3. The matrix product of any two elements in the set yields an element that is also in the set. -# -# The set can be more succinctly specified via a set of generators: a minimal set of operators in the stabilizer set that can produce all +# Notation: In the stabilizer formalism, we often omit explicit tensor product symbols (:math:`\otimes`) for brevity. +# For example, :math:`X_0 Z_1` denotes :math:`X \otimes Z` acting on qubits 0 and 1, respectively. +# When identity operators are omitted, we use subscripts to indicate which qubits the non-identity Pauli operators act on. +# If all positions are filled (e.g., :math:`XZI`), the position implicitly indicates the qubit index (qubit 0, 1, 2). + +# +# The stabilizer formalism is a powerful framework for constructing quantum error-correcting codes using the algebraic structure of *Pauli operators*. +# It focuses on subgroups of the *Pauli group* on :math:`n` qubits—denoted :math:`\mathcal{P}_n`—which consists of all tensor products of single-qubit Pauli operators :math:`\{I, X, Y, Z\}` (with overall phases :math:`\pm1, \pm i`). +# A **stabilizer group** :math:`S` is defined as a subgroup of :math:`\mathcal{P}_n` that does satisfies the following: +# +# 1. It contains the identity operator :math:`I_0 \otimes I_1 \otimes \cdots \otimes I_{n-1}`. +# 2. All elements of :math:`S` commute with each other. +# 3. The product of any two elements in :math:`S` is also in :math:`S` (i.e., it forms a group under matrix multiplication). +# 4. It does not contain the negative identity operator :math:`-I^{\otimes n}`. +# +# Rather than listing all the elements of a stabilizer group explicitly, the group can be more succinctly specified via a +# set of **generators**: a minimal set of operators in the stabilizer group that can produce all # the other elements through pairwise multiplication. As a simple example, consider the stabilizer set # # .. math:: # # S = \left\lbrace I_0 \otimes I_1 \otimes I_2, \ Z_0 \otimes Z_1 \otimes I_2, \ Z_0 \otimes I_1 \otimes Z_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rbrace. # -# We can check that it satisfies the defining properties 1. to 3. The most cumbersome to check is property 3, where we have to take all +# We can check that it satisfies the defining properties 1. to 4. The most cumbersome to check is property 3, where we have to take all # possible products of the elements and check whether the result is in :math:`S.` For example, -# +# # .. math:: -# +# # (Z_0 \otimes Z_1 \otimes I_2)\cdot (I_0 \otimes Z_1 \otimes Z_2) = Z_0 \otimes I_1 \otimes Z_2, \\ # (Z_0 \otimes Z_1 \otimes I_2 )^2 = I_0 \otimes I_1 \otimes I_2, -# +# # and so on. Note that we can obtain all the elements in :math:`S` just from :math:`Z_0 \otimes Z_1 \otimes I_2` and :math:`I_0 \otimes Z_1 \otimes Z_2.` Because # of this property, these elements are **stabilizer generators** for :math:`S`. We write this fact as # @@ -317,19 +339,22 @@ def error_correction(error_wire): # S = \left\langle Z_0 \otimes Z_1 \otimes I_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rangle, # # which reads ":math:`S` *is the stabilizer set generated by the elements* :math:`Z_0 \otimes Z_1 \otimes I_2` *and* :math:`I_0 \otimes Z_1 \otimes Z_2.`" -# It turns out that specifying these generators is enough to completely define an error correcting code. # -# Now that we know how stabilizer generators work, let us create a tool for later use that creates the full stabilizer set from its generators. +# It turns out that specifying these generators is sufficient to completely define the stabilizer group, and +# thereby the corresponding quantum error-correcting code. +# +# Now that we know how stabilizer generators work, let us create a tool for later use that creates the full stabilizer group from its generators. # import itertools from pennylane import X, Y, Z from pennylane import Identity as I + def generate_stabilizer_group(gens, num_wires): group = [] - init_op =I(0) - for i in range(1,num_wires): - init_op = init_op @ I(i) + init_op = I(0) + for i in range(1, num_wires): + init_op = init_op @ I(i) for bits in itertools.product([0, 1], repeat=len(gens)): op = init_op for i, bit in enumerate(bits): @@ -338,54 +363,54 @@ def generate_stabilizer_group(gens, num_wires): group.append(op) return set(group) -generators = [Z(0)@Z(1)@I(2), I(0)@Z(1)@Z(2)] + +generators = [Z(0) @ Z(1) @ I(2), I(0) @ Z(1) @ Z(2)] generate_stabilizer_group(generators, 3) + ############################################################################## # -# Indeed, obtain all the elements of the set by inputting the generators only. Feel free to try out this code with different generators! +# Indeed, obtain all the elements of the group by inputting the generators only. Feel free to try out this code with different generators! # # Defining the codespace # ~~~~~~~~~~~~~~~~~~~~~~~ # -# At this point, the pressing question is, given a set of stabilizer operators, how do we recover the error correction circuit? That is, -# we need to find a way to go from the operator picture to the state picture. Let us recall that the property that inspired using stabilizer generators +# At this point, the pressing question is, how to define the error correction code given a set of stabilizer generators. As far as +# we know stabilizer generators are only a bunch of Pauli strings satisfying some properties. Let us recall that the property that inspired using stabilizer generators # is that they must leave the codewords invariant. For any stabilizer element :math:`S` and codeword :math:`\vert \psi \rangle`, we must # have -# -# .. math:: -# -# S\vert \psi \rangle = \vert \psi \rangle. -# -# The **codespace** is defined as the set made up of all states such that :math:`S_i \vert\psi\rangle = \vert \psi\rangle` for all stabilizer -# generators :math:`S_i.` The **codewords** can then be recovered by choosing an orthogonal basis of the codespace. -# -# With this in mind, the error correcting code can be recovered for the set :math:`\left\lbrace S_i \right\rbrace`, as shown below # -# .. figure:: ../_static/demonstration_assets/stabilizer_codes/general_stabilizer_circuit.png -# :align: center -# :width: 100% +# .. math:: # -# .. +# S\vert \psi \rangle = \vert \psi \rangle. # -# The stabilizer generators act as the controlled operators in the codewords and the measurements in the auxiliary wires yield unique syndromes for the -# Pauli errors that the code deals with. +# The **codespace** is defined as the set made up of all states such that :math:`S_i \vert\psi\rangle = \vert \psi\rangle` for all stabilizer +# group :math:`S_i.` The **codewords** can then be recovered by choosing an orthogonal basis of the codespace. +# For example, for the three-qubit repetition code, the codewords (:math:`\vert 000 \rangle` and :math:`\vert 111 \rangle`) +# can be recovered by from the stabilizer generators :math:`Z_0 \otimes Z_1 \otimes I_2` and :math:`I_0 \otimes Z_1 \otimes Z_2` from table above. # +# There is a one-to-one correspondence between stabilizer groups and the quantum error-correcting codes they define. +# This means we can describe a code entirely by its stabilizer group, using operators rather than listing the codewords +# directly as state vectors. + + # Logical operators # ~~~~~~~~~~~~~~~~~~ # -# Thus far, we have defined the stabilizer generators, which correspond to the operators that implement syndrome measurements, and the -# codewords, which are states invariant under the stabilizers and hence usable as logical qubits. One missing ingredient are gates we can implement -# on logical qubits without leaving the codespace. Such operators would act non-trivially on the codewords, so they cannot be in the -# stabilizer set, but have to preserve the codespace, meaning they **must commute** with all the stabilizer generators. In particular -# we are interested in the logical Pauli :math:`\bar{X}` and :math:`\bar{Z}` operators, defined by +# +# So far, we have introduced the stabilizer generators, which define the syndrome measurements, and the codewords, +# which are the states left unchanged by all stabilizers. What remains is to understand how to perform computation on the +# encoded qubits, specifically, how to apply gates that act on logical qubits without leaving the codespace. +# These operators must act non-trivially on the codewords, so they cannot be part of the stabilizer group. +# However, to preserve the codespace, they **must commute** with all stabilizer generators. +# In particular, we are interested in the logical Pauli operators \:math:`\bar{X}` and \:math:`\bar{Z}`, defined by: # # .. math:: -# -# \bar{X}\vert \bar{0} \rangle = \vert {1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert {0} \rangle \\ -# \ \bar{Z}\vert \bar{0} \rangle = \vert {0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert {1} \rangle # -# For example, in the three qubit bit flip error correcting code, the logical operators are :math:`\bar{X} = X_0 X_1 X_2` and +# \bar{X}\vert \bar{0} \rangle = \vert \bar{1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert \bar{0} \rangle \\ +# \ \bar{Z}\vert \bar{0} \rangle = \vert \bar{0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert \bar{1} \rangle +# +# For example, in the three qubit bit flip error correcting code, the logical operators are :math:`\bar{X} = X_0 X_1 X_2` and # :math:`\bar{Z} = Z_0 Z_1 Z_2,` but they will not always be this simple. In general, given a stabilizer set $S$, the logical # operators for the code satisfy the following properties: # @@ -393,20 +418,36 @@ def generate_stabilizer_group(gens, num_wires): # 2. They are not in the stabilizer group, # 3. They anticommute with each other, which means they act in a non-trivial way on the codewords. # -# The codespace, the logical operators, and the syndrome measurements are all we need to define an error correcting code. We can even write -# a script to classify operators given a set of stabilizer generators. +# L.S.D Theorem +# ~~~~~~~~~~~~~~~~~~ +# Remember that stabilizer group is a subgroup of the Pauli group with some properties. In the stabilizer formalism, +# every Pauli operator acting on the qubits can be categorized based on how it interacts with the stabilizer group. +# The **LSD theorem** states that Pauli operators on encoded qubits can be divided into three types: + + +# * **L**: Logical operators – commute with all stabilizers but act non-trivially on the codewords. +# * **S**: Stabilizers – leave all codewords unchanged. +# * **D**: Destabilizers (errors) – do not commute with at least one stabilizer and take the state out of the codespace. # -def classify_pauli(operator, logical_ops, generators, n_wires): +# This classification helps distinguish between correctable errors, harmless stabilizer actions, and useful logical operations. +# +# So let us now write some code to classify Pauli operators based on the LSD theorem for a given stabilizer generators. + +def classify_pauli(operator, logical_ops, generators, n_wires): allowed_wires = set(range(n_wires)) operator_wires = set(operator.wires) - assert operator_wires.issubset(allowed_wires), "Operator has wires not allowed by the code" + assert operator_wires.issubset(allowed_wires), ( + "Operator has wires not allowed by the code" + ) operator_names = set([op.name for op in operator.decomposition()]) - allowed_operators = set(['Identity', 'PauliX', 'PauliY', 'PauliZ', 'SProd']) + allowed_operators = set(["Identity", "PauliX", "PauliY", "PauliZ", "SProd"]) - assert operator_names.issubset(allowed_operators), "Operator contains an illegal operation" + assert operator_names.issubset(allowed_operators), ( + "Operator contains an illegal operation" + ) stabilizer_group = generate_stabilizer_group(generators, n_wires) @@ -421,62 +462,58 @@ def classify_pauli(operator, logical_ops, generators, n_wires): return f"{operator} is an Error Operator (Destabilizer)." -generators = [Z(0)@Z(1)@I(2), I(0)@Z(1)@Z(2)] -logical_ops = [X(0)@X(1)@X(2), Z(0)@Z(1)@Z(2)] -print(classify_pauli(Z(0)@I(1)@Z(2), logical_ops, generators, 3)) -print(classify_pauli(Y(0)@Y(1)@Y(2), logical_ops, generators, 3)) -print(classify_pauli(X(0)@Y(1)@Z(2), logical_ops, generators, 3)) + +generators = [Z(0) @ Z(1) @ I(2), I(0) @ Z(1) @ Z(2)] +logical_ops = [X(0) @ X(1) @ X(2), Z(0) @ Z(1) @ Z(2)] +print(classify_pauli(Z(0) @ I(1) @ Z(2), logical_ops, generators, 3)) +print(classify_pauli(Y(0) @ Y(1) @ Y(2), logical_ops, generators, 3)) +print(classify_pauli(X(0) @ Y(1) @ Z(2), logical_ops, generators, 3)) ############################################################################## # # .. note:: # In the literature, you may have come across an error correction code being called an ":math:`[n,k]`-stabilizer code." In this notation, the number :math:`n` represents -# the number of physical qubit used to encode the logical qubit. The integer :math:`k` is the number of logical qubits and it is +# the number of physical qubit used to encode the logical qubit. The integer :math:`k` is the number of logical qubits and it is # equal to :math:`1` for all the codes in this demo. It is possible to show that the number of stabilizer generators :math:`m` # is related to :math:`n` and :math:`k` via :math:`m = n - k.` # -# Five-qubit stabilizer code -# --------------------------- +# Example: Five-qubit stabilizer code +# ----------------------------------- # # Unlike other error correcting codes, the 5-qubit code does not have a special name, but it holds a special place as the smallest -# error correcting protocol capable of correcting arbitrary Pauli Errors--unwanted applications of :math:`X,` :math:`Y,` or :math:`Z` gates. -# In this section, we will build it starting from its stabilizer generators: +# error correcting code capable of correcting arbitrary Pauli Errors--unwanted applications of :math:`X,` :math:`Y,` or :math:`Z` +# gates on a single qubit. In this section, we will build and implement the complete error correction procedure +# starting from its stabilizer generators: # # .. math:: # -# S = \langle S_0, \ S_1, \ S_2, \ S_3 \rangle, +# S = \langle S_0, \ S_1, \ S_2, \ S_3 \rangle, # -# with +# with # # .. math:: -# +# # S_0 = X_0 Z_1 Z_2 X_3 I_4,\\ # S_1 = I_0 X_1 Z_2 Z_3 X_4,\\ # S_2 = X_0 I_1 X_2 Z_3 Z_4,\\ # S_3 = Z_0 X_1 I_2 Z_3 X_4. # -# The calculations are a bit cumbersome, but with some patience we can find the common :math:`+1`-eigenspace of the stabilizer generators, -# spanned by the codewords +# Note: People usally drop the subscript of 0,1,2 etc as the position of the qubit carries that information. # -# .. math:: -# -# \begin{align*} -# \vert \bar{0}\rangle = &\frac{1}{4}\left(\vert 00000 \rangle \vert 10010 \rangle + \vert 01001 \rangle + \vert 10100 \rangle + \vert 01010 \rangle - \vert 11011 \rangle - \vert 00110 \rangle \right.\\ -# &\left. - \vert 11101 \rangle - \vert 00011\rangle - \vert 11110 \rangle - \vert 01111 \rangle - \vert 10001 \rangle - \vert 01100 \rangle - \vert 10111 \rangle + \vert 00101 \rangle \right) -# \end{align*} +# Encoding the data qubit +# ~~~~~~~~~~~~~~~~~~~~~~~ # -# .. math:: +# First, we need to prepare a data qubit that we want to protect. For this tutorial, we will use the qubit +# :math:`\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle, as our data qubit. # -# \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0} \rangle. -# -# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`\bar{Z}=Z^{\otimes 5}.` With these -# defining features in mind, let us build the five-qubit stabilizer code. First, we need to prepare an arbitrary logical qubit +# The next step is to encode this data qubit into a logical qubit. This is done by **encoding circuit** given below. Notice that we +# do not need to know the logicals to implement the encoding circuit. The circuit is completely determined by the stabilizer generators. +# It is beyond the scope of this tutorial to explain how the circuit is constructed from the stabilizer generators. The state after encoding +# is given by: # # .. math:: # -# \vert \bar{\psi}\rangle = \alpha \vert \bar{0} \rangle + \beta \vert \bar{1} \rangle -# -# starting from the qubit :math:`\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle,` which can be done using the circuit below. +# \vert \bar{\psi}\rangle = \alpha \vert \bar{0} \rangle + \beta \vert \bar{1} \rangle # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/five_qubit_encode.png # :align: center @@ -484,80 +521,156 @@ def classify_pauli(operator, logical_ops, generators, n_wires): # # .. # -# This is straightforward to implement in PennyLane. +# Let us implement this encoding circuit in PennyLane. # + def five_qubit_encode(alpha, beta): + qml.StatePrep([alpha, beta], wires=4) + qml.Hadamard(wires=0) + qml.S(wires=0) + qml.CZ(wires=[0, 1]) + qml.CZ(wires=[0, 3]) + qml.CY(wires=[0, 4]) + qml.Hadamard(wires=1) + qml.CZ(wires=[1, 2]) + qml.CZ(wires=[1, 3]) + qml.CNOT(wires=[1, 4]) + qml.Hadamard(wires=2) + qml.CZ(wires=[2, 0]) + qml.CZ(wires=[2, 1]) + qml.CNOT(wires=[2, 4]) + qml.Hadamard(wires=3) + qml.S(wires=3) + qml.CZ(wires=[3, 0]) + qml.CZ(wires=[3, 2]) + qml.CY(wires=[3, 4]) - qml.StatePrep([alpha, beta], wires = 4) - qml.Hadamard(wires = 0) - qml.S(wires = 0) - qml.CZ(wires = [0,1]) - qml.CZ(wires = [0,3]) - qml.CY(wires = [0,4]) - qml.Hadamard(wires = 1) - qml.CZ(wires = [1,2]) - qml.CZ(wires = [1,3]) - qml.CNOT(wires = [1,4]) - qml.Hadamard(wires = 2) - qml.CZ(wires = [2,0]) - qml.CZ(wires = [2,1]) - qml.CNOT(wires = [2,4]) - qml.Hadamard(wires = 3) - qml.S(wires = 3) - qml.CZ(wires = [3,0]) - qml.CZ(wires = [3,2]) - qml.CY(wires = [3,4]) ############################################################################## # -# Having encoded the logical state, we can implement the general circuit built from the stabilizers to obtain the syndrome table like we did with the three-qubit code. +# +# Applying noise +# ~~~~~~~~~~~~~~~~ +# After encoding, the qubit would normally be exposed to noise and decoherence in a real system. +# To simulate this, we introduce an artificial error by randomly flipping one of the physical +# qubits using a Pauli \:math:`X` operation. +# +# PennyLane code for applying noise +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# Syndrome measurement +# ~~~~~~~~~~~~~~~~~~~~ +# +# The next step is to detect the error by measuring the syndrome. The syndrome circuit is given below. +# +# .. figure:: ../_static/demonstration_assets/stabilizer_codes/five_qubit_syndrome.png +# :align: center +# :width: 100% +# +# .. +# +# Let us implement this syndrome measurement circuit in PennyLane. +# +# +# PennyLane code for syndrome measurement +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# +# Now we need to build the syndrome table like we did with the three-qubit code. The syndrome table maps measurement outcomes to +# specific errors, allowing us to detect and correct errors on the encoded qubits. + +dev = qml.device("default.qubit", wires=9, shots=1) -dev = qml.device("default.qubit", wires = 9, shots = 1) +stabilizers = [ + X(0) @ Z(1) @ Z(2) @ X(3) @ I(4), + I(0) @ X(1) @ Z(2) @ Z(3) @ X(4), + X(0) @ I(1) @ X(2) @ Z(3) @ Z(4), + Z(0) @ X(1) @ I(2) @ X(3) @ Z(4), +] -stabilizers = [X(0)@Z(1)@Z(2)@X(3)@I(4), I(0)@X(1)@Z(2)@Z(3)@X(4), - X(0)@I(1)@X(2)@Z(3)@Z(4), Z(0)@X(1)@I(2)@X(3)@Z(4)] @qml.qnode(dev) def five_qubit_code(alpha, beta, error_type, error_wire): - five_qubit_encode(alpha, beta) - if error_type == 'X': - qml.PauliX(wires = error_wire) + if error_type == "X": + qml.PauliX(wires=error_wire) - elif error_type == 'Y': - qml.PauliY(wires = error_wire) + elif error_type == "Y": + qml.PauliY(wires=error_wire) - elif error_type == 'Z': - qml.PauliZ(wires = error_wire) + elif error_type == "Z": + qml.PauliZ(wires=error_wire) - for wire in range(5,9): - qml.Hadamard(wires = wire) + for wire in range(5, 9): + qml.Hadamard(wires=wire) for i in range(len(stabilizers)): + qml.ctrl(stabilizers[i], control=[i + 5]) - qml.ctrl(stabilizers[i], control = [i + 5]) + for wire in range(5, 9): + qml.Hadamard(wires=wire) - for wire in range(5,9): - qml.Hadamard(wires = wire) + return qml.sample(wires=range(5, 9)) - return qml.sample(wires = range(5,9)) for wire in (0, 1, 2, 3, 4): - - for error in ('X', 'Y', 'Z'): - - print(f"{error} {wire}", five_qubit_code(1/2, np.sqrt(3)/2, error, wire)) + for error in ("X", "Y", "Z"): + print(f"{error} {wire}", five_qubit_code(1 / 2, np.sqrt(3) / 2, error, wire)) ############################################################################## # # The syndrome table is printed, and with we can apply the necessary operators to fix the corresponding Pauli errors. The script above is straightforward -# to generalize to any valid set of stabilizers. +# to generalize to any valid set of stabilizers. +# +# Error correction +# ~~~~~~~~~~~~~~~~ +# +# The last step is to correct the error by applying the appropriate Pauli operators to the encoded qubits. +# +# PennyLane code for error correction +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# +# +# +# Note that to build the encoding, syndrome measurement, and error correction circuits, we did only use the stabilizer generators. +# This is a powerful feature of the stabilizer formalism. It allows us to construct the code from its stabilizer generators +# and then use the code to correct errors. However, we can also find the codewords and logical operatorsdirectly from the stabilizer generators by +# finding the common +1-eigenspace of the stabilizer generators. +# +# The calculations are a bit cumbersome, but with some patience we can find the common :math:`+1`-eigenspace of the stabilizer generators, +# which are the codewords. +# +# .. math:: +# +# \begin{align*} +# \vert \bar{0}\rangle = &\frac{1}{4}\left(\vert 00000 \rangle \vert 10010 \rangle + \vert 01001 \rangle + \vert 10100 \rangle + \vert 01010 \rangle - \vert 11011 \rangle - \vert 00110 \rangle \right.\\ +# &\left. - \vert 11101 \rangle - \vert 00011\rangle - \vert 11110 \rangle - \vert 01111 \rangle - \vert 10001 \rangle - \vert 01100 \rangle - \vert 10111 \rangle + \vert 00101 \rangle \right) +# \end{align*} +# +# .. math:: +# +# \vert \bar{1}\rangle = X\otimes X \otimes X \otimes X \otimes X \vert \bar{0} \rangle. +# +# The logical operators bit-flip and phase-flip are for this code are :math:`\bar{X}= X^{\otimes 5}` and :math:`\bar{Z}=Z^{\otimes 5}. +# +# +# Conclusion +# ~~~~~~~~~~~ +# +# +# In this tutorial, we introduced the stabilizer formalism and showed how it can be used to construct quantum error correction codes. +# We applied it to a concrete example—the five-qubit code—using a PennyLane implementation. +# However, finding the codewords, logical operators, and the encoding circuit directly from the stabilizer generators +# is not straightforward. Take a look at the types of gates used in the encoding circuit: you’ll notice they are all Clifford gates. +# In fact, any standard stabilizer code can be encoded using only Clifford gates, which is a major advantage. Why? +# Because clifford gates are easy to implement and measure. # # References # ----------- # # About the author # ----------------- -# -# +# +# From 7224d60c9086111eaf08364250f8f5c7425f522a Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Mon, 30 Jun 2025 14:26:15 -0400 Subject: [PATCH 36/37] rendering issues --- demonstrations/tutorial_stabilizer_codes.py | 16 ++++------------ 1 file changed, 4 insertions(+), 12 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index f9adc585fa..a6c3b350bb 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -304,7 +304,6 @@ def error_correction(error_wire): # For example, :math:`X_0 Z_1` denotes :math:`X \otimes Z` acting on qubits 0 and 1, respectively. # When identity operators are omitted, we use subscripts to indicate which qubits the non-identity Pauli operators act on. # If all positions are filled (e.g., :math:`XZI`), the position implicitly indicates the qubit index (qubit 0, 1, 2). - # # The stabilizer formalism is a powerful framework for constructing quantum error-correcting codes using the algebraic structure of *Pauli operators*. # It focuses on subgroups of the *Pauli group* on :math:`n` qubits—denoted :math:`\mathcal{P}_n`—which consists of all tensor products of single-qubit Pauli operators :math:`\{I, X, Y, Z\}` (with overall phases :math:`\pm1, \pm i`). @@ -344,7 +343,7 @@ def error_correction(error_wire): # thereby the corresponding quantum error-correcting code. # # Now that we know how stabilizer generators work, let us create a tool for later use that creates the full stabilizer group from its generators. -# + import itertools from pennylane import X, Y, Z from pennylane import Identity as I @@ -392,8 +391,7 @@ def generate_stabilizer_group(gens, num_wires): # There is a one-to-one correspondence between stabilizer groups and the quantum error-correcting codes they define. # This means we can describe a code entirely by its stabilizer group, using operators rather than listing the codewords # directly as state vectors. - - +# # Logical operators # ~~~~~~~~~~~~~~~~~~ # @@ -423,8 +421,8 @@ def generate_stabilizer_group(gens, num_wires): # Remember that stabilizer group is a subgroup of the Pauli group with some properties. In the stabilizer formalism, # every Pauli operator acting on the qubits can be categorized based on how it interacts with the stabilizer group. # The **LSD theorem** states that Pauli operators on encoded qubits can be divided into three types: - - +# +# # * **L**: Logical operators – commute with all stabilizers but act non-trivially on the codewords. # * **S**: Stabilizers – leave all codewords unchanged. # * **D**: Destabilizers (errors) – do not commute with at least one stabilizer and take the state out of the codespace. @@ -522,8 +520,6 @@ def classify_pauli(operator, logical_ops, generators, n_wires): # .. # # Let us implement this encoding circuit in PennyLane. -# - def five_qubit_encode(alpha, beta): qml.StatePrep([alpha, beta], wires=4) @@ -631,9 +627,6 @@ def five_qubit_code(alpha, beta, error_type, error_wire): # PennyLane code for error correction # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # -# -# -# # Note that to build the encoding, syndrome measurement, and error correction circuits, we did only use the stabilizer generators. # This is a powerful feature of the stabilizer formalism. It allows us to construct the code from its stabilizer generators # and then use the code to correct errors. However, we can also find the codewords and logical operatorsdirectly from the stabilizer generators by @@ -659,7 +652,6 @@ def five_qubit_code(alpha, beta, error_type, error_wire): # Conclusion # ~~~~~~~~~~~ # -# # In this tutorial, we introduced the stabilizer formalism and showed how it can be used to construct quantum error correction codes. # We applied it to a concrete example—the five-qubit code—using a PennyLane implementation. # However, finding the codewords, logical operators, and the encoding circuit directly from the stabilizer generators From ed0f5110f5309f3feb05c0605a601947453b2576 Mon Sep 17 00:00:00 2001 From: alvaro-at-xanadu Date: Wed, 2 Jul 2025 15:34:27 -0400 Subject: [PATCH 37/37] some changes to text --- demonstrations/tutorial_stabilizer_codes.py | 71 ++++++++++++--------- 1 file changed, 41 insertions(+), 30 deletions(-) diff --git a/demonstrations/tutorial_stabilizer_codes.py b/demonstrations/tutorial_stabilizer_codes.py index a6c3b350bb..4d0862995d 100644 --- a/demonstrations/tutorial_stabilizer_codes.py +++ b/demonstrations/tutorial_stabilizer_codes.py @@ -15,10 +15,10 @@ **stabilizer codes**, such as the repetition, Shor, Steane, and surface codes, all fall under this formalism. Other codes, like the GKP codes used by Xanadu, lie outside of it. -In this demo, we will introduce the stabilizer formalism using bottom-up approach. We construct +In this demo, we will introduce the stabilizer formalism using bottom-up approach. We build some well-known codes using **stabilizer generators,** from which -the other elements of the code (codewords, syndrome measurements, etc.) can be reconstructed. This enables the construction of a wide range of error correction codes -directly from their stabilizer generators. We will use the quantum circuit formalism to construct the codes. +the other elements of the code (codewords, syndrome measurements, etc.) can be reconstructed. Then, we +represent these codes as quantum circuits and implement them in PennyLane. A toy example: the repetition code @@ -26,7 +26,7 @@ To start with, let us explore the general structure of error correction codes using a simple example: the **three-qubit repetition code.** We will introduce this code as a quantum circuit with definite steps to gain some intuition on how it corrects errors on qubits. -We represent the states of the qubits in the circuit using **state vectors** (the Schrödinger picture). In this formalism, error correction codes follow a simple structure: +We represent the states of the qubits in the circuit using **state vectors**. In this formalism, error correction codes follow a simple structure: - Qubit encoding - Error detection @@ -40,7 +40,7 @@ The first step in an error correction code is **encoding** one abstract or **logical qubit** into a set of many on-device **physical qubits.** The rationale is that, if some external factor changes the state of one of the qubits, the remaining qubits still provide information about the original logical qubit. For example, in the three-qubit repetition code, the logical basis-state -qubits, or **logical codewords**, :math:`\vert \bar{0}\rangle` and :math:`\vert \bar{1}\rangle` are encoded into three physical qubits via +qubits, or **logical codewords**, :math:`\vert \bar{0}\rangle` ("logical 0") and :math:`\vert \bar{1}\rangle` ("logical 1") are encoded into three physical qubits via .. math:: @@ -52,7 +52,7 @@ \alpha \vert \bar{0}\rangle + \beta \vert \bar{1}\rangle \mapsto \alpha \vert 000 \rangle + \beta \vert 111\rangle. -This encoding can be done via the following quantum circuit. It should be explicitly stated that the 0 bar is logical 0 and 1 bar is logical 1. +This encoding can be done via the following quantum circuit. .. figure:: ../_static/demonstration_assets/stabilizer_codes/three_qubit_encode.png :align: center @@ -106,15 +106,12 @@ def encoded_state(alpha, beta): # # X_2 \vert \bar{\psi}\rangle = \alpha \vert 010 \rangle + \beta \vert 101 \rangle # -# How do we detect this error? -# -# As you already know that measuring the state collapses it, we cannot measure the state to detect the error. -# +# How do we detect this error? As we already know, measuring the state collapses it, so we cannot measure the state to detect the error. # # To detect a bit-flip error on one of the physical qubits without disturbing the encoded logical state, we perform a **parity measurement.** # This checks whether all physical qubits are in the same state by comparing them two at a time, without directly measuring them. # Instead, auxiliary qubits are used and measured. For the three-qubit repetition code, this involves measuring two auxiliary qubits -# in the computational basis after applying a series of :math:\textrm{CNOT} gates, as illustrated in the circuit below. +# in the computational basis after applying a series of :math:`\textrm{CNOT}` gates, as illustrated in the circuit below. # # .. figure:: ../_static/demonstration_assets/stabilizer_codes/parity_measurements.png # :align: center @@ -240,13 +237,15 @@ def error_correction(error_wire): # --------------------------------------------------------------- # # We have worked with a simple example (repetition code), but it is quite limited. Indeed, the three-qubit code only works for -# a single bit flip error. There are more powerful codes but they also need more resources. For example, Shor's code -# can correct any error on a single logical qubit but it already needs 9 qubits. To make matters worse, to avoid errors at an acceptable level, +# a single bit flip error. There are more powerful codes, but they also need more resources. For example, Shor's code +# can correct any error on a single logical qubit, but it needs 9 qubits. To make matters worse, to avoid errors at an acceptable level, # the industry standard is about 1000 physical qubits per logical qubit. Even with a few qubits, the encoded states and protocols can become increasingly -# complex, so writing a 1000 qubit state vector seems quite daunting! To deal with these situations, we resort to a different +# complex. Writing a 1000 qubit state vector seems quite daunting! To deal with these situations, we resort to a different # representation of error correction codes, using **Pauli operators** instead of state vectors. # -# It is great time to look at Pauli operators and their properties now, if you are not familiar with them. +# .. note:: +# It is great time to look at `Pauli operators `_ and their properties now, +# if you are not familiar with them. # # To gain some intuition about the operator picture, let us express the three-qubit repetition code in a different way. Using the # identity below, @@ -278,7 +277,7 @@ def error_correction(error_wire): # I_0 Z_1 Z_2 \vert 000 \rangle = \vert 000 \rangle, \quad I_0 Z_1 Z_2 \vert 111 \rangle = \vert 111 \rangle. # # This is great news. As long as no error has occurred, the logical qubits will be left alone. Otherwise, there will be -# some operations applied on the state, but we will be able to fix them via an error correction scheme. The invariance property +# some operations applied on the state, but we will be able to fix them via an error correction scheme. This invariance property # **holds true for and only for the logical codewords.** For any other three-qubit basis states, at least one of these operators will have eigenvalue # :math:`-1`, as shown in the table below. Therefore measuring the eigenvalues of these operators will tell us if an error has occurred. # @@ -298,12 +297,24 @@ def error_correction(error_wire): # Stabilizer generators # ~~~~~~~~~~~~~~~~~~~~~~ # -# Note: It is important to have a minimal understanding of group theory to understand the stabilizer formalism. +# .. admonition:: Groups +# :class: note +# +# It is important to have a minimal understanding of group theory to understand the stabilizer formalism. As a refresher, here is +# the definition of a group. +# +# A group is a set of elements that has: +# 1. an operation that maps two elements a and b of the set into a third element of the set, for example c = a + b, +# 2. an "identity element" e such that e + a = a for any element a, and +# 3. an inverse -a for every element a, such that a + (-a) = e. # -# Notation: In the stabilizer formalism, we often omit explicit tensor product symbols (:math:`\otimes`) for brevity. -# For example, :math:`X_0 Z_1` denotes :math:`X \otimes Z` acting on qubits 0 and 1, respectively. -# When identity operators are omitted, we use subscripts to indicate which qubits the non-identity Pauli operators act on. -# If all positions are filled (e.g., :math:`XZI`), the position implicitly indicates the qubit index (qubit 0, 1, 2). +# .. admonition:: Notation +# :class: note +# +# In the stabilizer formalism, we often omit explicit tensor product symbols (:math:`\otimes`) for brevity. +# For example, :math:`X_0 Z_1` denotes :math:`X \otimes Z` acting on qubits 0 and 1, respectively. +# When identity operators are omitted, we use subscripts to indicate which qubits the non-identity Pauli operators act on. +# If all positions are filled (e.g., :math:`XZI`), the position implicitly indicates the qubit index (qubit 0, 1, 2). # # The stabilizer formalism is a powerful framework for constructing quantum error-correcting codes using the algebraic structure of *Pauli operators*. # It focuses on subgroups of the *Pauli group* on :math:`n` qubits—denoted :math:`\mathcal{P}_n`—which consists of all tensor products of single-qubit Pauli operators :math:`\{I, X, Y, Z\}` (with overall phases :math:`\pm1, \pm i`). @@ -337,7 +348,7 @@ def error_correction(error_wire): # # S = \left\langle Z_0 \otimes Z_1 \otimes I_2, \ I_0 \otimes Z_1 \otimes Z_2 \right\rangle, # -# which reads ":math:`S` *is the stabilizer set generated by the elements* :math:`Z_0 \otimes Z_1 \otimes I_2` *and* :math:`I_0 \otimes Z_1 \otimes Z_2.`" +# which reads ":math:`S` *is the stabilizer group generated by the elements* :math:`Z_0 \otimes Z_1 \otimes I_2` *and* :math:`I_0 \otimes Z_1 \otimes Z_2.`" # # It turns out that specifying these generators is sufficient to completely define the stabilizer group, and # thereby the corresponding quantum error-correcting code. @@ -374,8 +385,8 @@ def generate_stabilizer_group(gens, num_wires): # Defining the codespace # ~~~~~~~~~~~~~~~~~~~~~~~ # -# At this point, the pressing question is, how to define the error correction code given a set of stabilizer generators. As far as -# we know stabilizer generators are only a bunch of Pauli strings satisfying some properties. Let us recall that the property that inspired using stabilizer generators +# At this point, the pressing question is how to define the error correction code given a set of stabilizer generators. As far as +# we know, stabilizer generators are only a bunch of Pauli strings satisfying some properties. Let us recall that the property that inspired using stabilizer generators # is that they must leave the codewords invariant. For any stabilizer element :math:`S` and codeword :math:`\vert \psi \rangle`, we must # have # @@ -384,7 +395,7 @@ def generate_stabilizer_group(gens, num_wires): # S\vert \psi \rangle = \vert \psi \rangle. # # The **codespace** is defined as the set made up of all states such that :math:`S_i \vert\psi\rangle = \vert \psi\rangle` for all stabilizer -# group :math:`S_i.` The **codewords** can then be recovered by choosing an orthogonal basis of the codespace. +# group elements :math:`S_i.` The **codewords** can then be recovered by choosing an orthogonal basis of the codespace. # For example, for the three-qubit repetition code, the codewords (:math:`\vert 000 \rangle` and :math:`\vert 111 \rangle`) # can be recovered by from the stabilizer generators :math:`Z_0 \otimes Z_1 \otimes I_2` and :math:`I_0 \otimes Z_1 \otimes Z_2` from table above. # @@ -401,12 +412,12 @@ def generate_stabilizer_group(gens, num_wires): # encoded qubits, specifically, how to apply gates that act on logical qubits without leaving the codespace. # These operators must act non-trivially on the codewords, so they cannot be part of the stabilizer group. # However, to preserve the codespace, they **must commute** with all stabilizer generators. -# In particular, we are interested in the logical Pauli operators \:math:`\bar{X}` and \:math:`\bar{Z}`, defined by: +# In particular, we are interested in the logical Pauli operators :math:`\bar{X}` and :math:`\bar{Z}`, defined by: # # .. math:: # -# \bar{X}\vert \bar{0} \rangle = \vert \bar{1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert \bar{0} \rangle \\ -# \ \bar{Z}\vert \bar{0} \rangle = \vert \bar{0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert \bar{1} \rangle +# \bar{X}\vert \bar{0} \rangle = \vert \bar{1} \rangle, \quad \bar{X}\vert \bar{1} \rangle = \vert \bar{0} \rangle, \\ +# \ \bar{Z}\vert \bar{0} \rangle = \vert \bar{0} \rangle, \quad \bar{Z}\vert \bar{1} \rangle = - \vert \bar{1} \rangle. # # For example, in the three qubit bit flip error correcting code, the logical operators are :math:`\bar{X} = X_0 X_1 X_2` and # :math:`\bar{Z} = Z_0 Z_1 Z_2,` but they will not always be this simple. In general, given a stabilizer set $S$, the logical @@ -416,8 +427,8 @@ def generate_stabilizer_group(gens, num_wires): # 2. They are not in the stabilizer group, # 3. They anticommute with each other, which means they act in a non-trivial way on the codewords. # -# L.S.D Theorem -# ~~~~~~~~~~~~~~~~~~ +# Lloyd-Shor-Devetak (LSD) Theorem +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Remember that stabilizer group is a subgroup of the Pauli group with some properties. In the stabilizer formalism, # every Pauli operator acting on the qubits can be categorized based on how it interacts with the stabilizer group. # The **LSD theorem** states that Pauli operators on encoded qubits can be divided into three types:

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