From 6343dee369eb0c26688d27bec50c0af430f89aba Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 14:57:00 -0400 Subject: [PATCH 01/24] =?UTF-8?q?Demo=20files=20for=20"Decoding=20Quantum?= =?UTF-8?q?=20Errors=20on=20the=20Steane=E2=80=91code=20with=20Belief=20Pr?= =?UTF-8?q?opagation=20&=20Catalyst"?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- _static/authors/tom_ginsberg.jpg | Bin 0 -> 2850816 bytes _static/authors/tom_ginsberg.txt | 4 + .../demonstration_assets/bp_catalyst/css.svg | 4 + .../demonstration_assets/bp_catalyst/qec.svg | 4 + .../tutorial_bp_catalyst.metadata.json | 115 ++ demonstrations/tutorial_bp_catalyst.py | 1180 +++++++++++++++++ 6 files changed, 1307 insertions(+) create mode 100644 _static/authors/tom_ginsberg.jpg create mode 100644 _static/authors/tom_ginsberg.txt create mode 100644 _static/demonstration_assets/bp_catalyst/css.svg create mode 100644 _static/demonstration_assets/bp_catalyst/qec.svg 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\ No newline at end of file diff --git a/_static/demonstration_assets/bp_catalyst/qec.svg b/_static/demonstration_assets/bp_catalyst/qec.svg new file mode 100644 index 0000000000..a26de1e73d --- /dev/null +++ b/_static/demonstration_assets/bp_catalyst/qec.svg @@ -0,0 +1,4 @@ + + + +
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\ No newline at end of file diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json new file mode 100644 index 0000000000..740a045f3a --- /dev/null +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -0,0 +1,115 @@ +{ + "title": "Decoding Quantum Errors on the Steane‑code with Belief Propagation & Catalyst", + "authors": [ + { + "username": "tomginsberg" + } + ], + "dateOfPublication": "2025-05-14T14:49:55.155960-04:00", + "dateOfLastModification": "2025-05-14T14:49:55.155960-04:00", + "categories": ["Quantum Error Correction", "Catalyst"], + "tags": ["quantum error correction", "css codes", "belief propagation"], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_magic-state-distillation.png" + }, + { + "type": "large_thumbnail", + "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_magic-state-distillation.png" + } + ], + "seoDescription": "Learn how to decode CSS codes using belief propagation with JAX, Pennylane and Catalyst", + "doi": "", + "references": [ + { + "id": "Pesah2023", + "type": "webpage", + "title": "The stabilizer trilogy I — Stabilizer codes", + "authors": "Arthur Pesah", + "year": "2023", + "doi": "", + "url": "https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-1/" + }, + { + "id": "Wiberg2001", + "type": "thesis", + "title": "Codes and Decoding on General Graphs", + "authors": "Niclas Wiberg", + "year": "2001", + "doi": "", + "url": "https://www.essrl.wustl.edu/~jao/itrg/wiberg.pdf" + }, + { + "id": "Panteleev2019", + "type": "preprint", + "title": "Degenerate Quantum LDPC Codes With Good Finite Length Performance", + "authors": "Pavel Panteleev", + "year": "2019", + "doi": "", + "url": "https://arxiv.org/abs/1904.02703v3" + }, + { + "id": "Hillmann2024", + "type": "preprint", + "title": "Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes", + "authors": "Timo Hillmann", + "year": "2024", + "doi": "", + "url": "https://arxiv.org/abs/2406.18655v1" + }, + { + "id": "Wolanski2024", + "type": "preprint", + "title": "Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes", + "authors": "Stasiu Wolanski", + "year": "2024", + "doi": "", + "url": "https://arxiv.org/abs/2406.14527v2" + }, + { + "id": "Loeliger2004", + "type": "article-journal", + "title": "An introduction to factor graphs", + "authors": "Hans-Andrea Loeliger", + "year": "2004", + "doi": "", + "url": "https://www.isiweb.ee.ethz.ch/papers/arch/aloe-2004-spmagffg.pdf" + }, + { + "id": "Barber2012", + "type": "book", + "title": "Bayesian Reasoning and Machine Learning", + "authors": "David Barber", + "year": "2012", + "doi": "", + "url": "http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/180325.pdf#page=107.50" + } + ], + "basedOnPapers": [], + "doi": "", + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_magic_state_distillation", + "weight": 0.75 + }, + { + "type": "demonstration", + "id": "tutorial_how_to_use_noise_models", + "weight": 0.25 + }, + { + "type": "demonstration", + "id": "tutorial_mcm_introduction", + "weight": 0.75 + }, + { + "type": "demonstration", + "id": "tutorial_qjit_compile_grovers_algorithm_with_catalyst", + "weight": 0.5 + } + ] + ] +} \ No newline at end of file diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py new file mode 100644 index 0000000000..be40a517fe --- /dev/null +++ b/demonstrations/tutorial_bp_catalyst.py @@ -0,0 +1,1180 @@ +r"""Decoding Quantum Errors on the Steane‑code with Belief Propagation & Catalyst +============================================================================= + +*Learn how to build, simulate, and decode the Steane code using JAX and Catalyst, blending quantum +circuits with fast classical decoders in a seamless workflow.* + +-------------- + +Introduction +------------ + +This tutorial walks you through a simplified error correction cycle using the Steane $[[7,1,3]]$ +code. You’ll encode a logical qubit, introduce noise, extract syndromes, and apply decoding using +two different strategies: a simple lookup table and a belief propagation (BP) decoder. Both decoders +are implemented in JAX and JIT-compiled by Catalyst, allowing everything to run inside a single +``@qml.qjit`` circuit. + +Why is this exciting? Quantum error correction (QEC) is essential for building reliable quantum +computers, but it’s not enough to just run the quantum part. We need fast classical feedback too. +Catalyst helps by fuseing the classical and quantum workflows, giving us one unified, +hardware-agnostic kernel that runs on CPUs, GPUs, and beyond. + +What We’ll Build +---------------- + +By the end of this tutorial, you’ll have: + +- **Encoded** a logical $|0⟩$ using the Steane code. +- **Simulated noise** using configurable bit-flip and phase-flip channels. +- **Extracted syndromes** via ancilla-assisted stabilizer measurements. +- **Decoded errors** using: + + - a **Lookup Table (LUT)** decoder + - a **Belief Propagation (BP)** decoder + +- **Benchmarked performance** across different physical error rates. +""" + +###################################################################### +# Understanding Quantum Error Correction +# -------------------------------------- +# +# At its core, QEC protects quantum information by encoding it redundantly. Stabilizer codes are a +# foundational tool for this purpose. Each stabilizer is a multi-qubit Pauli operator (composed of I, +# X, Y, Z gates) that defines parity constraints the code space must satisfy. +# +# Symplectic Representation +# ~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# To formalize stabilizers and errors, we use **symplectic vectors**. For :math:`n` physical qubits, +# any :math:`n`-qubit Pauli operator can be represented as a binary vector of length :math:`2n`: +# +# .. math:: +# +# +# (v | u) = (v_1, v_2, ..., v_n | u_1, u_2, ..., u_n), +# +# where: +# +# - :math:`v_i = 1` if the Pauli has an X on qubit :math:`i` (0 otherwise) +# - :math:`u_i = 1` if the Pauli has a Z on qubit :math:`i` (0 otherwise) +# - :math:`v_i = u_i = 1` if the Pauli has a Y on qubit :math:`i`, since :math:`X \cdot Z \propto Y` +# +# For example, the operator :math:`XZIY` on 3 qubits corresponds to :math:`(1,0,0, 1 | 0,1,0, 1)`. +# The stabilizer group is then generated by a set of such symplectic vectors, forming a **stabilizer +# matrix** of size :math:`m \times 2n` (with :math:`m` generators). +# +# The **commutation condition** between two Pauli operators :math:`(v|u)` and :math:`(v'|u')` is +# captured by their **symplectic inner product**: +# +# .. math:: +# +# +# v \cdot u' + u \cdot v' \pmod{2}. +# +# For a valid stabilizer code, all generators must commute—so the symplectic inner product between any +# two rows of the stabilizer matrix must be zero. +# +# Measurement and Syndrome +# ~~~~~~~~~~~~~~~~~~~~~~~~ +# +# When an error :math:`e` occurs, represented as a symplectic vector :math:`(v_e | u_e)`, the syndrome +# is calculated by taking the symplectic inner product of :math:`e` with each stabilizer generator. +# This produces a syndrome bit :math:`s_i` for each generator: +# +# .. math:: +# +# +# s_i = v_g^{(i)} \cdot u_e + u_g^{(i)} \cdot v_e \pmod{2}, +# +# where :math:`(v_g^{(i)} | u_g^{(i)})` is the :math:`i`-th generator. +# +# Below we show the basic quantum circuit for extracting a syndrome value. The ancilla qubit is +# initialized in the :math:`|+\rangle` state, and controlled operations are applied based on the +# stabilizer generators. Finally, the ancilla is measured in the :math:`X`-basis to obtain the +# syndrome value. Check out Arthur Pesah’s excellent blog post series [#Pesah]\_ on Stabilizer codes +# for a deeper introduction. +# + +from typing import Callable + +import pennylane as qml + +syndromes = ['XXZIZIX', 'XXIIZZI'] +dev = qml.device("lightning.qubit", wires := max(map(len, syndromes)) + 1) + + +@qml.qnode(device=dev) +def ancilla_assisted_syndrome_extraction(syndromes: list[str]): + ancilla = wires - 1 + for i, syndrome in enumerate(syndromes): + qml.Hadamard(ancilla) + for i, s in enumerate(syndrome): + if s == 'X': + qml.CNOT(wires=[ancilla, i]) + elif s == 'Z': + qml.CZ(wires=[ancilla, i]) + qml.Hadamard(ancilla) + qml.measure(ancilla) + qml.Barrier() + ancilla += 1 + + +print(qml.draw(ancilla_assisted_syndrome_extraction, show_all_wires=True)(syndromes)) + +###################################################################### +# Decoding: The Classical Half of QEC +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# Once you have syndrome bits from your stabilizer measurements, you need to figure out what error +# likely occurred. This is the job of the decoder. Formally, given the syndrome, you’re solving for +# the most probable error (usually called maximum likelihood estimate, or MLE for the error). +# +# However, exact MLE decoding depends on the prcise information of your noise model and is generally +# computationally intractable (NP-Hard) because :math:`n` one bit syndrome measurements can take on +# :math:`2^n` unique values–in practice we rely on approximate methods tuned to assumptions about the +# noise model. +# +# .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg +# :: +# :align: center +# :width: 70% +# :alt: A complete quantum error correction (QEC) cycle. A logical state $|\psi\rangle_L$, experiences an error before stabilizers $\mathcal{S}$ are measured. The resulting syndrome is decoded classically to produce a correction ($\mathcal{R}$), which is applied to restore the logical state +# +# CSS Codes: Simplifying the Structure +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# CSS (Calderbank–Shor–Steane) codes are a special class where the stabilizer generators are split +# into **X-type** and **Z-type** operators. Their symplectic vectors look like this: +# +# - **X-type generator**: :math:`(v | 0)` (only Xs) +# - **Z-type generator**: :math:`(0 | u)` (only Zs) +# +# This allows us to represent the stabilizers with two **:math:`m \times n` matrices**: +# +# - :math:`H_X` for X-type generators +# - :math:`H_Z` for Z-type generators +# +# The commutation condition to ensure that all generators are simultaneously observable is: +# +# .. math:: +# +# +# H_X H_Z^T = 0 \pmod{2}, +# +# which ensures that all X and Z stabilizers commute pairwise. When measuring syndromes: +# +# - X-type stabilizers detect **Z errors** via :math:`s_X = H_X e_Z^T \pmod{2}` +# - Z-type stabilizers detect **X errors** via :math:`s_Z = H_Z e_X^T \pmod{2}` +# +# This separation makes decoding modular: you can handle X and Z errors independently. Later, when we +# introduce the Steane code, you’ll see these matrices explicitly and how they simplify syndrome +# calculation and decoding. See a similar diagram below for the CSS code cycle structure. +# +# .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg +# :: +# :align: center +# :width: 70% +# :alt: The Error Correction Cycle on a CSS Code +# + +###################################################################### +# The Steane Code +# --------------- +# +# The Steane code is one of the simplest quantum error correcting codes, a CSS code built from two +# classical Hamming codes that encodes one logical qubit into seven physical qubits that can correct +# any single-qubit error–traditionaly the error correcting ability of a code is referred as the +# distance or :math:`d` and the number of errors a code can correct is +# :math:`\lfloor (d-1)/2 \rfloor`. In this case where our code can correct a single error it is said +# to have distance :math:`3`. The Stean code uses six stabilizer generators: +# +# .. math:: +# +# +# H_X = \begin{bmatrix} +# 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ +# 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ +# 1 & 0 & 1 & 0 & 1 & 0 & 1 +# \end{bmatrix}, \quad +# H_Z = H_X. +# +# We’ll start by implementing two decoding strategies: +# +# - **Lookup Table (LUT)**: Pre-compute minimal corrections for every syndrome (possible for small +# codes like this one). +# - **Belief Propagation (BP)**: An iterative message-passing algorithm run on the code’s Tanner graph +# (a bipartite graph representing the relationships between qubits and stabilizers). It approximates +# the marginal probabilities of errors on each qubit and is more scalable to larger, sparser codes. +# +# We’ll start with the LUT decoder for its simplicity and then explore BP, which is more flexible for +# larger or sparser codes. +# + +###################################################################### +# Lookup‑table (LUT) decoding +# --------------------------- +# +# For the Steane code since there are :math:`3` :math:`X` and :math:`3` :math:`Z` stabilizer +# generators, there are :math:`2^3=8` possible syndromes for both :math:`X` and :math:`Z`. We can +# create a a small table maps each three‑bit syndrome to a weight‑1 error. +# + +import jax.numpy as jnp +from itertools import combinations +from jax.typing import ArrayLike +import jax +from tabulate import tabulate + + +def lookup_decoder(matrix: ArrayLike, max_weight: int = 1): + m, n = matrix.shape + lut = jnp.zeros((1 << m, n), dtype=jnp.int8) + + # fill table with the lowest‑weight correction for each syndrome + # we do this by iterating over all possible weight one errors and computing their corresponding syndromes + for w in range(1, max_weight + 1): + # iterate over all possible weight-w errors + for qs in combinations(range(n), w): + err = jnp.zeros(n, dtype=jnp.int8).at[jnp.array(qs)].set(1) # error mask + syn = (matrix @ err) % 2 # syndrome for this error + idx = jnp.dot(syn, 1 << jnp.arange(m, dtype=jnp.int8)) # syndrome bits to base 10 index + lut = lut.at[idx].set(err) + + @jax.jit + def _decode(syndrome: ArrayLike): + # convert the syndrome to base 10 and look it up in the table + idx = jnp.dot(syndrome, 1 << jnp.arange(m)) + return lut[idx] + + return _decode + + +H_stean = jnp.array([[0, 0, 0, 1, 1, 1, 1], + [0, 1, 1, 0, 0, 1, 1], + [1, 0, 1, 0, 1, 0, 1]], dtype=int) +lut_stean = lookup_decoder(H_stean) + +# we see that the stean code has a nice property where counting up in binary shifts the error to the right +table_data = [] +for i in range(8): + decoded = lut_stean(jnp.array([int(x) for x in f'{i:03b}'])) + table_data.append( + [f'{i:03b}', ''.join(map(str, decoded))] + ) + +print(tabulate(table_data, headers=['Syndrome', 'LUT Error'])) + +###################################################################### +# While this approach is optimal for small codes any larger examples for instance the distance +# :math:`30` rotated surface code which encodes only :math:`1` logical qubits has :math:`450` +# stabilizers for both :math:`X` and :math:`Z`. A full lookup table decoder just for one check type +# would require roughly :math:`2.9\times 10^{35}` entries. +# + +###################################################################### +# Belief-Propagation (BP) Decoder +# ------------------------------- +# +# Belief propagation is an iterative message-passing algorithm used to decode errors by working on the +# **Tanner graph** [#Wiberg]\_ of the code. This graph has two types of nodes: +# +# - **Variable nodes** represent the physical qubits, which may or may not have experienced an error. +# These correspond to the bits of the error vector :math:`e = (e_1, e_2, \dots, e_n)`. +# - **Check nodes** represent stabilizers, which enforce parity constraints on subsets of qubits. Each +# check node corresponds to a row of the parity-check matrix :math:`H`. +# +# There is an edge between a check node :math:`c` and a variable node :math:`v` if and only if +# :math:`H_{cv} = 1`, meaning that qubit :math:`v` participates in stabilizer :math:`c`. +# +# The goal is to estimate the probability that each qubit has been flipped (i.e., that +# :math:`e_v = 1`), given the observed syndrome bits :math:`s_c`. BP updates beliefs iteratively by +# exchanging messages between variable and check nodes. +# +# The Sum-Product Algorithm +# ~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# The BP decoder is based on the **sum-product algorithm**, which computes marginal probabilities over +# the graph. Here’s the procedure: +# +# 1. **Initialization** +# +# Each variable node :math:`v` sends an initial message to its neighboring checks that reflects the +# *intrinsic* belief about whether an error has occurred. This is the log-likelihood ratio (LLR) +# based on the physical error rate :math:`p`: +# +# .. math:: +# +# +# L_0 = \log\frac{1 - p}{p} +# +# This expresses the prior belief: if :math:`p` is small (e.g., 0.01), then :math:`L_0` is +# positive, favoring no error; if :math:`p` is close to 0.5, :math:`L_0` is near zero (no strong +# prior). In general :math:`p` is a parameter of the algorithm that can be tuned to your specific +# noise source. +# +# 2. **Variable-to-Check Update** +# +# Each variable node updates its message to a neighboring check :math:`c` by combining its +# intrinsic belief with the incoming messages from other connected checks: +# +# .. math:: +# +# +# m_{v \to c} = L_0 + \sum_{c' \in N(v) \setminus c} m_{c' \to v} +# +# Here: +# +# - :math:`m_{v \to c}` is the message from variable :math:`v` to check :math:`c`. +# - :math:`N(v)` is the set of checks connected to variable :math:`v`. +# - :math:`m_{c' \to v}` are messages received from neighboring checks other than :math:`c`. +# +# 3. **Check-to-Variable Update** +# +# Each check node updates its message to a neighboring variable :math:`v` based on the syndrome bit +# :math:`s_c` and the incoming messages from the other variables connected to it: +# +# .. math:: +# +# +# m_{c \to v} = (-1)^{s_c} \; 2 \, \operatorname{arctanh} \biggl( \prod_{v' \in N(c) \setminus v} \tanh\frac{m_{v' \to c}}{2} \biggr) +# +# Here: +# +# - :math:`m_{c \to v}` is the message from check :math:`c` to variable :math:`v`. +# - :math:`s_c` is the syndrome bit for check :math:`c` (0 if the stabilizer is satisfied, 1 if +# violated). +# - :math:`N(c)` is the set of variables connected to check :math:`c`. +# - The :math:`\tanh` and :math:`\operatorname{arctanh}` functions implement the **sum-product +# rule** for combining binary parity checks dervied from classical probability theory. +# +# **What’s going on?** This formula tells us: if the product of incoming :math:`\tanh` terms is +# close to +1 or -1, it means there is strong belief about whether the parity is satisfied or +# violated. The :math:`\operatorname{arctanh}` converts that back into an LLR-style message. The +# :math:`(-1)^{s_c}` factor flips the sign if the syndrome is 1, signaling that a parity error was +# detected. +# +# 4. **Iteration** +# +# Steps 2 and 3 are repeated for a fixed number of iterations (e.g., 10–20) or until the messages +# converge (i.e., stop changing significantly). Traditional theory and heuristics in error +# correction say to repeat :math:`BP` roughly on the order of :math:`O(n)`. +# +# 5. **Decision Rule** +# +# After the iterations, each variable node computes its **posterior LLR** by summing its intrinsic +# belief and all incoming messages: +# +# .. math:: +# +# +# L_v = L_0 + \sum_{c \in N(v)} m_{c \to v} +# +# The decoder makes a hard decision: +# +# - If :math:`L_v < 0`, guess :math:`e_v = 1` (error detected). +# - If :math:`L_v > 0`, guess :math:`e_v = 0` (no error). +# +# Why This Works +# ~~~~~~~~~~~~~~ +# +# Belief propagation is exact on **tree-like graphs**, where no cycles exist. However, even on Tanner +# graphs, which are never tree like, it provides a good approximation the maximum-likelihood decoder +# by using only local, iterative computations. However, its performance can degrade when the Tanner +# graph contains many short cycles—as is the case for many popular quantum codes leading to poor +# convergence. In practice further extensions like BP-OSD [#Panteleev]\ *, BP-LSD [#Hillmann]* or +# Ambiguity Clustering [#Wolanski]\_ are used to fix these issues. +# +# See the following summary article [#Loeliger]\_ as well as Chapter 5 in Bayesian Reasoning and +# Machine Learning [#Barber]\_ for a deeper dive into message passing algorithms on graphs. +# +# 4.2 BP in JAX +# ~~~~~~~~~~~~~ +# +# Below, we implement a BP decoder using Jax broken down into it’s core components. +# + +###################################################################### +# Before we can pass messages, we need to know who talks to whom. ``_build_graph`` scans the +# parity‑check matrix once and records, for every variable node, which checks touch it and vice‑versa. +# We convert the neighbour lists to tuples so they become immutable, hashable static data. JAX can +# then embed their values as compile‑time constants in the XLA program and reliably reuse the compiled +# kernel multiple times. A cool thing about JAX/XLA is when using simple static paramters like our +# below, the individual integers it contains are baked into the XLA program as compile‑time constants, +# so we can truely compile a high performance decoder for our specific parity check matrix. +# + +def _build_graph(pcm: ArrayLike) -> tuple[tuple[tuple[int, ...], ...], tuple[tuple[int, ...], ...]]: + """ + Pre‑compute variable‑node and check‑node neighbours. + + Returns + ------- + var_neighbors : tuple[tuple[int, ...], ...] # length = n + check_neighbors : tuple[tuple[int, ...], ...] # length = m + """ + m, n = pcm.shape + vars_, checks_ = [[] for _ in range(n)], [[] for _ in range(m)] + + for c in range(m): + for v in range(n): + if pcm[c, v]: + vars_[v].append(c) + checks_[c].append(v) + + return tuple(map(tuple, vars_)), tuple(map(tuple, checks_)) + +###################################################################### +# A nice way to visulaize this Tanner graph is using the ``networkx`` package. Below is an example on +# the Stean code. +# + +import matplotlib.pyplot as plt +import networkx as nx + +vars, checks = _build_graph(H_stean) +G = nx.Graph() +num_vars = len(vars) +num_checks = len(checks) + +# build the nx graph object from our vars and checks +for v in range(num_vars): + G.add_node(f'v{v}', bipartite=0) +for c in range(num_checks): + G.add_node(f'c{c}', bipartite=1) +for c in range(num_checks): + for v in checks[c]: + G.add_edge(f'c{c}', f'v{v}') + +pos = nx.bipartite_layout(G, nodes=[f'v{i}' for i in range(num_vars)]) + +plt.figure(figsize=(10, 7)) +nx.draw(G, pos, with_labels=True, node_color='skyblue', node_size=500, font_weight='bold') +plt.title('Bipartite Graph for H_stean', fontsize=16) +plt.show() + +###################################################################### +# This next helper ``_c2v_update`` performs one full sweep of check‑to‑variable updates (step 3 of the +# sum‑product algorithm). It takes the previous messages, the syndrome, the neighbour tables, and two +# scalars (``L_int`` for the intrinsic log‑likelihood ratio and ``eps`` for numerical safety). It +# loops only over edges that exist, multiplies the relevant :math:`\operatorname{tanh}` terms, clips +# the product, applies :math:`\operatorname{arctanh}`, and writes the new message into the next +# matrix. +# + +def _c2v_update( + m_c2v_prev: ArrayLike, + syndrome: ArrayLike, + var_nei: tuple[tuple[int, ...], ...], + check_nei: tuple[tuple[int, ...], ...], + L_int: float, + eps: float +) -> ArrayLike: + """ + Compute the next round of check‑to‑variable messages. + """ + m, n = m_c2v_prev.shape + m_c2v_next = jnp.zeros_like(m_c2v_prev) + + # Loop over checks (outer) then their vars (inner) + for c in range(m): + Vc = check_nei[c] + if len(Vc) < 2: + continue # degree‑1 checks carry no new info + + for v in Vc: + prod = 1.0 + # product over all v' ≠ v in this check + for v_p in Vc: + if v_p == v: + continue + incoming = L_int + for c_p in var_nei[v_p]: + if c_p != c: + incoming += m_c2v_prev[c_p, v_p] + prod *= jnp.tanh(incoming / 2.0) + + prod = jnp.clip(prod, -1.0 + eps, 1.0 - eps) + msg = ((-1) ** syndrome[c]) * 2.0 * jnp.arctanh(prod) + m_c2v_next = m_c2v_next.at[c, v].set(msg) + + return m_c2v_next + +###################################################################### +# Once the main loop finishes, we still need a hard decision. ``_posterior_llrs`` folds every final +# check‑to‑variable message for bit ``v`` into its intrinsic LLR, yielding the posterior belief for +# that bit. A negative value means “error likely,” a positive value means “no error.” +# + +from typing import Callable + + +def _posterior_llrs( + m_c2v_final: ArrayLike, + var_nei: tuple[tuple[int, ...], ...], + L_int: float +) -> ArrayLike: + """ + Combine intrinsic LLR with all incoming messages. + """ + n = m_c2v_final.shape[1] + llr = jnp.full(n, L_int) + for v in range(n): + for c in var_nei[v]: + llr = llr.at[v].add(m_c2v_final[c, v]) + return llr + +###################################################################### +# ``build_bp_decoder`` is the main entry point for compiling our decoder. It takes the parity‑check +# matrix and channel error rate, builds the parity graph, pre‑computes the intrinsic LLR, and returns +# a JIT‑compiled function ``_decode``. +# +# Inside ``_decode`` we: +# +# 1. Zero‑initialise all messages. +# 2. Call ``_c2v_update`` inside a ``jax.lax.fori_loop`` for ``max_iter`` rounds. +# 3. Convert final messages to posterior LLRs with ``_posterior_llrs``. +# 4. Output a binary error vector by thresholding the LLRs at zero. +# +# Because the whole ``_decode`` body is wrapped in ``@jax.jit``, the first call compiles everything +# into an XLA kernel; subsequent calls run at full device speed. +# + +def build_bp_decoder( + parity_check_matrix: ArrayLike, + error_rate: float, + max_iter: int = 10, + epsilon: float = 1e-9 +) -> Callable[[ArrayLike], ArrayLike]: + """ + Return a JIT‑compiled BP decoder for the given code and channel. + + Parameters + ---------- + parity_check_matrix : array‑like (m, n) + error_rate : float # BSC crossover probability p + max_iter : int + epsilon : float # numerical safety margin + """ + pcm = jnp.asarray(parity_check_matrix, dtype=jnp.int32) + m, n = pcm.shape + L_int = jnp.log((1.0 - error_rate) / error_rate) + + var_nei, check_nei = _build_graph(pcm) + + @jax.jit + def _decode(syndrome: ArrayLike) -> ArrayLike: + syndrome = jnp.asarray(syndrome, dtype=jnp.int32) + + # Initialise all messages to zero + m_c2v = jnp.zeros((m, n), dtype=jnp.float32) + + # BP loop + def body(_, msgs): + return _c2v_update(msgs, syndrome, var_nei, check_nei, L_int, epsilon) + + m_c2v = jax.lax.fori_loop(0, max_iter, body, m_c2v) + + # Hard decision from posterior LLRs + llr = _posterior_llrs(m_c2v, var_nei, L_int) + return (llr < 0).astype(jnp.int32) + + # optionally we can force our decoder to compile right away by calling it on a test input + _decode(jnp.zeros(m, dtype=jnp.int32)) + + return _decode + +###################################################################### +# Let’s test the performance of the BP decoder on the Stean code compared to the LUT decoder. +# + +bp_stean = build_bp_decoder(H_stean, error_rate=.05, max_iter=7) + +n_bits = H_stean.shape[0] +correct = 0 +total_syndromes = 2 ** n_bits + +table_data = [] +headers = ["Syndrome", "BP Estimated Error", "LUT Exact Error", "Match"] + +for i in range(total_syndromes): + syndrome_binary_string = f'{i:0{n_bits}b}' + s_array = jnp.array([int(x) for x in syndrome_binary_string]) + + # Get error patterns from BP decoder and LUT + bp_pattern = bp_stean(s_array) + lut_pattern = lut_stean(s_array) + match = jnp.all(bp_pattern == lut_pattern) + + bp_pattern_str = "".join(map(str, bp_pattern.tolist())) + lut_pattern_str = "".join(map(str, lut_pattern.tolist())) + + table_data.append([ + syndrome_binary_string, + bp_pattern_str, + lut_pattern_str, + str(match) + ]) + + # Increment correct count if patterns match + if match: + correct += 1 + +print(tabulate(table_data, headers=headers)) + +# Calculate and print the BP accuracy +accuracy = (correct / total_syndromes) * 100 if total_syndromes > 0 else 0 +print(f'\nBP Accuracy: {accuracy:.2f}%') + +###################################################################### +# Before diving into the code, let’s test our belief‑propagation (BP) decoder on a bigger example: the +# `n‑bit repetition code `__. This code stores each +# logical bit by repeating it :math:`n` times—:math:`0 \mapsto 00\ldots0` and +# :math:`1 \mapsto 11\ldots1`. Its parity‑check matrix consists of :math:`n-1` rows, each enforcing +# that two neighbouring bits are equal. Below, we measure how often the BP decoder corrects random +# errors on a 50‑bit repetition code and compare that success rate to an optimal maximum‑likelihood +# (ML) decoder, which simply picks the lower‑weight error pattern consistent with the observed +# syndrome. +# + +import jax +import jax.numpy as jnp +from jax.typing import ArrayLike + + +def rep_code(n: int) -> jnp.ndarray: + """ + Build the (n − 1) × n parity‑check matrix H for the [n, 1] repetition code. + + Each row enforces equality between two neighbouring bits: + H[i] has 1s in positions i and i+1, zeros elsewhere. + """ + # First row: parity check on bits 0 and 1 → [1, 1, 0, 0, …, 0] + first_row = jnp.zeros(n, dtype=jnp.int8).at[jnp.array([0, 1])].set(1) + rows = [first_row] + + # Remaining rows: slide the two‑bit “window” to the right + for _ in range(n - 2): + rows.append(jnp.roll(rows[-1], 1)) # shift previous row by 1 position + + return jnp.stack(rows) # shape = (n‑1, n) + + +@jax.jit +def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: + """ + Minimum‑weight decoder for the repetition code. + + Parameters + ---------- + syndrome : ArrayLike, shape (n‑1,) + The syndrome s = H e (mod 2). + + Returns + ------- + error : ArrayLike, shape (n,) + A lowest‑weight error vector consistent with `syndrome`. + """ + # Candidate 1: assume e[0] = 0, then recover the rest via cumulative XOR. + # e[k+1] = e[k] ⊕ s[k] ⇒ e = [0, cumsum(s) mod 2] + e0 = jnp.concatenate( + (jnp.array([0], dtype=jnp.int32), + jnp.mod(jnp.cumsum(syndrome), 2)) + ) + + # Candidate 2: flip every bit (equivalent to choosing e[0] = 1). + e1 = (e0 + 1) & 1 # fast “add‑one then mod 2” + + # Compare Hamming weights. + w0, w1 = jnp.sum(e0), jnp.sum(e1) + + # Return the lighter candidate (ties resolved in favour of e0). + return jax.lax.cond(w0 <= w1, lambda _: e0, lambda _: e1, operand=None) + +###################################################################### +# We run a short experiment on a :math:`50` bit repetition code. We sample 10,000 random syndromes +# vectors and compute the accuracy of our BP decoder compared to our baseline ``ml_rep_decoder`` +# + +H_rep = rep_code(n := 50) +bp_rep = build_bp_decoder(parity_check_matrix=H_rep, error_rate=.1, max_iter=n) + +# sample random syndromes +N = 10_000 +key = jax.random.PRNGKey(0) +syndromes = jax.random.randint(key, shape=(N, n - 1), minval=0, maxval=2) + +# use jax to map the decoder over the syndromes +# since our decoders are jit compiled jax functions they can be used with jax.vmap +success_rate = jnp.mean( + jnp.all( + jax.vmap(ml_rep_decoder)(syndromes) == jax.vmap(bp_rep)(syndromes), + axis=1 + ) +) + +print(f'Decoding success rate: {success_rate * 100:.2f}%') + +###################################################################### +# 4 Catalyst hybrid kernel +# ------------------------ +# +# Now that we understand a good chunk of theory behind CSS codes, the Steane code and decoding +# algorithms let’s put this into action with Catalyst! +# +# Catalyst lets us build hybrid quantum-classical workflows, compiling both quantum operations and +# classical decoding logic into a single, efficient kernel. We’ll start with a quantum-classical +# circuit to prepare the logical zero state :math:`|0\rangle_L` for our Steane code. This method is +# also general for intializing logical zero states for any CSS codes. +# +# Start with a :math:`+1` eigen state (or stabilizer state) of all the :math:`Z` type stabilizers +# (:math:`|0\ldots 0\rangle` is always stabilized by any :math:`Z` type Pauli operator so is a +# suityable choice). +# +# Then for each X-type generator: - Prepare an ancilla qubits in the :math:`|+\rangle` state. - +# Measure X-type stabilizers using CNOT operations onto an ancilla. - Measure in the :math:`X` basis +# +# Then - Use measurement outcomes (syndromes) to determine necessary corrections using our decoder. - +# Apply Z-type corrections based on decoding results. +# +# This procedures uses projective measurements to force the data qubits to all be in the :math:`+1` +# eigen state of our :math:`X` type generators. Since the state was already a :math:`+1` eigen state +# of our :math:`Z` type generators and my virtue of the CSS code all :math:`X` and :math:`Z` +# generators simulataneously commute we are left with a state in the :math:`+1` eigen space of all the +# generators. +# + +import pennylane as qml +from jax import random +import catalyst + +r, n = H_stean.shape +n_wires = n + r + +dev = qml.device("lightning.qubit", wires=n_wires) + + +def measure_x_stabilizers(H: ArrayLike): + """ + Measure all X stype stabilizersbased on the parity check matrix X then apply Z type corrections from our decoder + :param H: Parity check X matrix + """ + r, n = H.shape + + # Encode logical |0> + # (Hadamard on ancillas, controlled X stabilizers) + for a in range(r): + qml.H(wires=n + a) + for a, row in enumerate(H): + for q, x in enumerate(row): + if x: + qml.CNOT(wires=[n + a, q]) + for a in range(r): + qml.H(wires=n + a) + + # Measure + reset ancillas (X stabilizers) + sx = jnp.stack([catalyst.measure(n + a) for a in range(r)]) + for a, bit in enumerate(sx): + if bit: + qml.PauliX(wires=n + a) # reset ancilla + + # Z‑correction + # Since the BP and LUT decoder + # we're both perfect on the Stean code + # well use the LUT for simplicity + rec_z = lut_stean(sx) + for q, bit in enumerate(rec_z): + if bit: + qml.PauliZ(wires=q) + + +@qml.qjit(autograph=True) +@qml.qnode(dev) +def encode_zero_stean(): + measure_x_stabilizers(H_stean) + return qml.state() + +###################################################################### +# A simple utility function to display the state +# +# @TODO can we move this to a utility file (or better yet make it part of pennylane 😁) +# + +from typing import Optional + + +def state_vector_to_dict( + sv: ArrayLike, + eps: float = 1e-8, + wires: Optional[list[int]] = None, + probability: bool = False, + pretty_print: bool = False +) -> dict[str, float | complex]: + """ + Converts a state vector to a dictionary representation, optionally + filtering by wires and displaying probabilities or amplitudes. + + Args: + sv: The input state vector. + eps: Values with absolute square less than eps are ignored. + wires: An optional list of wire indices to include. + If None, all wires are included. + probability: If True, the dictionary values are probabilities (amplitude squared). + If False, the dictionary values are amplitudes. + pretty_print: If True, prints the state in a table format using tabulate. + + Returns: + A dictionary where keys are computational basis states (strings) + and values are either probabilities (float) or amplitudes (complex or float). + """ + + n_qubits = int(jnp.log2(sv.shape[0])) + d: dict[str, float | complex] = {} + + processed_wires: Optional[set[int]] = set(wires) if wires is not None else None + + for i, amplitude_val in enumerate(sv): + if probability: + val_to_check = jnp.abs(amplitude_val) ** 2 + else: + val_to_check = jnp.abs(amplitude_val) + + if val_to_check > eps: + full_bits = f'{i:0{n_qubits}b}' + + if processed_wires is not None: + # Ensure wires are within valid range [0, n_qubits-1] + if not all(0 <= w < n_qubits for w in processed_wires): + raise ValueError(f"Wire indices in {wires} are out of bounds for {n_qubits} qubits.") + current_bits = "".join( + [bit_char for idx, bit_char in enumerate(full_bits) if idx in processed_wires] + ) + else: + current_bits = full_bits + + if probability: + prob_val = (jnp.abs(amplitude_val) ** 2).item() + d[current_bits] = d.get(current_bits, 0.0) + prob_val # Ensure float for sum + else: + # Warning this may lead to errors when using wires != None and probability=False + amp_item = amplitude_val.item() + d[current_bits] = amp_item + + if pretty_print: + table_data = [] + if probability: + headers = ["State |k〉", "Probability (%)"] + # Sort by probability descending for better readability + sorted_items = sorted(d.items(), key=lambda item: item[1], reverse=True) + for k, v_prob in sorted_items: + if v_prob > eps: # Apply eps again for final display after potential summation + table_data.append([f"|{k}〉", f"{v_prob * 100:.3f}"]) + print("\nState Probabilities:") + print(tabulate(table_data, headers=headers, tablefmt="grid", numalign="decimal", stralign="center")) + else: + headers = ["State |k〉", "Amplitude (real)", "Amplitude (imag)"] + # Sort by bitstring for consistent output + sorted_items = sorted(d.items(), key=lambda item: item[0]) + has_complex = False + has_real = False + for k, v_amp in sorted_items: + amp = complex(v_amp) # Ensure it's complex for consistent processing + if abs(amp) > eps: # Apply eps again for final display + table_data.append([f"|{k}〉", f"{amp.real:.5f}", f"{amp.imag:+.5f}"]) + if abs(amp.imag) > eps: # Check if there's a significant imaginary part + has_complex = True + if abs(amp.real) > eps: + has_real = True + + print("\nState Amplitudes:") + if not has_complex: # Simplify table if all imaginary parts are zero (or very small) + headers = ["State |k〉", "Amplitude"] + simple_table_data = [[row[0], row[1]] for row in table_data] + print(tabulate(simple_table_data, headers=headers, numalign="decimal", stralign="center")) + elif not has_real: # Simplify table if all imaginary parts are zero (or very small) + headers = ["State |k〉", "Amplitude"] + simple_table_data = [[row[0], row[2]] for row in table_data] + print(tabulate(simple_table_data, headers=headers, numalign="decimal", stralign="center")) + else: + print(tabulate(table_data, headers=headers, numalign="decimal", stralign="center")) + + return d + +###################################################################### +# We run the ``encode_zero`` function and see that we recover the `correct logical zero state for the +# Stean code `__: +# +# .. math:: \begin{aligned}|\overline{0}\rangle= & \frac{1}{\sqrt{8}}(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle \\ & +|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle)\end{aligned} +# + +sv_clean = encode_zero_stean() +state_vector_to_dict(sv_clean, pretty_print=True, wires=range(n)); + +###################################################################### +# Simulating Errors and Full Correction +# ------------------------------------- +# +# We’re now ready to wrap everything together: +# +# - Prepare the zero state. +# - Simulate noise using a depolarizing channel. +# - Perform one complete round of stabilizer measurements and corrections. +# + +def noise_channel(n: int, p_err: float, key: random.PRNGKey): + """ + Apply a single‑qubit Pauli noise channel independently to each of `n` qubits. + + For every qubit the channel does: + 0 → I with probability 1 - p_err + 1 → X with probability p_err / 3 + 2 → Z with probability p_err / 3 + 3 → Y with probability p_err / 3 + """ + probs = jnp.array([1.0 - p_err, p_err / 3, p_err / 3, p_err / 3]) + outcomes = random.choice(key, 4, shape=(n,), p=probs) + + for idx, outcome in enumerate(outcomes): + if outcome == 1: + qml.X(wires=idx) + elif outcome == 2: + qml.Z(wires=idx) + elif outcome == 3: + qml.Y(wires=idx) + + +# this is a helper function to get the specific error we used in a given round based on the key +def get_error(n: int, p_err: float, key: random.PRNGKey): + err = [] + probs = jnp.array([1.0 - p_err, p_err / 3, p_err / 3, p_err / 3]) + outcomes = random.choice(key, 4, shape=(n,), p=probs) + + for idx, outcome in enumerate(outcomes): + if outcome == 1: + err.append(qml.X(wires=idx)) + elif outcome == 2: + err.append(qml.Z(wires=idx)) + elif outcome == 3: + err.append(qml.Y(wires=idx)) + return qml.ops.prod(*err) + + +###################################################################### +# Similar to ``measure_x_stabilizers`` but instead now we apply CNOT from data to an ancilla prepared +# in the :math:`|0\rangle` state and perform a :math:`Z` basis measurement. +# + +def measure_z_stabilizers(H): + r, n = H.shape + for a, row in enumerate(H): + for q, x in enumerate(row): + if x: + qml.CNOT(wires=[q, n + a]) + + sz = jnp.stack([catalyst.measure(n + a) for a in range(r)]) + for a, bit in enumerate(sz): + if bit: + qml.PauliX(wires=n + a) + + rec_x = lut_stean(sz) + for q, bit in enumerate(rec_x): + if bit: + qml.PauliX(wires=q) + +###################################################################### +# Now we run the ``qec_round`` using state preparation, one round of noise and one round of :math:`X` +# and :math:`Z` correction. We’ll print the error was was using in our noisy channel and show that the +# output state still looks like the noiseless state we saw above. +# + +@qml.qjit(autograph=True) +@qml.qnode(dev, interface="jax") +def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): + """One round of Steane‑code QEC with LUT decoding.""" + + measure_x_stabilizers(H) # prepare 0 state + noise_channel(n, p_err, key) # inject IID pauli noise + measure_x_stabilizers(H) # correct X errors + measure_z_stabilizers(H) # correct Z errors + + return qml.state() + + +p_err = 0.1 +key = random.PRNGKey(10) +print( + f'Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}' +) +state_vector_to_dict( + qec_round( + H_stean, p_err=p_err, key=key + ), pretty_print=True, wires=range(n) +); + +###################################################################### +# If we increase the likelihood of errors we are more likely to end up with an error pattern that +# can’t be corrected. +# + +p_err = 0.3 +key = random.PRNGKey(8) +print( + f'Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}' +) +state_vector_to_dict( + qec_round( + H_stean, p_err=p_err, key=key + ), pretty_print=True, wires=range(n) +); + +###################################################################### +# Benchmarking logical vs. physical error rates +# --------------------------------------------- +# +# In the final section of this demo we will compute the average performance of our Stean code error +# correction circuit for a range of possible error rates. We’ll define logical error rates by +# comparing the state vector from our noisy simulation to with the clean state vector ``sv_clean`` of +# the Stean code logical zero state. +# + +import pandas as pd +import seaborn as sns +import matplotlib.pyplot as plt +from tqdm.notebook import tqdm + +clean_idx = jnp.where(sv_clean)[0] + + +def logical_error(sv): + st = sv[clean_idx] + return 1 - jnp.all(jnp.isclose(st / st[0], 1)) + +###################################################################### +# Simulate 1000x noisy shots for several noise levels, we’ll use ``jax.vmap`` to efficently map our +# catalyst kernel over a set of random keys. +# + +@jax.jit +def single_trial_error(key, p_err, H_stean_val): + """Performs one QEC round and checks for a logical error.""" + round_output = qec_round(H_stean_val, p_err, key) + err = logical_error(round_output) + return err + + +batch_trial_errors = jax.vmap(single_trial_error, in_axes=(0, None, None)) + +N = 1000 +p_rng = 2**jnp.arange(-5, -1.75, .25, dtype=jnp.float32) +res = [] +master_key = random.PRNGKey(0) + +for p in tqdm(p_rng): + keys_for_batch, master_key = random.split(master_key) + all_keys = random.split(keys_for_batch, N) + + errors_batch = batch_trial_errors(all_keys, p, H_stean) + + p_value = p.item() + for idx, err in enumerate(errors_batch): + res.append( + {'p': p_value, 'seed': idx, 'err': err.item()} + ) + +df = pd.DataFrame(res) + +###################################################################### +# Plot the results using seaborn +# + +p_rng_min = p_rng[0] +p_rng_max = p_rng[-1] + +sns.set_theme(style="whitegrid", context="talk") + +plt.figure(figsize=(10, 7)) +sns.lineplot( + data=df, + x='p', + y='err', + marker='o', + markersize=8, + linewidth=2.5, + label='Simulated Logical Error Rate' +) + +plt.plot( + [p_rng_min, p_rng_max], + [p_rng_min, p_rng_max], + linestyle='--', + color='gray', + linewidth=1.5, + label='$p_{physical} = p_{logical}$' # Label for legend +) +plt.xlabel('Physical Error Rate ($p$)', fontsize=16) +plt.ylabel('Logical Error Rate ($P_L$)', fontsize=16) +plt.xscale('log', base=2) +plt.yscale('log', base=2) +plt.title('Logical vs. Physical Error Rate', fontsize=18, pad=20) +plt.legend(fontsize=14) +plt.grid(True, which="both", ls="--", c='lightgray', alpha=0.7) # 'both' for major and minor ticks +sns.despine() +plt.tight_layout() +plt.show() + +###################################################################### +# Conclusion and Limitations +# -------------------------- +# +# In this tutorial, we’ve successfully built, simulated, and decoded a simple quantum error correction +# cycle using the Steane code. We’ve demonstrated encoding a logical qubit, introduced errors through +# noise simulation, and performed error correction using stabilizer measurements combined with +# classical decoding. The performance was benchmarked by measuring the logical versus physical error +# rates. +# +# However, our approach had a significant simplifying assumption known as the code capacity model, +# where errors are assumed to occur at only one stage of the circuit, with perfect encoding and +# syndrome extraction otherwise. A more realistic approach—called circuit-level noise—accounts for +# potential errors at every gate and measurement within the circuit. This model significantly +# complicates decoding because it requires mapping every possible error location not only in space but +# also across multiple syndrome measurement rounds, forming a complex space-time hypergraph. Decoding +# then involves interpreting error events over both spatial and temporal dimensions. +# +# Nevertheless, the fundamental decoding principles explored here, particularly the Belief Propagation +# (BP) algorithm, remain highly relevant. BP is flexible enough to operate effectively on more full +# circuit level decoding hypergraphs. +# + +###################################################################### +# References +# ---------- +# +# .. [#Pesah] Pesah, Arthur. “The stabilizer trilogy I — Stabilizer codes.” Arthur Pesah, 31 +# Jan. 2023, https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-1/. +# +# .. [#Wiberg] Wiberg, Niclas. (2001). Codes and Decoding on General Graphs. +# https://www.essrl.wustl.edu/~jao/itrg/wiberg.pdf +# +# .. [#Panteleev] Panteleev, Pavel. “Degenerate Quantum LDPC Codes With Good Finite Length +# Performance.” arXiv.org, 04 Apr. 2019, https://arxiv.org/abs/1904.02703v3. +# +# .. [#Hillmann] Hillmann, Timo. “Localized statistics decoding: A parallel decoding algorithm for +# quantum low-density parity-check codes.” arXiv.org, 26 Jun. 2024, +# https://arxiv.org/abs/2406.18655v1. +# +# .. [#Wolanski] Wolanski, Stasiu. “Ambiguity Clustering: an accurate and efficient decoder for qLDPC +# codes.” arXiv.org, 20 Jun. 2024, https://arxiv.org/abs/2406.14527v2. +# +# .. [#Loeliger] Loeliger, Hans-Andrea. “An introduction to factor graphs” in IEEE Signal Processing +# Magazine, vol. 21, no. 1, pp. 28-41, Jan. 2004, +# https://www.isiweb.ee.ethz.ch/papers/arch/aloe-2004-spmagffg.pdf. +# +# .. [#Barber] Barber, David. “Bayesian Reasoning and Machine Learning”. Cambridge University Press, +# USA. 2012, http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/180325.pdf#page=107.50 +# + +###################################################################### +# About the author +# ---------------- +# # .. include:: ../_static/authors/tom_ginsberg.txt From 336b80ae9e7f87639fd8916769145ae34bbaa09e Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:00:53 -0400 Subject: [PATCH 02/24] Ran black code formatting --- demonstrations/tutorial_bp_catalyst.py | 526 ++++++++++++++----------- 1 file changed, 288 insertions(+), 238 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index be40a517fe..088d4d953c 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -39,69 +39,69 @@ ###################################################################### # Understanding Quantum Error Correction # -------------------------------------- -# +# # At its core, QEC protects quantum information by encoding it redundantly. Stabilizer codes are a # foundational tool for this purpose. Each stabilizer is a multi-qubit Pauli operator (composed of I, # X, Y, Z gates) that defines parity constraints the code space must satisfy. -# +# # Symplectic Representation # ~~~~~~~~~~~~~~~~~~~~~~~~~ -# +# # To formalize stabilizers and errors, we use **symplectic vectors**. For :math:`n` physical qubits, # any :math:`n`-qubit Pauli operator can be represented as a binary vector of length :math:`2n`: -# +# # .. math:: -# -# +# +# # (v | u) = (v_1, v_2, ..., v_n | u_1, u_2, ..., u_n), -# +# # where: -# +# # - :math:`v_i = 1` if the Pauli has an X on qubit :math:`i` (0 otherwise) # - :math:`u_i = 1` if the Pauli has a Z on qubit :math:`i` (0 otherwise) # - :math:`v_i = u_i = 1` if the Pauli has a Y on qubit :math:`i`, since :math:`X \cdot Z \propto Y` -# +# # For example, the operator :math:`XZIY` on 3 qubits corresponds to :math:`(1,0,0, 1 | 0,1,0, 1)`. # The stabilizer group is then generated by a set of such symplectic vectors, forming a **stabilizer # matrix** of size :math:`m \times 2n` (with :math:`m` generators). -# +# # The **commutation condition** between two Pauli operators :math:`(v|u)` and :math:`(v'|u')` is # captured by their **symplectic inner product**: -# +# # .. math:: -# -# +# +# # v \cdot u' + u \cdot v' \pmod{2}. -# +# # For a valid stabilizer code, all generators must commute—so the symplectic inner product between any # two rows of the stabilizer matrix must be zero. -# +# # Measurement and Syndrome # ~~~~~~~~~~~~~~~~~~~~~~~~ -# +# # When an error :math:`e` occurs, represented as a symplectic vector :math:`(v_e | u_e)`, the syndrome # is calculated by taking the symplectic inner product of :math:`e` with each stabilizer generator. # This produces a syndrome bit :math:`s_i` for each generator: -# +# # .. math:: -# -# +# +# # s_i = v_g^{(i)} \cdot u_e + u_g^{(i)} \cdot v_e \pmod{2}, -# +# # where :math:`(v_g^{(i)} | u_g^{(i)})` is the :math:`i`-th generator. -# +# # Below we show the basic quantum circuit for extracting a syndrome value. The ancilla qubit is # initialized in the :math:`|+\rangle` state, and controlled operations are applied based on the # stabilizer generators. Finally, the ancilla is measured in the :math:`X`-basis to obtain the # syndrome value. Check out Arthur Pesah’s excellent blog post series [#Pesah]\_ on Stabilizer codes # for a deeper introduction. -# +# from typing import Callable import pennylane as qml -syndromes = ['XXZIZIX', 'XXIIZZI'] +syndromes = ["XXZIZIX", "XXIIZZI"] dev = qml.device("lightning.qubit", wires := max(map(len, syndromes)) + 1) @@ -111,9 +111,9 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): for i, syndrome in enumerate(syndromes): qml.Hadamard(ancilla) for i, s in enumerate(syndrome): - if s == 'X': + if s == "X": qml.CNOT(wires=[ancilla, i]) - elif s == 'Z': + elif s == "Z": qml.CZ(wires=[ancilla, i]) qml.Hadamard(ancilla) qml.measure(ancilla) @@ -126,100 +126,100 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): ###################################################################### # Decoding: The Classical Half of QEC # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# +# # Once you have syndrome bits from your stabilizer measurements, you need to figure out what error # likely occurred. This is the job of the decoder. Formally, given the syndrome, you’re solving for # the most probable error (usually called maximum likelihood estimate, or MLE for the error). -# +# # However, exact MLE decoding depends on the prcise information of your noise model and is generally # computationally intractable (NP-Hard) because :math:`n` one bit syndrome measurements can take on # :math:`2^n` unique values–in practice we rely on approximate methods tuned to assumptions about the # noise model. -# +# # .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg # :: # :align: center # :width: 70% # :alt: A complete quantum error correction (QEC) cycle. A logical state $|\psi\rangle_L$, experiences an error before stabilizers $\mathcal{S}$ are measured. The resulting syndrome is decoded classically to produce a correction ($\mathcal{R}$), which is applied to restore the logical state -# +# # CSS Codes: Simplifying the Structure # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -# +# # CSS (Calderbank–Shor–Steane) codes are a special class where the stabilizer generators are split # into **X-type** and **Z-type** operators. Their symplectic vectors look like this: -# +# # - **X-type generator**: :math:`(v | 0)` (only Xs) # - **Z-type generator**: :math:`(0 | u)` (only Zs) -# +# # This allows us to represent the stabilizers with two **:math:`m \times n` matrices**: -# +# # - :math:`H_X` for X-type generators # - :math:`H_Z` for Z-type generators -# +# # The commutation condition to ensure that all generators are simultaneously observable is: -# +# # .. math:: -# -# +# +# # H_X H_Z^T = 0 \pmod{2}, -# +# # which ensures that all X and Z stabilizers commute pairwise. When measuring syndromes: -# +# # - X-type stabilizers detect **Z errors** via :math:`s_X = H_X e_Z^T \pmod{2}` # - Z-type stabilizers detect **X errors** via :math:`s_Z = H_Z e_X^T \pmod{2}` -# +# # This separation makes decoding modular: you can handle X and Z errors independently. Later, when we # introduce the Steane code, you’ll see these matrices explicitly and how they simplify syndrome # calculation and decoding. See a similar diagram below for the CSS code cycle structure. -# +# # .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg # :: # :align: center # :width: 70% # :alt: The Error Correction Cycle on a CSS Code -# +# ###################################################################### # The Steane Code # --------------- -# +# # The Steane code is one of the simplest quantum error correcting codes, a CSS code built from two # classical Hamming codes that encodes one logical qubit into seven physical qubits that can correct # any single-qubit error–traditionaly the error correcting ability of a code is referred as the # distance or :math:`d` and the number of errors a code can correct is # :math:`\lfloor (d-1)/2 \rfloor`. In this case where our code can correct a single error it is said # to have distance :math:`3`. The Stean code uses six stabilizer generators: -# +# # .. math:: -# -# +# +# # H_X = \begin{bmatrix} # 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ # 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ # 1 & 0 & 1 & 0 & 1 & 0 & 1 # \end{bmatrix}, \quad # H_Z = H_X. -# +# # We’ll start by implementing two decoding strategies: -# +# # - **Lookup Table (LUT)**: Pre-compute minimal corrections for every syndrome (possible for small # codes like this one). # - **Belief Propagation (BP)**: An iterative message-passing algorithm run on the code’s Tanner graph # (a bipartite graph representing the relationships between qubits and stabilizers). It approximates # the marginal probabilities of errors on each qubit and is more scalable to larger, sparser codes. -# +# # We’ll start with the LUT decoder for its simplicity and then explore BP, which is more flexible for # larger or sparser codes. -# +# ###################################################################### # Lookup‑table (LUT) decoding # --------------------------- -# +# # For the Steane code since there are :math:`3` :math:`X` and :math:`3` :math:`Z` stabilizer # generators, there are :math:`2^3=8` possible syndromes for both :math:`X` and :math:`Z`. We can # create a a small table maps each three‑bit syndrome to a weight‑1 error. -# +# import jax.numpy as jnp from itertools import combinations @@ -239,7 +239,9 @@ def lookup_decoder(matrix: ArrayLike, max_weight: int = 1): for qs in combinations(range(n), w): err = jnp.zeros(n, dtype=jnp.int8).at[jnp.array(qs)].set(1) # error mask syn = (matrix @ err) % 2 # syndrome for this error - idx = jnp.dot(syn, 1 << jnp.arange(m, dtype=jnp.int8)) # syndrome bits to base 10 index + idx = jnp.dot( + syn, 1 << jnp.arange(m, dtype=jnp.int8) + ) # syndrome bits to base 10 index lut = lut.at[idx].set(err) @jax.jit @@ -251,149 +253,147 @@ def _decode(syndrome: ArrayLike): return _decode -H_stean = jnp.array([[0, 0, 0, 1, 1, 1, 1], - [0, 1, 1, 0, 0, 1, 1], - [1, 0, 1, 0, 1, 0, 1]], dtype=int) +H_stean = jnp.array( + [[0, 0, 0, 1, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 0, 1]], dtype=int +) lut_stean = lookup_decoder(H_stean) # we see that the stean code has a nice property where counting up in binary shifts the error to the right table_data = [] for i in range(8): - decoded = lut_stean(jnp.array([int(x) for x in f'{i:03b}'])) - table_data.append( - [f'{i:03b}', ''.join(map(str, decoded))] - ) + decoded = lut_stean(jnp.array([int(x) for x in f"{i:03b}"])) + table_data.append([f"{i:03b}", "".join(map(str, decoded))]) -print(tabulate(table_data, headers=['Syndrome', 'LUT Error'])) +print(tabulate(table_data, headers=["Syndrome", "LUT Error"])) ###################################################################### # While this approach is optimal for small codes any larger examples for instance the distance # :math:`30` rotated surface code which encodes only :math:`1` logical qubits has :math:`450` # stabilizers for both :math:`X` and :math:`Z`. A full lookup table decoder just for one check type # would require roughly :math:`2.9\times 10^{35}` entries. -# +# ###################################################################### # Belief-Propagation (BP) Decoder # ------------------------------- -# +# # Belief propagation is an iterative message-passing algorithm used to decode errors by working on the # **Tanner graph** [#Wiberg]\_ of the code. This graph has two types of nodes: -# +# # - **Variable nodes** represent the physical qubits, which may or may not have experienced an error. # These correspond to the bits of the error vector :math:`e = (e_1, e_2, \dots, e_n)`. # - **Check nodes** represent stabilizers, which enforce parity constraints on subsets of qubits. Each # check node corresponds to a row of the parity-check matrix :math:`H`. -# +# # There is an edge between a check node :math:`c` and a variable node :math:`v` if and only if # :math:`H_{cv} = 1`, meaning that qubit :math:`v` participates in stabilizer :math:`c`. -# +# # The goal is to estimate the probability that each qubit has been flipped (i.e., that # :math:`e_v = 1`), given the observed syndrome bits :math:`s_c`. BP updates beliefs iteratively by # exchanging messages between variable and check nodes. -# +# # The Sum-Product Algorithm # ~~~~~~~~~~~~~~~~~~~~~~~~~ -# +# # The BP decoder is based on the **sum-product algorithm**, which computes marginal probabilities over # the graph. Here’s the procedure: -# +# # 1. **Initialization** -# +# # Each variable node :math:`v` sends an initial message to its neighboring checks that reflects the # *intrinsic* belief about whether an error has occurred. This is the log-likelihood ratio (LLR) # based on the physical error rate :math:`p`: -# +# # .. math:: -# -# +# +# # L_0 = \log\frac{1 - p}{p} -# +# # This expresses the prior belief: if :math:`p` is small (e.g., 0.01), then :math:`L_0` is # positive, favoring no error; if :math:`p` is close to 0.5, :math:`L_0` is near zero (no strong # prior). In general :math:`p` is a parameter of the algorithm that can be tuned to your specific # noise source. -# +# # 2. **Variable-to-Check Update** -# +# # Each variable node updates its message to a neighboring check :math:`c` by combining its # intrinsic belief with the incoming messages from other connected checks: -# +# # .. math:: -# -# +# +# # m_{v \to c} = L_0 + \sum_{c' \in N(v) \setminus c} m_{c' \to v} -# +# # Here: -# +# # - :math:`m_{v \to c}` is the message from variable :math:`v` to check :math:`c`. # - :math:`N(v)` is the set of checks connected to variable :math:`v`. # - :math:`m_{c' \to v}` are messages received from neighboring checks other than :math:`c`. -# +# # 3. **Check-to-Variable Update** -# +# # Each check node updates its message to a neighboring variable :math:`v` based on the syndrome bit # :math:`s_c` and the incoming messages from the other variables connected to it: -# +# # .. math:: -# -# +# +# # m_{c \to v} = (-1)^{s_c} \; 2 \, \operatorname{arctanh} \biggl( \prod_{v' \in N(c) \setminus v} \tanh\frac{m_{v' \to c}}{2} \biggr) -# +# # Here: -# +# # - :math:`m_{c \to v}` is the message from check :math:`c` to variable :math:`v`. # - :math:`s_c` is the syndrome bit for check :math:`c` (0 if the stabilizer is satisfied, 1 if # violated). # - :math:`N(c)` is the set of variables connected to check :math:`c`. # - The :math:`\tanh` and :math:`\operatorname{arctanh}` functions implement the **sum-product # rule** for combining binary parity checks dervied from classical probability theory. -# +# # **What’s going on?** This formula tells us: if the product of incoming :math:`\tanh` terms is # close to +1 or -1, it means there is strong belief about whether the parity is satisfied or # violated. The :math:`\operatorname{arctanh}` converts that back into an LLR-style message. The # :math:`(-1)^{s_c}` factor flips the sign if the syndrome is 1, signaling that a parity error was # detected. -# +# # 4. **Iteration** -# +# # Steps 2 and 3 are repeated for a fixed number of iterations (e.g., 10–20) or until the messages # converge (i.e., stop changing significantly). Traditional theory and heuristics in error # correction say to repeat :math:`BP` roughly on the order of :math:`O(n)`. -# +# # 5. **Decision Rule** -# +# # After the iterations, each variable node computes its **posterior LLR** by summing its intrinsic # belief and all incoming messages: -# +# # .. math:: -# -# +# +# # L_v = L_0 + \sum_{c \in N(v)} m_{c \to v} -# +# # The decoder makes a hard decision: -# +# # - If :math:`L_v < 0`, guess :math:`e_v = 1` (error detected). # - If :math:`L_v > 0`, guess :math:`e_v = 0` (no error). -# +# # Why This Works # ~~~~~~~~~~~~~~ -# +# # Belief propagation is exact on **tree-like graphs**, where no cycles exist. However, even on Tanner # graphs, which are never tree like, it provides a good approximation the maximum-likelihood decoder # by using only local, iterative computations. However, its performance can degrade when the Tanner # graph contains many short cycles—as is the case for many popular quantum codes leading to poor # convergence. In practice further extensions like BP-OSD [#Panteleev]\ *, BP-LSD [#Hillmann]* or # Ambiguity Clustering [#Wolanski]\_ are used to fix these issues. -# +# # See the following summary article [#Loeliger]\_ as well as Chapter 5 in Bayesian Reasoning and # Machine Learning [#Barber]\_ for a deeper dive into message passing algorithms on graphs. -# +# # 4.2 BP in JAX # ~~~~~~~~~~~~~ -# +# # Below, we implement a BP decoder using Jax broken down into it’s core components. -# +# ###################################################################### # Before we can pass messages, we need to know who talks to whom. ``_build_graph`` scans the @@ -403,9 +403,12 @@ def _decode(syndrome: ArrayLike): # kernel multiple times. A cool thing about JAX/XLA is when using simple static paramters like our # below, the individual integers it contains are baked into the XLA program as compile‑time constants, # so we can truely compile a high performance decoder for our specific parity check matrix. -# +# + -def _build_graph(pcm: ArrayLike) -> tuple[tuple[tuple[int, ...], ...], tuple[tuple[int, ...], ...]]: +def _build_graph( + pcm: ArrayLike, +) -> tuple[tuple[tuple[int, ...], ...], tuple[tuple[int, ...], ...]]: """ Pre‑compute variable‑node and check‑node neighbours. @@ -425,10 +428,11 @@ def _build_graph(pcm: ArrayLike) -> tuple[tuple[tuple[int, ...], ...], tuple[tup return tuple(map(tuple, vars_)), tuple(map(tuple, checks_)) + ###################################################################### # A nice way to visulaize this Tanner graph is using the ``networkx`` package. Below is an example on # the Stean code. -# +# import matplotlib.pyplot as plt import networkx as nx @@ -440,18 +444,20 @@ def _build_graph(pcm: ArrayLike) -> tuple[tuple[tuple[int, ...], ...], tuple[tup # build the nx graph object from our vars and checks for v in range(num_vars): - G.add_node(f'v{v}', bipartite=0) + G.add_node(f"v{v}", bipartite=0) for c in range(num_checks): - G.add_node(f'c{c}', bipartite=1) + G.add_node(f"c{c}", bipartite=1) for c in range(num_checks): for v in checks[c]: - G.add_edge(f'c{c}', f'v{v}') + G.add_edge(f"c{c}", f"v{v}") -pos = nx.bipartite_layout(G, nodes=[f'v{i}' for i in range(num_vars)]) +pos = nx.bipartite_layout(G, nodes=[f"v{i}" for i in range(num_vars)]) plt.figure(figsize=(10, 7)) -nx.draw(G, pos, with_labels=True, node_color='skyblue', node_size=500, font_weight='bold') -plt.title('Bipartite Graph for H_stean', fontsize=16) +nx.draw( + G, pos, with_labels=True, node_color="skyblue", node_size=500, font_weight="bold" +) +plt.title("Bipartite Graph for H_stean", fontsize=16) plt.show() ###################################################################### @@ -461,15 +467,16 @@ def _build_graph(pcm: ArrayLike) -> tuple[tuple[tuple[int, ...], ...], tuple[tup # loops only over edges that exist, multiplies the relevant :math:`\operatorname{tanh}` terms, clips # the product, applies :math:`\operatorname{arctanh}`, and writes the new message into the next # matrix. -# +# + def _c2v_update( - m_c2v_prev: ArrayLike, - syndrome: ArrayLike, - var_nei: tuple[tuple[int, ...], ...], - check_nei: tuple[tuple[int, ...], ...], - L_int: float, - eps: float + m_c2v_prev: ArrayLike, + syndrome: ArrayLike, + var_nei: tuple[tuple[int, ...], ...], + check_nei: tuple[tuple[int, ...], ...], + L_int: float, + eps: float, ) -> ArrayLike: """ Compute the next round of check‑to‑variable messages. @@ -501,19 +508,18 @@ def _c2v_update( return m_c2v_next + ###################################################################### # Once the main loop finishes, we still need a hard decision. ``_posterior_llrs`` folds every final # check‑to‑variable message for bit ``v`` into its intrinsic LLR, yielding the posterior belief for # that bit. A negative value means “error likely,” a positive value means “no error.” -# +# from typing import Callable def _posterior_llrs( - m_c2v_final: ArrayLike, - var_nei: tuple[tuple[int, ...], ...], - L_int: float + m_c2v_final: ArrayLike, var_nei: tuple[tuple[int, ...], ...], L_int: float ) -> ArrayLike: """ Combine intrinsic LLR with all incoming messages. @@ -525,27 +531,29 @@ def _posterior_llrs( llr = llr.at[v].add(m_c2v_final[c, v]) return llr + ###################################################################### # ``build_bp_decoder`` is the main entry point for compiling our decoder. It takes the parity‑check # matrix and channel error rate, builds the parity graph, pre‑computes the intrinsic LLR, and returns # a JIT‑compiled function ``_decode``. -# +# # Inside ``_decode`` we: -# +# # 1. Zero‑initialise all messages. # 2. Call ``_c2v_update`` inside a ``jax.lax.fori_loop`` for ``max_iter`` rounds. # 3. Convert final messages to posterior LLRs with ``_posterior_llrs``. # 4. Output a binary error vector by thresholding the LLRs at zero. -# +# # Because the whole ``_decode`` body is wrapped in ``@jax.jit``, the first call compiles everything # into an XLA kernel; subsequent calls run at full device speed. -# +# + def build_bp_decoder( - parity_check_matrix: ArrayLike, - error_rate: float, - max_iter: int = 10, - epsilon: float = 1e-9 + parity_check_matrix: ArrayLike, + error_rate: float, + max_iter: int = 10, + epsilon: float = 1e-9, ) -> Callable[[ArrayLike], ArrayLike]: """ Return a JIT‑compiled BP decoder for the given code and channel. @@ -585,21 +593,22 @@ def body(_, msgs): return _decode + ###################################################################### # Let’s test the performance of the BP decoder on the Stean code compared to the LUT decoder. -# +# -bp_stean = build_bp_decoder(H_stean, error_rate=.05, max_iter=7) +bp_stean = build_bp_decoder(H_stean, error_rate=0.05, max_iter=7) n_bits = H_stean.shape[0] correct = 0 -total_syndromes = 2 ** n_bits +total_syndromes = 2**n_bits table_data = [] headers = ["Syndrome", "BP Estimated Error", "LUT Exact Error", "Match"] for i in range(total_syndromes): - syndrome_binary_string = f'{i:0{n_bits}b}' + syndrome_binary_string = f"{i:0{n_bits}b}" s_array = jnp.array([int(x) for x in syndrome_binary_string]) # Get error patterns from BP decoder and LUT @@ -610,12 +619,9 @@ def body(_, msgs): bp_pattern_str = "".join(map(str, bp_pattern.tolist())) lut_pattern_str = "".join(map(str, lut_pattern.tolist())) - table_data.append([ - syndrome_binary_string, - bp_pattern_str, - lut_pattern_str, - str(match) - ]) + table_data.append( + [syndrome_binary_string, bp_pattern_str, lut_pattern_str, str(match)] + ) # Increment correct count if patterns match if match: @@ -625,7 +631,7 @@ def body(_, msgs): # Calculate and print the BP accuracy accuracy = (correct / total_syndromes) * 100 if total_syndromes > 0 else 0 -print(f'\nBP Accuracy: {accuracy:.2f}%') +print(f"\nBP Accuracy: {accuracy:.2f}%") ###################################################################### # Before diving into the code, let’s test our belief‑propagation (BP) decoder on a bigger example: the @@ -636,7 +642,7 @@ def body(_, msgs): # errors on a 50‑bit repetition code and compare that success rate to an optimal maximum‑likelihood # (ML) decoder, which simply picks the lower‑weight error pattern consistent with the observed # syndrome. -# +# import jax import jax.numpy as jnp @@ -679,8 +685,7 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: # Candidate 1: assume e[0] = 0, then recover the rest via cumulative XOR. # e[k+1] = e[k] ⊕ s[k] ⇒ e = [0, cumsum(s) mod 2] e0 = jnp.concatenate( - (jnp.array([0], dtype=jnp.int32), - jnp.mod(jnp.cumsum(syndrome), 2)) + (jnp.array([0], dtype=jnp.int32), jnp.mod(jnp.cumsum(syndrome), 2)) ) # Candidate 2: flip every bit (equivalent to choosing e[0] = 1). @@ -692,13 +697,14 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: # Return the lighter candidate (ties resolved in favour of e0). return jax.lax.cond(w0 <= w1, lambda _: e0, lambda _: e1, operand=None) + ###################################################################### # We run a short experiment on a :math:`50` bit repetition code. We sample 10,000 random syndromes # vectors and compute the accuracy of our BP decoder compared to our baseline ``ml_rep_decoder`` -# +# H_rep = rep_code(n := 50) -bp_rep = build_bp_decoder(parity_check_matrix=H_rep, error_rate=.1, max_iter=n) +bp_rep = build_bp_decoder(parity_check_matrix=H_rep, error_rate=0.1, max_iter=n) # sample random syndromes N = 10_000 @@ -708,42 +714,39 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: # use jax to map the decoder over the syndromes # since our decoders are jit compiled jax functions they can be used with jax.vmap success_rate = jnp.mean( - jnp.all( - jax.vmap(ml_rep_decoder)(syndromes) == jax.vmap(bp_rep)(syndromes), - axis=1 - ) + jnp.all(jax.vmap(ml_rep_decoder)(syndromes) == jax.vmap(bp_rep)(syndromes), axis=1) ) -print(f'Decoding success rate: {success_rate * 100:.2f}%') +print(f"Decoding success rate: {success_rate * 100:.2f}%") ###################################################################### # 4 Catalyst hybrid kernel # ------------------------ -# +# # Now that we understand a good chunk of theory behind CSS codes, the Steane code and decoding # algorithms let’s put this into action with Catalyst! -# +# # Catalyst lets us build hybrid quantum-classical workflows, compiling both quantum operations and # classical decoding logic into a single, efficient kernel. We’ll start with a quantum-classical # circuit to prepare the logical zero state :math:`|0\rangle_L` for our Steane code. This method is # also general for intializing logical zero states for any CSS codes. -# +# # Start with a :math:`+1` eigen state (or stabilizer state) of all the :math:`Z` type stabilizers # (:math:`|0\ldots 0\rangle` is always stabilized by any :math:`Z` type Pauli operator so is a # suityable choice). -# +# # Then for each X-type generator: - Prepare an ancilla qubits in the :math:`|+\rangle` state. - # Measure X-type stabilizers using CNOT operations onto an ancilla. - Measure in the :math:`X` basis -# +# # Then - Use measurement outcomes (syndromes) to determine necessary corrections using our decoder. - # Apply Z-type corrections based on decoding results. -# +# # This procedures uses projective measurements to force the data qubits to all be in the :math:`+1` # eigen state of our :math:`X` type generators. Since the state was already a :math:`+1` eigen state # of our :math:`Z` type generators and my virtue of the CSS code all :math:`X` and :math:`Z` # generators simulataneously commute we are left with a state in the :math:`+1` eigen space of all the # generators. -# +# import pennylane as qml from jax import random @@ -762,7 +765,7 @@ def measure_x_stabilizers(H: ArrayLike): """ r, n = H.shape - # Encode logical |0> + # Encode logical |0> # (Hadamard on ancillas, controlled X stabilizers) for a in range(r): qml.H(wires=n + a) @@ -773,7 +776,7 @@ def measure_x_stabilizers(H: ArrayLike): for a in range(r): qml.H(wires=n + a) - # Measure + reset ancillas (X stabilizers) + # Measure + reset ancillas (X stabilizers) sx = jnp.stack([catalyst.measure(n + a) for a in range(r)]) for a, bit in enumerate(sx): if bit: @@ -795,21 +798,22 @@ def encode_zero_stean(): measure_x_stabilizers(H_stean) return qml.state() + ###################################################################### # A simple utility function to display the state -# +# # @TODO can we move this to a utility file (or better yet make it part of pennylane 😁) -# +# from typing import Optional def state_vector_to_dict( - sv: ArrayLike, - eps: float = 1e-8, - wires: Optional[list[int]] = None, - probability: bool = False, - pretty_print: bool = False + sv: ArrayLike, + eps: float = 1e-8, + wires: Optional[list[int]] = None, + probability: bool = False, + pretty_print: bool = False, ) -> dict[str, float | complex]: """ Converts a state vector to a dictionary representation, optionally @@ -841,21 +845,29 @@ def state_vector_to_dict( val_to_check = jnp.abs(amplitude_val) if val_to_check > eps: - full_bits = f'{i:0{n_qubits}b}' + full_bits = f"{i:0{n_qubits}b}" if processed_wires is not None: # Ensure wires are within valid range [0, n_qubits-1] if not all(0 <= w < n_qubits for w in processed_wires): - raise ValueError(f"Wire indices in {wires} are out of bounds for {n_qubits} qubits.") + raise ValueError( + f"Wire indices in {wires} are out of bounds for {n_qubits} qubits." + ) current_bits = "".join( - [bit_char for idx, bit_char in enumerate(full_bits) if idx in processed_wires] + [ + bit_char + for idx, bit_char in enumerate(full_bits) + if idx in processed_wires + ] ) else: current_bits = full_bits if probability: prob_val = (jnp.abs(amplitude_val) ** 2).item() - d[current_bits] = d.get(current_bits, 0.0) + prob_val # Ensure float for sum + d[current_bits] = ( + d.get(current_bits, 0.0) + prob_val + ) # Ensure float for sum else: # Warning this may lead to errors when using wires != None and probability=False amp_item = amplitude_val.item() @@ -868,10 +880,20 @@ def state_vector_to_dict( # Sort by probability descending for better readability sorted_items = sorted(d.items(), key=lambda item: item[1], reverse=True) for k, v_prob in sorted_items: - if v_prob > eps: # Apply eps again for final display after potential summation + if ( + v_prob > eps + ): # Apply eps again for final display after potential summation table_data.append([f"|{k}〉", f"{v_prob * 100:.3f}"]) print("\nState Probabilities:") - print(tabulate(table_data, headers=headers, tablefmt="grid", numalign="decimal", stralign="center")) + print( + tabulate( + table_data, + headers=headers, + tablefmt="grid", + numalign="decimal", + stralign="center", + ) + ) else: headers = ["State |k〉", "Amplitude (real)", "Amplitude (imag)"] # Sort by bitstring for consistent output @@ -881,46 +903,77 @@ def state_vector_to_dict( for k, v_amp in sorted_items: amp = complex(v_amp) # Ensure it's complex for consistent processing if abs(amp) > eps: # Apply eps again for final display - table_data.append([f"|{k}〉", f"{amp.real:.5f}", f"{amp.imag:+.5f}"]) - if abs(amp.imag) > eps: # Check if there's a significant imaginary part + table_data.append( + [f"|{k}〉", f"{amp.real:.5f}", f"{amp.imag:+.5f}"] + ) + if ( + abs(amp.imag) > eps + ): # Check if there's a significant imaginary part has_complex = True if abs(amp.real) > eps: has_real = True print("\nState Amplitudes:") - if not has_complex: # Simplify table if all imaginary parts are zero (or very small) + if ( + not has_complex + ): # Simplify table if all imaginary parts are zero (or very small) headers = ["State |k〉", "Amplitude"] simple_table_data = [[row[0], row[1]] for row in table_data] - print(tabulate(simple_table_data, headers=headers, numalign="decimal", stralign="center")) - elif not has_real: # Simplify table if all imaginary parts are zero (or very small) + print( + tabulate( + simple_table_data, + headers=headers, + numalign="decimal", + stralign="center", + ) + ) + elif ( + not has_real + ): # Simplify table if all imaginary parts are zero (or very small) headers = ["State |k〉", "Amplitude"] simple_table_data = [[row[0], row[2]] for row in table_data] - print(tabulate(simple_table_data, headers=headers, numalign="decimal", stralign="center")) + print( + tabulate( + simple_table_data, + headers=headers, + numalign="decimal", + stralign="center", + ) + ) else: - print(tabulate(table_data, headers=headers, numalign="decimal", stralign="center")) + print( + tabulate( + table_data, + headers=headers, + numalign="decimal", + stralign="center", + ) + ) return d + ###################################################################### # We run the ``encode_zero`` function and see that we recover the `correct logical zero state for the # Stean code `__: -# +# # .. math:: \begin{aligned}|\overline{0}\rangle= & \frac{1}{\sqrt{8}}(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle \\ & +|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle)\end{aligned} -# +# sv_clean = encode_zero_stean() -state_vector_to_dict(sv_clean, pretty_print=True, wires=range(n)); +state_vector_to_dict(sv_clean, pretty_print=True, wires=range(n)) ###################################################################### # Simulating Errors and Full Correction # ------------------------------------- -# +# # We’re now ready to wrap everything together: -# +# # - Prepare the zero state. # - Simulate noise using a depolarizing channel. # - Perform one complete round of stabilizer measurements and corrections. -# +# + def noise_channel(n: int, p_err: float, key: random.PRNGKey): """ @@ -963,7 +1016,8 @@ def get_error(n: int, p_err: float, key: random.PRNGKey): ###################################################################### # Similar to ``measure_x_stabilizers`` but instead now we apply CNOT from data to an ancilla prepared # in the :math:`|0\rangle` state and perform a :math:`Z` basis measurement. -# +# + def measure_z_stabilizers(H): r, n = H.shape @@ -982,11 +1036,13 @@ def measure_z_stabilizers(H): if bit: qml.PauliX(wires=q) + ###################################################################### # Now we run the ``qec_round`` using state preparation, one round of noise and one round of :math:`X` # and :math:`Z` correction. We’ll print the error was was using in our noisy channel and show that the # output state still looks like the noiseless state we saw above. -# +# + @qml.qjit(autograph=True) @qml.qnode(dev, interface="jax") @@ -1003,40 +1059,32 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.1 key = random.PRNGKey(10) -print( - f'Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}' -) +print(f"Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") state_vector_to_dict( - qec_round( - H_stean, p_err=p_err, key=key - ), pretty_print=True, wires=range(n) -); + qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n) +) ###################################################################### # If we increase the likelihood of errors we are more likely to end up with an error pattern that # can’t be corrected. -# +# p_err = 0.3 key = random.PRNGKey(8) -print( - f'Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}' -) +print(f"Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") state_vector_to_dict( - qec_round( - H_stean, p_err=p_err, key=key - ), pretty_print=True, wires=range(n) -); + qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n) +) ###################################################################### # Benchmarking logical vs. physical error rates # --------------------------------------------- -# +# # In the final section of this demo we will compute the average performance of our Stean code error # correction circuit for a range of possible error rates. We’ll define logical error rates by # comparing the state vector from our noisy simulation to with the clean state vector ``sv_clean`` of # the Stean code logical zero state. -# +# import pandas as pd import seaborn as sns @@ -1050,10 +1098,12 @@ def logical_error(sv): st = sv[clean_idx] return 1 - jnp.all(jnp.isclose(st / st[0], 1)) + ###################################################################### # Simulate 1000x noisy shots for several noise levels, we’ll use ``jax.vmap`` to efficently map our # catalyst kernel over a set of random keys. -# +# + @jax.jit def single_trial_error(key, p_err, H_stean_val): @@ -1066,7 +1116,7 @@ def single_trial_error(key, p_err, H_stean_val): batch_trial_errors = jax.vmap(single_trial_error, in_axes=(0, None, None)) N = 1000 -p_rng = 2**jnp.arange(-5, -1.75, .25, dtype=jnp.float32) +p_rng = 2 ** jnp.arange(-5, -1.75, 0.25, dtype=jnp.float32) res = [] master_key = random.PRNGKey(0) @@ -1078,15 +1128,13 @@ def single_trial_error(key, p_err, H_stean_val): p_value = p.item() for idx, err in enumerate(errors_batch): - res.append( - {'p': p_value, 'seed': idx, 'err': err.item()} - ) + res.append({"p": p_value, "seed": idx, "err": err.item()}) df = pd.DataFrame(res) ###################################################################### # Plot the results using seaborn -# +# p_rng_min = p_rng[0] p_rng_max = p_rng[-1] @@ -1096,29 +1144,31 @@ def single_trial_error(key, p_err, H_stean_val): plt.figure(figsize=(10, 7)) sns.lineplot( data=df, - x='p', - y='err', - marker='o', + x="p", + y="err", + marker="o", markersize=8, linewidth=2.5, - label='Simulated Logical Error Rate' + label="Simulated Logical Error Rate", ) plt.plot( [p_rng_min, p_rng_max], [p_rng_min, p_rng_max], - linestyle='--', - color='gray', + linestyle="--", + color="gray", linewidth=1.5, - label='$p_{physical} = p_{logical}$' # Label for legend + label="$p_{physical} = p_{logical}$", # Label for legend ) -plt.xlabel('Physical Error Rate ($p$)', fontsize=16) -plt.ylabel('Logical Error Rate ($P_L$)', fontsize=16) -plt.xscale('log', base=2) -plt.yscale('log', base=2) -plt.title('Logical vs. Physical Error Rate', fontsize=18, pad=20) +plt.xlabel("Physical Error Rate ($p$)", fontsize=16) +plt.ylabel("Logical Error Rate ($P_L$)", fontsize=16) +plt.xscale("log", base=2) +plt.yscale("log", base=2) +plt.title("Logical vs. Physical Error Rate", fontsize=18, pad=20) plt.legend(fontsize=14) -plt.grid(True, which="both", ls="--", c='lightgray', alpha=0.7) # 'both' for major and minor ticks +plt.grid( + True, which="both", ls="--", c="lightgray", alpha=0.7 +) # 'both' for major and minor ticks sns.despine() plt.tight_layout() plt.show() @@ -1126,13 +1176,13 @@ def single_trial_error(key, p_err, H_stean_val): ###################################################################### # Conclusion and Limitations # -------------------------- -# +# # In this tutorial, we’ve successfully built, simulated, and decoded a simple quantum error correction # cycle using the Steane code. We’ve demonstrated encoding a logical qubit, introduced errors through # noise simulation, and performed error correction using stabilizer measurements combined with # classical decoding. The performance was benchmarked by measuring the logical versus physical error # rates. -# +# # However, our approach had a significant simplifying assumption known as the code capacity model, # where errors are assumed to occur at only one stage of the circuit, with perfect encoding and # syndrome extraction otherwise. A more realistic approach—called circuit-level noise—accounts for @@ -1140,39 +1190,39 @@ def single_trial_error(key, p_err, H_stean_val): # complicates decoding because it requires mapping every possible error location not only in space but # also across multiple syndrome measurement rounds, forming a complex space-time hypergraph. Decoding # then involves interpreting error events over both spatial and temporal dimensions. -# +# # Nevertheless, the fundamental decoding principles explored here, particularly the Belief Propagation # (BP) algorithm, remain highly relevant. BP is flexible enough to operate effectively on more full # circuit level decoding hypergraphs. -# +# ###################################################################### # References # ---------- -# +# # .. [#Pesah] Pesah, Arthur. “The stabilizer trilogy I — Stabilizer codes.” Arthur Pesah, 31 # Jan. 2023, https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-1/. -# +# # .. [#Wiberg] Wiberg, Niclas. (2001). Codes and Decoding on General Graphs. # https://www.essrl.wustl.edu/~jao/itrg/wiberg.pdf -# +# # .. [#Panteleev] Panteleev, Pavel. “Degenerate Quantum LDPC Codes With Good Finite Length # Performance.” arXiv.org, 04 Apr. 2019, https://arxiv.org/abs/1904.02703v3. -# +# # .. [#Hillmann] Hillmann, Timo. “Localized statistics decoding: A parallel decoding algorithm for # quantum low-density parity-check codes.” arXiv.org, 26 Jun. 2024, # https://arxiv.org/abs/2406.18655v1. -# +# # .. [#Wolanski] Wolanski, Stasiu. “Ambiguity Clustering: an accurate and efficient decoder for qLDPC # codes.” arXiv.org, 20 Jun. 2024, https://arxiv.org/abs/2406.14527v2. -# +# # .. [#Loeliger] Loeliger, Hans-Andrea. “An introduction to factor graphs” in IEEE Signal Processing # Magazine, vol. 21, no. 1, pp. 28-41, Jan. 2004, # https://www.isiweb.ee.ethz.ch/papers/arch/aloe-2004-spmagffg.pdf. -# +# # .. [#Barber] Barber, David. “Bayesian Reasoning and Machine Learning”. Cambridge University Press, # USA. 2012, http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/180325.pdf#page=107.50 -# +# ###################################################################### # About the author From 02751a63cafbcdec6682352af960616c1c7ffb70 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:06:37 -0400 Subject: [PATCH 03/24] Prefer using pennylane user name over manual author metadata --- demonstrations/tutorial_bp_catalyst.py | 60 +++++++------------------- 1 file changed, 15 insertions(+), 45 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 088d4d953c..2d3bef8b49 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -239,9 +239,7 @@ def lookup_decoder(matrix: ArrayLike, max_weight: int = 1): for qs in combinations(range(n), w): err = jnp.zeros(n, dtype=jnp.int8).at[jnp.array(qs)].set(1) # error mask syn = (matrix @ err) % 2 # syndrome for this error - idx = jnp.dot( - syn, 1 << jnp.arange(m, dtype=jnp.int8) - ) # syndrome bits to base 10 index + idx = jnp.dot(syn, 1 << jnp.arange(m, dtype=jnp.int8)) # syndrome bits to base 10 index lut = lut.at[idx].set(err) @jax.jit @@ -454,9 +452,7 @@ def _build_graph( pos = nx.bipartite_layout(G, nodes=[f"v{i}" for i in range(num_vars)]) plt.figure(figsize=(10, 7)) -nx.draw( - G, pos, with_labels=True, node_color="skyblue", node_size=500, font_weight="bold" -) +nx.draw(G, pos, with_labels=True, node_color="skyblue", node_size=500, font_weight="bold") plt.title("Bipartite Graph for H_stean", fontsize=16) plt.show() @@ -619,9 +615,7 @@ def body(_, msgs): bp_pattern_str = "".join(map(str, bp_pattern.tolist())) lut_pattern_str = "".join(map(str, lut_pattern.tolist())) - table_data.append( - [syndrome_binary_string, bp_pattern_str, lut_pattern_str, str(match)] - ) + table_data.append([syndrome_binary_string, bp_pattern_str, lut_pattern_str, str(match)]) # Increment correct count if patterns match if match: @@ -684,9 +678,7 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: """ # Candidate 1: assume e[0] = 0, then recover the rest via cumulative XOR. # e[k+1] = e[k] ⊕ s[k] ⇒ e = [0, cumsum(s) mod 2] - e0 = jnp.concatenate( - (jnp.array([0], dtype=jnp.int32), jnp.mod(jnp.cumsum(syndrome), 2)) - ) + e0 = jnp.concatenate((jnp.array([0], dtype=jnp.int32), jnp.mod(jnp.cumsum(syndrome), 2))) # Candidate 2: flip every bit (equivalent to choosing e[0] = 1). e1 = (e0 + 1) & 1 # fast “add‑one then mod 2” @@ -854,20 +846,14 @@ def state_vector_to_dict( f"Wire indices in {wires} are out of bounds for {n_qubits} qubits." ) current_bits = "".join( - [ - bit_char - for idx, bit_char in enumerate(full_bits) - if idx in processed_wires - ] + [bit_char for idx, bit_char in enumerate(full_bits) if idx in processed_wires] ) else: current_bits = full_bits if probability: prob_val = (jnp.abs(amplitude_val) ** 2).item() - d[current_bits] = ( - d.get(current_bits, 0.0) + prob_val - ) # Ensure float for sum + d[current_bits] = d.get(current_bits, 0.0) + prob_val # Ensure float for sum else: # Warning this may lead to errors when using wires != None and probability=False amp_item = amplitude_val.item() @@ -880,9 +866,7 @@ def state_vector_to_dict( # Sort by probability descending for better readability sorted_items = sorted(d.items(), key=lambda item: item[1], reverse=True) for k, v_prob in sorted_items: - if ( - v_prob > eps - ): # Apply eps again for final display after potential summation + if v_prob > eps: # Apply eps again for final display after potential summation table_data.append([f"|{k}〉", f"{v_prob * 100:.3f}"]) print("\nState Probabilities:") print( @@ -903,20 +887,14 @@ def state_vector_to_dict( for k, v_amp in sorted_items: amp = complex(v_amp) # Ensure it's complex for consistent processing if abs(amp) > eps: # Apply eps again for final display - table_data.append( - [f"|{k}〉", f"{amp.real:.5f}", f"{amp.imag:+.5f}"] - ) - if ( - abs(amp.imag) > eps - ): # Check if there's a significant imaginary part + table_data.append([f"|{k}〉", f"{amp.real:.5f}", f"{amp.imag:+.5f}"]) + if abs(amp.imag) > eps: # Check if there's a significant imaginary part has_complex = True if abs(amp.real) > eps: has_real = True print("\nState Amplitudes:") - if ( - not has_complex - ): # Simplify table if all imaginary parts are zero (or very small) + if not has_complex: # Simplify table if all imaginary parts are zero (or very small) headers = ["State |k〉", "Amplitude"] simple_table_data = [[row[0], row[1]] for row in table_data] print( @@ -927,9 +905,7 @@ def state_vector_to_dict( stralign="center", ) ) - elif ( - not has_real - ): # Simplify table if all imaginary parts are zero (or very small) + elif not has_real: # Simplify table if all imaginary parts are zero (or very small) headers = ["State |k〉", "Amplitude"] simple_table_data = [[row[0], row[2]] for row in table_data] print( @@ -1060,9 +1036,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.1 key = random.PRNGKey(10) print(f"Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") -state_vector_to_dict( - qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n) -) +state_vector_to_dict(qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n)) ###################################################################### # If we increase the likelihood of errors we are more likely to end up with an error pattern that @@ -1072,9 +1046,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.3 key = random.PRNGKey(8) print(f"Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") -state_vector_to_dict( - qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n) -) +state_vector_to_dict(qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n)) ###################################################################### # Benchmarking logical vs. physical error rates @@ -1166,9 +1138,7 @@ def single_trial_error(key, p_err, H_stean_val): plt.yscale("log", base=2) plt.title("Logical vs. Physical Error Rate", fontsize=18, pad=20) plt.legend(fontsize=14) -plt.grid( - True, which="both", ls="--", c="lightgray", alpha=0.7 -) # 'both' for major and minor ticks +plt.grid(True, which="both", ls="--", c="lightgray", alpha=0.7) # 'both' for major and minor ticks sns.despine() plt.tight_layout() plt.show() @@ -1227,4 +1197,4 @@ def single_trial_error(key, p_err, H_stean_val): ###################################################################### # About the author # ---------------- -# # .. include:: ../_static/authors/tom_ginsberg.txt +# From 826727697cb48418e802eabc7fbe641274ed3799 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:07:13 -0400 Subject: [PATCH 04/24] Removed non standard whitespace charaters --- demonstrations/tutorial_bp_catalyst.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 2d3bef8b49..050d34c042 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -1049,7 +1049,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): state_vector_to_dict(qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n)) ###################################################################### -# Benchmarking logical vs. physical error rates +# Benchmarking logical vs. physical error rates # --------------------------------------------- # # In the final section of this demo we will compute the average performance of our Stean code error @@ -1171,23 +1171,23 @@ def single_trial_error(key, p_err, H_stean_val): # ---------- # # .. [#Pesah] Pesah, Arthur. “The stabilizer trilogy I — Stabilizer codes.” Arthur Pesah, 31 -# Jan. 2023, https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-1/. +# Jan. 2023, https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-1/. # # .. [#Wiberg] Wiberg, Niclas. (2001). Codes and Decoding on General Graphs. # https://www.essrl.wustl.edu/~jao/itrg/wiberg.pdf # # .. [#Panteleev] Panteleev, Pavel. “Degenerate Quantum LDPC Codes With Good Finite Length -# Performance.” arXiv.org, 04 Apr. 2019, https://arxiv.org/abs/1904.02703v3. +# Performance.” arXiv.org, 04 Apr. 2019, https://arxiv.org/abs/1904.02703v3. # # .. [#Hillmann] Hillmann, Timo. “Localized statistics decoding: A parallel decoding algorithm for -# quantum low-density parity-check codes.” arXiv.org, 26 Jun. 2024, +# quantum low-density parity-check codes.” arXiv.org, 26 Jun. 2024, # https://arxiv.org/abs/2406.18655v1. # # .. [#Wolanski] Wolanski, Stasiu. “Ambiguity Clustering: an accurate and efficient decoder for qLDPC -# codes.” arXiv.org, 20 Jun. 2024, https://arxiv.org/abs/2406.14527v2. +# codes.” arXiv.org, 20 Jun. 2024, https://arxiv.org/abs/2406.14527v2. # # .. [#Loeliger] Loeliger, Hans-Andrea. “An introduction to factor graphs” in IEEE Signal Processing -# Magazine, vol. 21, no. 1, pp. 28-41, Jan. 2004, +# Magazine, vol. 21, no. 1, pp. 28-41, Jan. 2004, # https://www.isiweb.ee.ethz.ch/papers/arch/aloe-2004-spmagffg.pdf. # # .. [#Barber] Barber, David. “Bayesian Reasoning and Machine Learning”. 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zbe_<>cGbf=N!&*gjS-FsCkLPSY`h2zNE9Ot7rud9M>Zz@z|Mz$o;f00e*l5C8%|00;m9AOHk_01yBIKmZ5;0U!VbfB+Bx0zd!=00AHX z1b_e#00KY&2mk>f00e*l5C8%|00;m9AOHk_01yBIKmZ5;0U!VbfB+Bx0zd!=00AHX c1b_e#00KY&2mk>f00e*l5C8%|;QvzKKfs|Yl>h($ diff --git a/_static/authors/tom_ginsberg.txt b/_static/authors/tom_ginsberg.txt deleted file mode 100644 index ec244f4694..0000000000 --- a/_static/authors/tom_ginsberg.txt +++ /dev/null @@ -1,4 +0,0 @@ -.. bio:: Tom Ginsberg - :photo: ../_static/authors/tom_ginsberg.jpg - - Tom leads BEIT's efforts in quantum error correction & fault tolerant compilation. \ No newline at end of file From 16822853b5d67ccecbd362e6bb7b5a61b40b6ba7 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:08:23 -0400 Subject: [PATCH 06/24] Fixed categories to match pennylane approved categories --- demonstrations/tutorial_bp_catalyst.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json index 740a045f3a..137762f983 100644 --- a/demonstrations/tutorial_bp_catalyst.metadata.json +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -7,7 +7,7 @@ ], "dateOfPublication": "2025-05-14T14:49:55.155960-04:00", "dateOfLastModification": "2025-05-14T14:49:55.155960-04:00", - "categories": ["Quantum Error Correction", "Catalyst"], + "categories": ["Quantum Computing"], "tags": ["quantum error correction", "css codes", "belief propagation"], "previewImages": [ { From 8e8a13c55d028ad85750479820f354bd570ae7df Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:11:53 -0400 Subject: [PATCH 07/24] Changed relatedContent, set all weights to 1 per instructions, removed preview images check-jsonschema \ --schemafile metadata_schemas/demo.metadata.schema.0.1.5.json \ demonstrations/tutorial_bp_catalyst.metadata.json fails but does not return any error messages --- .../tutorial_bp_catalyst.metadata.json | 20 ++++--------------- 1 file changed, 4 insertions(+), 16 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json index 137762f983..3b44aacada 100644 --- a/demonstrations/tutorial_bp_catalyst.metadata.json +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -10,14 +10,7 @@ "categories": ["Quantum Computing"], "tags": ["quantum error correction", "css codes", "belief propagation"], "previewImages": [ - { - "type": "thumbnail", - "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_magic-state-distillation.png" - }, - { - "type": "large_thumbnail", - "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_magic-state-distillation.png" - } + ], "seoDescription": "Learn how to decode CSS codes using belief propagation with JAX, Pennylane and Catalyst", "doi": "", @@ -93,22 +86,17 @@ { "type": "demonstration", "id": "tutorial_magic_state_distillation", - "weight": 0.75 - }, - { - "type": "demonstration", - "id": "tutorial_how_to_use_noise_models", - "weight": 0.25 + "weight": 1 }, { "type": "demonstration", "id": "tutorial_mcm_introduction", - "weight": 0.75 + "weight": 1 }, { "type": "demonstration", "id": "tutorial_qjit_compile_grovers_algorithm_with_catalyst", - "weight": 0.5 + "weight": 1 } ] ] From fcfb999c79719f021d3dd6a508b34f558297aa55 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:14:21 -0400 Subject: [PATCH 08/24] rename "Stean-code" to "Stean code" --- demonstrations/tutorial_bp_catalyst.metadata.json | 2 +- demonstrations/tutorial_bp_catalyst.py | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json index 3b44aacada..86d9c5f57b 100644 --- a/demonstrations/tutorial_bp_catalyst.metadata.json +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -1,5 +1,5 @@ { - "title": "Decoding Quantum Errors on the Steane‑code with Belief Propagation & Catalyst", + "title": "Decoding Quantum Errors on the Steane code with Belief Propagation & Catalyst", "authors": [ { "username": "tomginsberg" diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 050d34c042..aaa461fb96 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -1,4 +1,4 @@ -r"""Decoding Quantum Errors on the Steane‑code with Belief Propagation & Catalyst +r"""Decoding Quantum Errors on the Steane code with Belief Propagation & Catalyst ============================================================================= *Learn how to build, simulate, and decode the Steane code using JAX and Catalyst, blending quantum From a6ef3d6f20f9cde82d013e3ccbf1ad48de299a48 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:16:40 -0400 Subject: [PATCH 09/24] replace tqdm notebook with tqdm --- demonstrations/tutorial_bp_catalyst.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index aaa461fb96..404a52dfcf 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -1061,7 +1061,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): import pandas as pd import seaborn as sns import matplotlib.pyplot as plt -from tqdm.notebook import tqdm +from tqdm import tqdm clean_idx = jnp.where(sv_clean)[0] From 7ecdf4c3b8b7de787af98cf9bcb2de50009e49d2 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Wed, 14 May 2025 15:33:02 -0400 Subject: [PATCH 10/24] spellcheck --- demonstrations/tutorial_bp_catalyst.py | 84 +++++++++++++------------- 1 file changed, 41 insertions(+), 43 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 404a52dfcf..6e47eb4672 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -17,7 +17,7 @@ Why is this exciting? Quantum error correction (QEC) is essential for building reliable quantum computers, but it’s not enough to just run the quantum part. We need fast classical feedback too. -Catalyst helps by fuseing the classical and quantum workflows, giving us one unified, +Catalyst helps by fusing the classical and quantum workflows, giving us one unified, hardware-agnostic kernel that runs on CPUs, GPUs, and beyond. What We’ll Build @@ -131,16 +131,15 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # likely occurred. This is the job of the decoder. Formally, given the syndrome, you’re solving for # the most probable error (usually called maximum likelihood estimate, or MLE for the error). # -# However, exact MLE decoding depends on the prcise information of your noise model and is generally +# However, exact MLE decoding depends on the precise information of your noise model and is generally # computationally intractable (NP-Hard) because :math:`n` one bit syndrome measurements can take on # :math:`2^n` unique values–in practice we rely on approximate methods tuned to assumptions about the # noise model. # # .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg -# :: # :align: center # :width: 70% -# :alt: A complete quantum error correction (QEC) cycle. A logical state $|\psi\rangle_L$, experiences an error before stabilizers $\mathcal{S}$ are measured. The resulting syndrome is decoded classically to produce a correction ($\mathcal{R}$), which is applied to restore the logical state +# :alt: A complete quantum error correction (QEC) cycle. A logical state (:math: `|\psi\rangle_L`), experiences an error before stabilizers (:math: `\mathcal{S}`) are measured. The resulting syndrome is decoded classically to produce a correction (:math: `\mathcal{R}`), which is applied to restore the logical state # # CSS Codes: Simplifying the Structure # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -172,8 +171,7 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # introduce the Steane code, you’ll see these matrices explicitly and how they simplify syndrome # calculation and decoding. See a similar diagram below for the CSS code cycle structure. # -# .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg -# :: +# .. figure ../_static/demonstration_assets/bp_catalyst/css.svg # :align: center # :width: 70% # :alt: The Error Correction Cycle on a CSS Code @@ -185,10 +183,10 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # # The Steane code is one of the simplest quantum error correcting codes, a CSS code built from two # classical Hamming codes that encodes one logical qubit into seven physical qubits that can correct -# any single-qubit error–traditionaly the error correcting ability of a code is referred as the +# any single-qubit error–traditionally the error correcting ability of a code is referred as the # distance or :math:`d` and the number of errors a code can correct is # :math:`\lfloor (d-1)/2 \rfloor`. In this case where our code can correct a single error it is said -# to have distance :math:`3`. The Stean code uses six stabilizer generators: +# to have distance :math:`3`. The Steane code uses six stabilizer generators: # # .. math:: # @@ -251,15 +249,15 @@ def _decode(syndrome: ArrayLike): return _decode -H_stean = jnp.array( +H_steane= jnp.array( [[0, 0, 0, 1, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 0, 1, 0, 1, 0, 1]], dtype=int ) -lut_stean = lookup_decoder(H_stean) +lut_steane= lookup_decoder(H_steane) -# we see that the stean code has a nice property where counting up in binary shifts the error to the right +# we see that the steane code has a nice property where counting up in binary shifts the error to the right table_data = [] for i in range(8): - decoded = lut_stean(jnp.array([int(x) for x in f"{i:03b}"])) + decoded = lut_steane(jnp.array([int(x) for x in f"{i:03b}"])) table_data.append([f"{i:03b}", "".join(map(str, decoded))]) print(tabulate(table_data, headers=["Syndrome", "LUT Error"])) @@ -435,7 +433,7 @@ def _build_graph( import matplotlib.pyplot as plt import networkx as nx -vars, checks = _build_graph(H_stean) +vars, checks = _build_graph(H_steane) G = nx.Graph() num_vars = len(vars) num_checks = len(checks) @@ -453,7 +451,7 @@ def _build_graph( plt.figure(figsize=(10, 7)) nx.draw(G, pos, with_labels=True, node_color="skyblue", node_size=500, font_weight="bold") -plt.title("Bipartite Graph for H_stean", fontsize=16) +plt.title("Bipartite Graph for H_steane", fontsize=16) plt.show() ###################################################################### @@ -591,12 +589,12 @@ def body(_, msgs): ###################################################################### -# Let’s test the performance of the BP decoder on the Stean code compared to the LUT decoder. +# Let’s test the performance of the BP decoder on the Steane code compared to the LUT decoder. # -bp_stean = build_bp_decoder(H_stean, error_rate=0.05, max_iter=7) +bp_steane = build_bp_decoder(H_steane, error_rate=0.05, max_iter=7) -n_bits = H_stean.shape[0] +n_bits = H_steane.shape[0] correct = 0 total_syndromes = 2**n_bits @@ -608,8 +606,8 @@ def body(_, msgs): s_array = jnp.array([int(x) for x in syndrome_binary_string]) # Get error patterns from BP decoder and LUT - bp_pattern = bp_stean(s_array) - lut_pattern = lut_stean(s_array) + bp_pattern = bp_steane(s_array) + lut_pattern = lut_steane(s_array) match = jnp.all(bp_pattern == lut_pattern) bp_pattern_str = "".join(map(str, bp_pattern.tolist())) @@ -643,11 +641,11 @@ def body(_, msgs): from jax.typing import ArrayLike -def rep_code(n: int) -> jnp.ndarray: +def rep_code(n: int) -> ArrayLike: """ Build the (n − 1) × n parity‑check matrix H for the [n, 1] repetition code. - Each row enforces equality between two neighbouring bits: + Each row enforces equality between two neighboring bits: H[i] has 1s in positions i and i+1, zeros elsewhere. """ # First row: parity check on bits 0 and 1 → [1, 1, 0, 0, …, 0] @@ -721,11 +719,11 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: # Catalyst lets us build hybrid quantum-classical workflows, compiling both quantum operations and # classical decoding logic into a single, efficient kernel. We’ll start with a quantum-classical # circuit to prepare the logical zero state :math:`|0\rangle_L` for our Steane code. This method is -# also general for intializing logical zero states for any CSS codes. +# also general for initializing logical zero states for any CSS codes. # # Start with a :math:`+1` eigen state (or stabilizer state) of all the :math:`Z` type stabilizers # (:math:`|0\ldots 0\rangle` is always stabilized by any :math:`Z` type Pauli operator so is a -# suityable choice). +# suitable choice). # # Then for each X-type generator: - Prepare an ancilla qubits in the :math:`|+\rangle` state. - # Measure X-type stabilizers using CNOT operations onto an ancilla. - Measure in the :math:`X` basis @@ -736,7 +734,7 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: # This procedures uses projective measurements to force the data qubits to all be in the :math:`+1` # eigen state of our :math:`X` type generators. Since the state was already a :math:`+1` eigen state # of our :math:`Z` type generators and my virtue of the CSS code all :math:`X` and :math:`Z` -# generators simulataneously commute we are left with a state in the :math:`+1` eigen space of all the +# generators simultaneously commute we are left with a state in the :math:`+1` eigen space of all the # generators. # @@ -744,7 +742,7 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: from jax import random import catalyst -r, n = H_stean.shape +r, n = H_steane.shape n_wires = n + r dev = qml.device("lightning.qubit", wires=n_wires) @@ -752,7 +750,7 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: def measure_x_stabilizers(H: ArrayLike): """ - Measure all X stype stabilizersbased on the parity check matrix X then apply Z type corrections from our decoder + Measure all X type stabilizers based on the parity check matrix X then apply Z type corrections from our decoder :param H: Parity check X matrix """ r, n = H.shape @@ -776,9 +774,9 @@ def measure_x_stabilizers(H: ArrayLike): # Z‑correction # Since the BP and LUT decoder - # we're both perfect on the Stean code + # we're both perfect on the Steane code # well use the LUT for simplicity - rec_z = lut_stean(sx) + rec_z = lut_steane(sx) for q, bit in enumerate(rec_z): if bit: qml.PauliZ(wires=q) @@ -786,8 +784,8 @@ def measure_x_stabilizers(H: ArrayLike): @qml.qjit(autograph=True) @qml.qnode(dev) -def encode_zero_stean(): - measure_x_stabilizers(H_stean) +def encode_zero_steane(): + measure_x_stabilizers(H_steane) return qml.state() @@ -936,7 +934,7 @@ def state_vector_to_dict( # .. math:: \begin{aligned}|\overline{0}\rangle= & \frac{1}{\sqrt{8}}(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle \\ & +|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle)\end{aligned} # -sv_clean = encode_zero_stean() +sv_clean = encode_zero_steane() state_vector_to_dict(sv_clean, pretty_print=True, wires=range(n)) ###################################################################### @@ -1007,7 +1005,7 @@ def measure_z_stabilizers(H): if bit: qml.PauliX(wires=n + a) - rec_x = lut_stean(sz) + rec_x = lut_steane(sz) for q, bit in enumerate(rec_x): if bit: qml.PauliX(wires=q) @@ -1023,7 +1021,7 @@ def measure_z_stabilizers(H): @qml.qjit(autograph=True) @qml.qnode(dev, interface="jax") def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): - """One round of Steane‑code QEC with LUT decoding.""" + """One round of Steane code QEC with LUT decoding.""" measure_x_stabilizers(H) # prepare 0 state noise_channel(n, p_err, key) # inject IID pauli noise @@ -1035,8 +1033,8 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.1 key = random.PRNGKey(10) -print(f"Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") -state_vector_to_dict(qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n)) +print(f"Running Steane Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") +state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), pretty_print=True, wires=range(n)) ###################################################################### # If we increase the likelihood of errors we are more likely to end up with an error pattern that @@ -1045,17 +1043,17 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.3 key = random.PRNGKey(8) -print(f"Running Stean Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") -state_vector_to_dict(qec_round(H_stean, p_err=p_err, key=key), pretty_print=True, wires=range(n)) +print(f"Running Steane Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") +state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), pretty_print=True, wires=range(n)) ###################################################################### # Benchmarking logical vs. physical error rates # --------------------------------------------- # -# In the final section of this demo we will compute the average performance of our Stean code error +# In the final section of this demo we will compute the average performance of our Steane code error # correction circuit for a range of possible error rates. We’ll define logical error rates by # comparing the state vector from our noisy simulation to with the clean state vector ``sv_clean`` of -# the Stean code logical zero state. +# the Steane code logical zero state. # import pandas as pd @@ -1072,15 +1070,15 @@ def logical_error(sv): ###################################################################### -# Simulate 1000x noisy shots for several noise levels, we’ll use ``jax.vmap`` to efficently map our +# Simulate 1000x noisy shots for several noise levels, we’ll use ``jax.vmap`` to efficiently map our # catalyst kernel over a set of random keys. # @jax.jit -def single_trial_error(key, p_err, H_stean_val): +def single_trial_error(key, p_err, H_steane): """Performs one QEC round and checks for a logical error.""" - round_output = qec_round(H_stean_val, p_err, key) + round_output = qec_round(H_steane, p_err, key) err = logical_error(round_output) return err @@ -1096,7 +1094,7 @@ def single_trial_error(key, p_err, H_stean_val): keys_for_batch, master_key = random.split(master_key) all_keys = random.split(keys_for_batch, N) - errors_batch = batch_trial_errors(all_keys, p, H_stean) + errors_batch = batch_trial_errors(all_keys, p, H_steane) p_value = p.item() for idx, err in enumerate(errors_batch): From 84ca8ef1aad1774507104c99936a1380992520c0 Mon Sep 17 00:00:00 2001 From: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com> Date: Wed, 14 May 2025 17:04:00 -0400 Subject: [PATCH 11/24] Update demonstrations/tutorial_bp_catalyst.metadata.json Co-authored-by: Paul Finlay <50180049+doctorperceptron@users.noreply.github.com> --- demonstrations/tutorial_bp_catalyst.metadata.json | 1 - 1 file changed, 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json index 86d9c5f57b..80bea86c81 100644 --- a/demonstrations/tutorial_bp_catalyst.metadata.json +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -99,5 +99,4 @@ "weight": 1 } ] - ] } \ No newline at end of file From 197b06f19abdff7f270d3e89f63aa525b6c837d8 Mon Sep 17 00:00:00 2001 From: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com> Date: Wed, 14 May 2025 17:18:14 -0400 Subject: [PATCH 12/24] Apply suggestions from code review --- .../tutorial_bp_catalyst.metadata.json | 23 ++++++++++--------- 1 file changed, 12 insertions(+), 11 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json index 80bea86c81..da92443186 100644 --- a/demonstrations/tutorial_bp_catalyst.metadata.json +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -5,11 +5,19 @@ "username": "tomginsberg" } ], - "dateOfPublication": "2025-05-14T14:49:55.155960-04:00", - "dateOfLastModification": "2025-05-14T14:49:55.155960-04:00", + "dateOfPublication": "2025-05-14T00:00:00+00:00", + "dateOfLastModification": "2025-05-14T00:00:00+00:00", "categories": ["Quantum Computing"], "tags": ["quantum error correction", "css codes", "belief propagation"], "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demonstration_assets/neutral_atoms/thumbnail_tutorial_neutral_atoms.png" + }, + { + "type": "large_thumbnail", + "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_neutral_atoms.png" + } ], "seoDescription": "Learn how to decode CSS codes using belief propagation with JAX, Pennylane and Catalyst", @@ -21,16 +29,14 @@ "title": "The stabilizer trilogy I — Stabilizer codes", "authors": "Arthur Pesah", "year": "2023", - "doi": "", "url": "https://arthurpesah.me/blog/2023-01-31-stabilizer-formalism-1/" }, { "id": "Wiberg2001", - "type": "thesis", + "type": "phdthesis", "title": "Codes and Decoding on General Graphs", "authors": "Niclas Wiberg", "year": "2001", - "doi": "", "url": "https://www.essrl.wustl.edu/~jao/itrg/wiberg.pdf" }, { @@ -39,7 +45,6 @@ "title": "Degenerate Quantum LDPC Codes With Good Finite Length Performance", "authors": "Pavel Panteleev", "year": "2019", - "doi": "", "url": "https://arxiv.org/abs/1904.02703v3" }, { @@ -48,7 +53,6 @@ "title": "Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes", "authors": "Timo Hillmann", "year": "2024", - "doi": "", "url": "https://arxiv.org/abs/2406.18655v1" }, { @@ -57,12 +61,11 @@ "title": "Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes", "authors": "Stasiu Wolanski", "year": "2024", - "doi": "", "url": "https://arxiv.org/abs/2406.14527v2" }, { "id": "Loeliger2004", - "type": "article-journal", + "type": "article", "title": "An introduction to factor graphs", "authors": "Hans-Andrea Loeliger", "year": "2004", @@ -75,12 +78,10 @@ "title": "Bayesian Reasoning and Machine Learning", "authors": "David Barber", "year": "2012", - "doi": "", "url": "http://web4.cs.ucl.ac.uk/staff/D.Barber/textbook/180325.pdf#page=107.50" } ], "basedOnPapers": [], - "doi": "", "referencedByPapers": [], "relatedContent": [ { From 8354eeaf623599965858acf976fd2e5267b1750f Mon Sep 17 00:00:00 2001 From: Alvaro Ballon <91897656+alvaro-at-xanadu@users.noreply.github.com> Date: Wed, 14 May 2025 17:20:31 -0400 Subject: [PATCH 13/24] Update demonstrations/tutorial_bp_catalyst.metadata.json --- demonstrations/tutorial_bp_catalyst.metadata.json | 1 - 1 file changed, 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.metadata.json b/demonstrations/tutorial_bp_catalyst.metadata.json index da92443186..f0a7b747ed 100644 --- a/demonstrations/tutorial_bp_catalyst.metadata.json +++ b/demonstrations/tutorial_bp_catalyst.metadata.json @@ -69,7 +69,6 @@ "title": "An introduction to factor graphs", "authors": "Hans-Andrea Loeliger", "year": "2004", - "doi": "", "url": "https://www.isiweb.ee.ethz.ch/papers/arch/aloe-2004-spmagffg.pdf" }, { From 0951f48a118aed6a684c748b3bd39537450dc663 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Fri, 4 Jul 2025 13:53:04 -0400 Subject: [PATCH 14/24] Update demonstrations/tutorial_bp_catalyst.py Co-authored-by: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> --- demonstrations/tutorial_bp_catalyst.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 6e47eb4672..d9406b77c1 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -1,4 +1,4 @@ -r"""Decoding Quantum Errors on the Steane code with Belief Propagation & Catalyst +r"""Decoding Quantum Errors on the Steane code with Belief Propagation and Catalyst ============================================================================= *Learn how to build, simulate, and decode the Steane code using JAX and Catalyst, blending quantum From 4e5464df6c9c9af661f28b855999fa697de9b2ea Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Thu, 10 Jul 2025 10:30:32 -0400 Subject: [PATCH 15/24] Apply suggestions from code review --- demonstrations/tutorial_bp_catalyst.py | 147 +++++++++++++------------ 1 file changed, 75 insertions(+), 72 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index d9406b77c1..152504ba2c 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -16,8 +16,8 @@ ``@qml.qjit`` circuit. Why is this exciting? Quantum error correction (QEC) is essential for building reliable quantum -computers, but it’s not enough to just run the quantum part. We need fast classical feedback too. -Catalyst helps by fusing the classical and quantum workflows, giving us one unified, +computers, but it requires more than just quantum operations. Fast classical feedback is needed as well. +Catalyst addresses this by fusing the classical and quantum workflows, giving us one unified, hardware-agnostic kernel that runs on CPUs, GPUs, and beyond. What We’ll Build @@ -40,7 +40,7 @@ # Understanding Quantum Error Correction # -------------------------------------- # -# At its core, QEC protects quantum information by encoding it redundantly. Stabilizer codes are a +# At its core, QEC protects quantum information through redundant encoding. Stabilizer codes are a # foundational tool for this purpose. Each stabilizer is a multi-qubit Pauli operator (composed of I, # X, Y, Z gates) that defines parity constraints the code space must satisfy. # @@ -57,9 +57,9 @@ # # where: # -# - :math:`v_i = 1` if the Pauli has an X on qubit :math:`i` (0 otherwise) -# - :math:`u_i = 1` if the Pauli has a Z on qubit :math:`i` (0 otherwise) -# - :math:`v_i = u_i = 1` if the Pauli has a Y on qubit :math:`i`, since :math:`X \cdot Z \propto Y` +# - :math:`v_i = 1` if the Pauli has an X on qubit :math:`i` (0 otherwise), +# - :math:`u_i = 1` if the Pauli has a Z on qubit :math:`i` (0 otherwise), +# - and :math:`v_i = u_i = 1` if the Pauli has a Y on qubit :math:`i`, since :math:`X \cdot Z \propto Y`. # # For example, the operator :math:`XZIY` on 3 qubits corresponds to :math:`(1,0,0, 1 | 0,1,0, 1)`. # The stabilizer group is then generated by a set of such symplectic vectors, forming a **stabilizer @@ -73,7 +73,7 @@ # # v \cdot u' + u \cdot v' \pmod{2}. # -# For a valid stabilizer code, all generators must commute—so the symplectic inner product between any +# For a valid stabilizer code, all generators must commute, which implies that the symplectic inner product between any # two rows of the stabilizer matrix must be zero. # # Measurement and Syndrome @@ -90,7 +90,7 @@ # # where :math:`(v_g^{(i)} | u_g^{(i)})` is the :math:`i`-th generator. # -# Below we show the basic quantum circuit for extracting a syndrome value. The ancilla qubit is +# Below, we show the basic quantum circuit for extracting a syndrome value. The ancilla qubit is # initialized in the :math:`|+\rangle` state, and controlled operations are applied based on the # stabilizer generators. Finally, the ancilla is measured in the :math:`X`-basis to obtain the # syndrome value. Check out Arthur Pesah’s excellent blog post series [#Pesah]\_ on Stabilizer codes @@ -128,12 +128,12 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # # Once you have syndrome bits from your stabilizer measurements, you need to figure out what error -# likely occurred. This is the job of the decoder. Formally, given the syndrome, you’re solving for +# likely occurred-this is the job of the decoder. Formally, given the syndrome, you’re solving for # the most probable error (usually called maximum likelihood estimate, or MLE for the error). # # However, exact MLE decoding depends on the precise information of your noise model and is generally -# computationally intractable (NP-Hard) because :math:`n` one bit syndrome measurements can take on -# :math:`2^n` unique values–in practice we rely on approximate methods tuned to assumptions about the +# computationally intractable (NP-Hard) because :math:`n` one-bit syndrome measurements can take on +# :math:`2^n` unique values. In practice, we rely on approximate methods tuned to assumptions about the # noise model. # # .. figure:: ../_static/demonstration_assets/bp_catalyst/css.svg @@ -144,13 +144,13 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # CSS Codes: Simplifying the Structure # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # -# CSS (Calderbank–Shor–Steane) codes are a special class where the stabilizer generators are split +# CSS (Calderbank–Shor–Steane) codes are a special class of stabilizer code where the generators are split # into **X-type** and **Z-type** operators. Their symplectic vectors look like this: # # - **X-type generator**: :math:`(v | 0)` (only Xs) # - **Z-type generator**: :math:`(0 | u)` (only Zs) # -# This allows us to represent the stabilizers with two **:math:`m \times n` matrices**: +# This allows us to represent the stabilizers with two :math:`m \times n` **matrices**: # # - :math:`H_X` for X-type generators # - :math:`H_Z` for Z-type generators @@ -164,11 +164,11 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # # which ensures that all X and Z stabilizers commute pairwise. When measuring syndromes: # -# - X-type stabilizers detect **Z errors** via :math:`s_X = H_X e_Z^T \pmod{2}` -# - Z-type stabilizers detect **X errors** via :math:`s_Z = H_Z e_X^T \pmod{2}` +# - X-type stabilizers detect **Z errors** via :math:`s_X = H_X e_Z^T \pmod{2}`. +# - Z-type stabilizers detect **X errors** via :math:`s_Z = H_Z e_X^T \pmod{2}`. # -# This separation makes decoding modular: you can handle X and Z errors independently. Later, when we -# introduce the Steane code, you’ll see these matrices explicitly and how they simplify syndrome +# This separation makes decoding modular, allowing you to handle X and Z errors independently. When we +# introduce the Steane code later, you’ll see these matrices explicitly and how they simplify syndrome # calculation and decoding. See a similar diagram below for the CSS code cycle structure. # # .. figure ../_static/demonstration_assets/bp_catalyst/css.svg @@ -182,10 +182,10 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # --------------- # # The Steane code is one of the simplest quantum error correcting codes, a CSS code built from two -# classical Hamming codes that encodes one logical qubit into seven physical qubits that can correct -# any single-qubit error–traditionally the error correcting ability of a code is referred as the +# classical Hamming codes. It encodes one logical qubit into seven physical qubits and can correct +# any single-qubit error. Traditionally, the error correcting ability of a code is referred as the # distance or :math:`d` and the number of errors a code can correct is -# :math:`\lfloor (d-1)/2 \rfloor`. In this case where our code can correct a single error it is said +# :math:`\lfloor (d-1)/2 \rfloor`. Since the Steane code can correct a single error, it is said # to have distance :math:`3`. The Steane code uses six stabilizer generators: # # .. math:: @@ -202,9 +202,9 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # # - **Lookup Table (LUT)**: Pre-compute minimal corrections for every syndrome (possible for small # codes like this one). -# - **Belief Propagation (BP)**: An iterative message-passing algorithm run on the code’s Tanner graph +# - **Belief Propagation (BP)**: An iterative message-passing algorithm that operates on the code’s Tanner graph # (a bipartite graph representing the relationships between qubits and stabilizers). It approximates -# the marginal probabilities of errors on each qubit and is more scalable to larger, sparser codes. +# the marginal probabilities of errors on each qubit, offering greater scalability for larger, sparser codes. # # We’ll start with the LUT decoder for its simplicity and then explore BP, which is more flexible for # larger or sparser codes. @@ -214,9 +214,9 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # Lookup‑table (LUT) decoding # --------------------------- # -# For the Steane code since there are :math:`3` :math:`X` and :math:`3` :math:`Z` stabilizer +# For the Steane code, with :math:`3` :math:`X` and :math:`3` :math:`Z` stabilizer # generators, there are :math:`2^3=8` possible syndromes for both :math:`X` and :math:`Z`. We can -# create a a small table maps each three‑bit syndrome to a weight‑1 error. +# create a small table that maps each three‑bit syndrome to a weight‑1 error. # import jax.numpy as jnp @@ -263,10 +263,10 @@ def _decode(syndrome: ArrayLike): print(tabulate(table_data, headers=["Syndrome", "LUT Error"])) ###################################################################### -# While this approach is optimal for small codes any larger examples for instance the distance -# :math:`30` rotated surface code which encodes only :math:`1` logical qubits has :math:`450` -# stabilizers for both :math:`X` and :math:`Z`. A full lookup table decoder just for one check type -# would require roughly :math:`2.9\times 10^{35}` entries. +# While this approach is optimal for small codes, it rapidly becomes infeasible for larger examples. +# For instance, the distance-:math:`30` rotated surface code, which encodes only :math:`1` logical qubits, has :math:`450` +# stabilizers for both :math:`X` and :math:`Z`. A full lookup table decoder for just one check type +# would require approximately :math:`2.9\times 10^{35}` entries. # ###################################################################### @@ -285,7 +285,7 @@ def _decode(syndrome: ArrayLike): # :math:`H_{cv} = 1`, meaning that qubit :math:`v` participates in stabilizer :math:`c`. # # The goal is to estimate the probability that each qubit has been flipped (i.e., that -# :math:`e_v = 1`), given the observed syndrome bits :math:`s_c`. BP updates beliefs iteratively by +# :math:`e_v = 1`), given the observed syndrome bits :math:`s_c`. BP updates there beliefs iteratively by # exchanging messages between variable and check nodes. # # The Sum-Product Algorithm @@ -343,10 +343,10 @@ def _decode(syndrome: ArrayLike): # violated). # - :math:`N(c)` is the set of variables connected to check :math:`c`. # - The :math:`\tanh` and :math:`\operatorname{arctanh}` functions implement the **sum-product -# rule** for combining binary parity checks dervied from classical probability theory. +# rule** for combining binary parity checks derived from classical probability theory. # -# **What’s going on?** This formula tells us: if the product of incoming :math:`\tanh` terms is -# close to +1 or -1, it means there is strong belief about whether the parity is satisfied or +# **What’s going on?** This formula indicates that if the product of incoming :math:`\tanh` terms is +# close to +1 or -1, it means there is a strong belief about whether the parity is satisfied or # violated. The :math:`\operatorname{arctanh}` converts that back into an LLR-style message. The # :math:`(-1)^{s_c}` factor flips the sign if the syndrome is 1, signaling that a parity error was # detected. @@ -367,18 +367,18 @@ def _decode(syndrome: ArrayLike): # # L_v = L_0 + \sum_{c \in N(v)} m_{c \to v} # -# The decoder makes a hard decision: +# The decoder then makes a hard decision: # -# - If :math:`L_v < 0`, guess :math:`e_v = 1` (error detected). -# - If :math:`L_v > 0`, guess :math:`e_v = 0` (no error). +# - If :math:`L_v < 0`, it guesses :math:`e_v = 1` (error detected). +# - If :math:`L_v > 0`, it guesses :math:`e_v = 0` (no error). # # Why This Works # ~~~~~~~~~~~~~~ # # Belief propagation is exact on **tree-like graphs**, where no cycles exist. However, even on Tanner -# graphs, which are never tree like, it provides a good approximation the maximum-likelihood decoder -# by using only local, iterative computations. However, its performance can degrade when the Tanner -# graph contains many short cycles—as is the case for many popular quantum codes leading to poor +# graphs, which are never tree-like, it provides a good approximation to the maximum-likelihood decoder +# by using only local, iterative computations. Nevertheless, its performance can degrade when the Tanner +# graph contains many short cycles—a common characteristic of many popular quantum codes, which can lead to poor # convergence. In practice further extensions like BP-OSD [#Panteleev]\ *, BP-LSD [#Hillmann]* or # Ambiguity Clustering [#Wolanski]\_ are used to fix these issues. # @@ -427,7 +427,7 @@ def _build_graph( ###################################################################### # A nice way to visulaize this Tanner graph is using the ``networkx`` package. Below is an example on -# the Stean code. +# the Steane code. # import matplotlib.pyplot as plt @@ -504,7 +504,7 @@ def _c2v_update( ###################################################################### -# Once the main loop finishes, we still need a hard decision. ``_posterior_llrs`` folds every final +# Once the main loop finishes, we still need a hard decision. The function ``_posterior_llrs`` folds every final # check‑to‑variable message for bit ``v`` into its intrinsic LLR, yielding the posterior belief for # that bit. A negative value means “error likely,” a positive value means “no error.” # @@ -531,12 +531,12 @@ def _posterior_llrs( # matrix and channel error rate, builds the parity graph, pre‑computes the intrinsic LLR, and returns # a JIT‑compiled function ``_decode``. # -# Inside ``_decode`` we: +# Inside ``_decode``, the following steps are executed: # -# 1. Zero‑initialise all messages. -# 2. Call ``_c2v_update`` inside a ``jax.lax.fori_loop`` for ``max_iter`` rounds. -# 3. Convert final messages to posterior LLRs with ``_posterior_llrs``. -# 4. Output a binary error vector by thresholding the LLRs at zero. +# 1. All messages are zero-initialized. +# 2. ``_c2v_update`` is called inside a ``jax.lax.fori_loop`` for ``max_iter`` rounds. +# 3. Final messages are converted to posterior LLRs with ``_posterior_llrs``. +# 4. A binary error vector is output by thresholding the LLRs at zero. # # Because the whole ``_decode`` body is wrapped in ``@jax.jit``, the first call compiles everything # into an XLA kernel; subsequent calls run at full device speed. @@ -628,10 +628,10 @@ def body(_, msgs): ###################################################################### # Before diving into the code, let’s test our belief‑propagation (BP) decoder on a bigger example: the # `n‑bit repetition code `__. This code stores each -# logical bit by repeating it :math:`n` times—:math:`0 \mapsto 00\ldots0` and -# :math:`1 \mapsto 11\ldots1`. Its parity‑check matrix consists of :math:`n-1` rows, each enforcing +# logical bit by repeating it :math:`n` times (e.g. :math:`0 \mapsto 00\ldots0` and +# :math:`1 \mapsto 11\ldots1`). Its parity‑check matrix consists of :math:`n-1` rows, each enforcing # that two neighbouring bits are equal. Below, we measure how often the BP decoder corrects random -# errors on a 50‑bit repetition code and compare that success rate to an optimal maximum‑likelihood +# errors on a 50‑bit repetition code and compare its success rate to an optimal maximum‑likelihood # (ML) decoder, which simply picks the lower‑weight error pattern consistent with the observed # syndrome. # @@ -714,27 +714,30 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: # ------------------------ # # Now that we understand a good chunk of theory behind CSS codes, the Steane code and decoding -# algorithms let’s put this into action with Catalyst! +# algorithms, let’s put this into action with Catalyst! # # Catalyst lets us build hybrid quantum-classical workflows, compiling both quantum operations and # classical decoding logic into a single, efficient kernel. We’ll start with a quantum-classical # circuit to prepare the logical zero state :math:`|0\rangle_L` for our Steane code. This method is # also general for initializing logical zero states for any CSS codes. # -# Start with a :math:`+1` eigen state (or stabilizer state) of all the :math:`Z` type stabilizers -# (:math:`|0\ldots 0\rangle` is always stabilized by any :math:`Z` type Pauli operator so is a -# suitable choice). +# Start with a :math:`+1` eigenstate (or stabilizer state) of all the :math:`Z` type stabilizers. +# The :math:`|0\ldots 0\rangle` is always stabilized by any :math:`Z` type Pauli operator, making it a +# suitable choice. # -# Then for each X-type generator: - Prepare an ancilla qubits in the :math:`|+\rangle` state. - -# Measure X-type stabilizers using CNOT operations onto an ancilla. - Measure in the :math:`X` basis +# Then, for each X-type generator: +# - Prepare an ancilla qubit in the :math:`|+\rangle` state. +# - Measure X-type stabilizers using CNOT operations onto an ancilla. +# - Measure in the :math:`X` basis. # -# Then - Use measurement outcomes (syndromes) to determine necessary corrections using our decoder. - -# Apply Z-type corrections based on decoding results. +# Next: +# - Use measurement outcomes (syndromes) to determine necessary corrections using our decoder. +# - Apply Z-type corrections based on decoding results. # -# This procedures uses projective measurements to force the data qubits to all be in the :math:`+1` -# eigen state of our :math:`X` type generators. Since the state was already a :math:`+1` eigen state -# of our :math:`Z` type generators and my virtue of the CSS code all :math:`X` and :math:`Z` -# generators simultaneously commute we are left with a state in the :math:`+1` eigen space of all the +# This procedure uses projective measurements to force the data qubits to be in the :math:`+1` +# eigenstate of our :math:`X` type generators. Since the state was already a :math:`+1` eigenstate +# of our :math:`Z` type generators, and by virtue of the CSS code all :math:`X` and :math:`Z` +# generators simultaneously commute, we are left with a state in the :math:`+1` eigenspace of all the # generators. # @@ -1050,9 +1053,9 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): # Benchmarking logical vs. physical error rates # --------------------------------------------- # -# In the final section of this demo we will compute the average performance of our Steane code error +# In the final section of this demo, we will compute the average performance of our Steane code error # correction circuit for a range of possible error rates. We’ll define logical error rates by -# comparing the state vector from our noisy simulation to with the clean state vector ``sv_clean`` of +# comparing the state vector from our noisy simulation with the clean state vector ``sv_clean`` of # the Steane code logical zero state. # @@ -1145,23 +1148,23 @@ def single_trial_error(key, p_err, H_steane): # Conclusion and Limitations # -------------------------- # -# In this tutorial, we’ve successfully built, simulated, and decoded a simple quantum error correction -# cycle using the Steane code. We’ve demonstrated encoding a logical qubit, introduced errors through +# In this tutorial, we successfully built, simulated, and decoded a simple quantum error correction +# cycle using the Steane code. We demonstrated encoding a logical qubit, introduced errors through # noise simulation, and performed error correction using stabilizer measurements combined with -# classical decoding. The performance was benchmarked by measuring the logical versus physical error +# classical decoding. Performance was benchmarked by measuring the logical versus physical error # rates. # -# However, our approach had a significant simplifying assumption known as the code capacity model, -# where errors are assumed to occur at only one stage of the circuit, with perfect encoding and -# syndrome extraction otherwise. A more realistic approach—called circuit-level noise—accounts for +# However, our approach relied on a significant simplifying assumption known as the code capacity model, +# where errors are assumed to occur at only one stage of the circuit, with otherwise perfect encoding and +# syndrome extraction. A more realistic approach—called circuit-level noise—accounts for # potential errors at every gate and measurement within the circuit. This model significantly -# complicates decoding because it requires mapping every possible error location not only in space but +# complicates decoding, as it requires mapping every possible error location not only in space but # also across multiple syndrome measurement rounds, forming a complex space-time hypergraph. Decoding # then involves interpreting error events over both spatial and temporal dimensions. # -# Nevertheless, the fundamental decoding principles explored here, particularly the Belief Propagation -# (BP) algorithm, remain highly relevant. BP is flexible enough to operate effectively on more full -# circuit level decoding hypergraphs. +# Nevertheless, the fundamental decoding principles explored here, particularly the Belief Propagation +# algorithm, remain highly relevant. BP is flexible enough to operate effectively on more comprehensive +# circuit-level decoding hypergraphs. # ###################################################################### From 7c906e30107f775694d24ce86b24743253efac7e Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Thu, 10 Jul 2025 10:37:02 -0400 Subject: [PATCH 16/24] Apply suggestions from code review_2 --- demonstrations/tutorial_bp_catalyst.py | 21 ++++++++++----------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 152504ba2c..b169db21fe 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -129,7 +129,7 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # # Once you have syndrome bits from your stabilizer measurements, you need to figure out what error # likely occurred-this is the job of the decoder. Formally, given the syndrome, you’re solving for -# the most probable error (usually called maximum likelihood estimate, or MLE for the error). +# the most probable error, usually called the maximum likelihood estimate (MLE) for the error. # # However, exact MLE decoding depends on the precise information of your noise model and is generally # computationally intractable (NP-Hard) because :math:`n` one-bit syndrome measurements can take on @@ -206,7 +206,7 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # (a bipartite graph representing the relationships between qubits and stabilizers). It approximates # the marginal probabilities of errors on each qubit, offering greater scalability for larger, sparser codes. # -# We’ll start with the LUT decoder for its simplicity and then explore BP, which is more flexible for +# We’ll begin with the LUT decoder due to its simplicity and then explore BP, which is more flexible for # larger or sparser codes. # @@ -455,10 +455,10 @@ def _build_graph( plt.show() ###################################################################### -# This next helper ``_c2v_update`` performs one full sweep of check‑to‑variable updates (step 3 of the +# The ``_c2v_update`` helper function performs one full sweep of check‑to‑variable updates (step 3 of the # sum‑product algorithm). It takes the previous messages, the syndrome, the neighbour tables, and two # scalars (``L_int`` for the intrinsic log‑likelihood ratio and ``eps`` for numerical safety). It -# loops only over edges that exist, multiplies the relevant :math:`\operatorname{tanh}` terms, clips +# loops only over existing edges, multiplies the relevant :math:`\operatorname{tanh}` terms, clips # the product, applies :math:`\operatorname{arctanh}`, and writes the new message into the next # matrix. # @@ -527,7 +527,7 @@ def _posterior_llrs( ###################################################################### -# ``build_bp_decoder`` is the main entry point for compiling our decoder. It takes the parity‑check +# ``build_bp_decoder`` serves as the main entry point for compiling our decoder. It takes the parity‑check # matrix and channel error rate, builds the parity graph, pre‑computes the intrinsic LLR, and returns # a JIT‑compiled function ``_decode``. # @@ -991,8 +991,8 @@ def get_error(n: int, p_err: float, key: random.PRNGKey): ###################################################################### -# Similar to ``measure_x_stabilizers`` but instead now we apply CNOT from data to an ancilla prepared -# in the :math:`|0\rangle` state and perform a :math:`Z` basis measurement. +# Similar to ``measure_x_stabilizers``, however, we now apply CNOT from data to an ancilla prepared +# in the :math:`|0\rangle` state and perform a :math:`Z`-basis measurement. # @@ -1015,9 +1015,8 @@ def measure_z_stabilizers(H): ###################################################################### -# Now we run the ``qec_round`` using state preparation, one round of noise and one round of :math:`X` -# and :math:`Z` correction. We’ll print the error was was using in our noisy channel and show that the -# output state still looks like the noiseless state we saw above. +# Now, let's run the ``qec_round`` using state preparation, followed by one round of noise injection and one round of :math:`X` +# and :math:`Z` correction. We’ll print the error that occurred in our noisy channel and demonstrate that the output state closely resembles the noiseless state we observed previously. # @@ -1040,7 +1039,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), pretty_print=True, wires=range(n)) ###################################################################### -# If we increase the likelihood of errors we are more likely to end up with an error pattern that +# If we increase the likelihood of errors, we are more likely to end up with an error pattern that # can’t be corrected. # From ad8ac8c5a6c1a4acf97694ac92aaa885eae660f8 Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Thu, 10 Jul 2025 10:43:47 -0400 Subject: [PATCH 17/24] Apply suggestions from code review --- demonstrations/tutorial_bp_catalyst.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index b169db21fe..821743b2f3 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -186,7 +186,7 @@ def ancilla_assisted_syndrome_extraction(syndromes: list[str]): # any single-qubit error. Traditionally, the error correcting ability of a code is referred as the # distance or :math:`d` and the number of errors a code can correct is # :math:`\lfloor (d-1)/2 \rfloor`. Since the Steane code can correct a single error, it is said -# to have distance :math:`3`. The Steane code uses six stabilizer generators: +# to have distance :math:`3`. This code uses six stabilizer generators: # # .. math:: # From 8fa8c4170ba993c8053d200e7fd3235c14564ae8 Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Fri, 11 Jul 2025 10:09:09 -0400 Subject: [PATCH 18/24] Update demonstrations/tutorial_bp_catalyst.py --- demonstrations/tutorial_bp_catalyst.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 821743b2f3..22c5bc55c0 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -61,7 +61,7 @@ # - :math:`u_i = 1` if the Pauli has a Z on qubit :math:`i` (0 otherwise), # - and :math:`v_i = u_i = 1` if the Pauli has a Y on qubit :math:`i`, since :math:`X \cdot Z \propto Y`. # -# For example, the operator :math:`XZIY` on 3 qubits corresponds to :math:`(1,0,0, 1 | 0,1,0, 1)`. +# For example, the operator :math:`XZIY` on 4 qubits corresponds to :math:`(1,0,0, 1 | 0,1,0, 1)`. # The stabilizer group is then generated by a set of such symplectic vectors, forming a **stabilizer # matrix** of size :math:`m \times 2n` (with :math:`m` generators). # From 5b9ef18c7918b52431bad82e4b16490b8ff04488 Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Fri, 11 Jul 2025 10:20:15 -0400 Subject: [PATCH 19/24] Update demonstrations/tutorial_bp_catalyst.py --- demonstrations/tutorial_bp_catalyst.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 22c5bc55c0..cd60647934 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -932,7 +932,7 @@ def state_vector_to_dict( ###################################################################### # We run the ``encode_zero`` function and see that we recover the `correct logical zero state for the -# Stean code `__: +# Steane code `__: # # .. math:: \begin{aligned}|\overline{0}\rangle= & \frac{1}{\sqrt{8}}(|0000000\rangle+|1010101\rangle+|0110011\rangle+|1100110\rangle \\ & +|0001111\rangle+|1011010\rangle+|0111100\rangle+|1101001\rangle)\end{aligned} # From 5436a418d3f50941d7937e4547ec7bdfe41ae231 Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Fri, 11 Jul 2025 14:17:24 -0400 Subject: [PATCH 20/24] Update demonstrations/tutorial_bp_catalyst.py --- demonstrations/tutorial_bp_catalyst.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index cd60647934..9ee1b3a040 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -379,7 +379,7 @@ def _decode(syndrome: ArrayLike): # graphs, which are never tree-like, it provides a good approximation to the maximum-likelihood decoder # by using only local, iterative computations. Nevertheless, its performance can degrade when the Tanner # graph contains many short cycles—a common characteristic of many popular quantum codes, which can lead to poor -# convergence. In practice further extensions like BP-OSD [#Panteleev]\ *, BP-LSD [#Hillmann]* or +# convergence. In practice, further extensions like BP-OSD [#Panteleev]_, BP-LSD [#Hillmann]_ or # Ambiguity Clustering [#Wolanski]\_ are used to fix these issues. # # See the following summary article [#Loeliger]\_ as well as Chapter 5 in Bayesian Reasoning and From adb8368d058862231593afac8acfc143dadcff36 Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Fri, 11 Jul 2025 14:54:41 -0400 Subject: [PATCH 21/24] Update tutorial_bp_catalyst.py --- demonstrations/tutorial_bp_catalyst.py | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 9ee1b3a040..24d4d51e4a 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -93,7 +93,7 @@ # Below, we show the basic quantum circuit for extracting a syndrome value. The ancilla qubit is # initialized in the :math:`|+\rangle` state, and controlled operations are applied based on the # stabilizer generators. Finally, the ancilla is measured in the :math:`X`-basis to obtain the -# syndrome value. Check out Arthur Pesah’s excellent blog post series [#Pesah]\_ on Stabilizer codes +# syndrome value. Check out Arthur Pesah’s excellent blog post series [#Pesah]_ on Stabilizer codes # for a deeper introduction. # @@ -274,7 +274,7 @@ def _decode(syndrome: ArrayLike): # ------------------------------- # # Belief propagation is an iterative message-passing algorithm used to decode errors by working on the -# **Tanner graph** [#Wiberg]\_ of the code. This graph has two types of nodes: +# **Tanner graph** [#Wiberg]_ of the code. This graph has two types of nodes: # # - **Variable nodes** represent the physical qubits, which may or may not have experienced an error. # These correspond to the bits of the error vector :math:`e = (e_1, e_2, \dots, e_n)`. @@ -380,10 +380,10 @@ def _decode(syndrome: ArrayLike): # by using only local, iterative computations. Nevertheless, its performance can degrade when the Tanner # graph contains many short cycles—a common characteristic of many popular quantum codes, which can lead to poor # convergence. In practice, further extensions like BP-OSD [#Panteleev]_, BP-LSD [#Hillmann]_ or -# Ambiguity Clustering [#Wolanski]\_ are used to fix these issues. +# Ambiguity Clustering [#Wolanski]_ are used to fix these issues. # -# See the following summary article [#Loeliger]\_ as well as Chapter 5 in Bayesian Reasoning and -# Machine Learning [#Barber]\_ for a deeper dive into message passing algorithms on graphs. +# See the following summary article [#Loeliger]_ as well as Chapter 5 in Bayesian Reasoning and +# Machine Learning [#Barber]_ for a deeper dive into message passing algorithms on graphs. # # 4.2 BP in JAX # ~~~~~~~~~~~~~ From 22dc3623929b42ec4f2405d4ef5c09b4266d47ac Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Mon, 21 Jul 2025 14:24:33 -0400 Subject: [PATCH 22/24] * Simplify state_vector_to_dict function in tutorial_bp_catalyst.py * Cleanup imports --- demonstrations/tutorial_bp_catalyst.py | 159 +++++-------------------- 1 file changed, 28 insertions(+), 131 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 24d4d51e4a..dbdd93643c 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -97,7 +97,7 @@ # for a deeper introduction. # -from typing import Callable +from typing import Callable, Optional, Dict, Union, Sequence import pennylane as qml @@ -406,7 +406,7 @@ def _build_graph( pcm: ArrayLike, ) -> tuple[tuple[tuple[int, ...], ...], tuple[tuple[int, ...], ...]]: """ - Pre‑compute variable‑node and check‑node neighbours. + Pre‑compute variable‑node and check‑node neighbors. Returns ------- @@ -456,7 +456,7 @@ def _build_graph( ###################################################################### # The ``_c2v_update`` helper function performs one full sweep of check‑to‑variable updates (step 3 of the -# sum‑product algorithm). It takes the previous messages, the syndrome, the neighbour tables, and two +# sum‑product algorithm). It takes the previous messages, the syndrome, the neighbor tables, and two # scalars (``L_int`` for the intrinsic log‑likelihood ratio and ``eps`` for numerical safety). It # loops only over existing edges, multiplies the relevant :math:`\operatorname{tanh}` terms, clips # the product, applies :math:`\operatorname{arctanh}`, and writes the new message into the next @@ -509,9 +509,6 @@ def _c2v_update( # that bit. A negative value means “error likely,” a positive value means “no error.” # -from typing import Callable - - def _posterior_llrs( m_c2v_final: ArrayLike, var_nei: tuple[tuple[int, ...], ...], L_int: float ) -> ArrayLike: @@ -636,10 +633,6 @@ def body(_, msgs): # syndrome. # -import jax -import jax.numpy as jnp -from jax.typing import ArrayLike - def rep_code(n: int) -> ArrayLike: """ @@ -798,136 +791,40 @@ def encode_zero_steane(): # @TODO can we move this to a utility file (or better yet make it part of pennylane 😁) # -from typing import Optional - +from pprint import pprint def state_vector_to_dict( sv: ArrayLike, + wires: Optional[Sequence[int]], eps: float = 1e-8, - wires: Optional[list[int]] = None, probability: bool = False, - pretty_print: bool = False, -) -> dict[str, float | complex]: + display: bool = True, +) -> Dict[str, Union[float, complex]]: """ - Converts a state vector to a dictionary representation, optionally - filtering by wires and displaying probabilities or amplitudes. - - Args: - sv: The input state vector. - eps: Values with absolute square less than eps are ignored. - wires: An optional list of wire indices to include. - If None, all wires are included. - probability: If True, the dictionary values are probabilities (amplitude squared). - If False, the dictionary values are amplitudes. - pretty_print: If True, prints the state in a table format using tabulate. - - Returns: - A dictionary where keys are computational basis states (strings) - and values are either probabilities (float) or amplitudes (complex or float). + Convert a state vector into {bitstring: amplitude | probability}. """ + n_qubits = int(jnp.log2(len(sv))) - n_qubits = int(jnp.log2(sv.shape[0])) - d: dict[str, float | complex] = {} - processed_wires: Optional[set[int]] = set(wires) if wires is not None else None + out: Dict[str, Union[float, complex]] = {} + + for idx, amp in enumerate(sv): + mag = jnp.abs(amp) ** 2 if probability else jnp.abs(amp) + if mag <= eps: + continue + + bitstring = f"{idx:0{n_qubits}b}" + key = "".join(b for i, b in enumerate(bitstring) if wires is None or i in wires) - for i, amplitude_val in enumerate(sv): - if probability: - val_to_check = jnp.abs(amplitude_val) ** 2 - else: - val_to_check = jnp.abs(amplitude_val) - - if val_to_check > eps: - full_bits = f"{i:0{n_qubits}b}" - - if processed_wires is not None: - # Ensure wires are within valid range [0, n_qubits-1] - if not all(0 <= w < n_qubits for w in processed_wires): - raise ValueError( - f"Wire indices in {wires} are out of bounds for {n_qubits} qubits." - ) - current_bits = "".join( - [bit_char for idx, bit_char in enumerate(full_bits) if idx in processed_wires] - ) - else: - current_bits = full_bits - - if probability: - prob_val = (jnp.abs(amplitude_val) ** 2).item() - d[current_bits] = d.get(current_bits, 0.0) + prob_val # Ensure float for sum - else: - # Warning this may lead to errors when using wires != None and probability=False - amp_item = amplitude_val.item() - d[current_bits] = amp_item - - if pretty_print: - table_data = [] if probability: - headers = ["State |k〉", "Probability (%)"] - # Sort by probability descending for better readability - sorted_items = sorted(d.items(), key=lambda item: item[1], reverse=True) - for k, v_prob in sorted_items: - if v_prob > eps: # Apply eps again for final display after potential summation - table_data.append([f"|{k}〉", f"{v_prob * 100:.3f}"]) - print("\nState Probabilities:") - print( - tabulate( - table_data, - headers=headers, - tablefmt="grid", - numalign="decimal", - stralign="center", - ) - ) + out[key] = out.get(key, 0.0) + float(mag) else: - headers = ["State |k〉", "Amplitude (real)", "Amplitude (imag)"] - # Sort by bitstring for consistent output - sorted_items = sorted(d.items(), key=lambda item: item[0]) - has_complex = False - has_real = False - for k, v_amp in sorted_items: - amp = complex(v_amp) # Ensure it's complex for consistent processing - if abs(amp) > eps: # Apply eps again for final display - table_data.append([f"|{k}〉", f"{amp.real:.5f}", f"{amp.imag:+.5f}"]) - if abs(amp.imag) > eps: # Check if there's a significant imaginary part - has_complex = True - if abs(amp.real) > eps: - has_real = True - - print("\nState Amplitudes:") - if not has_complex: # Simplify table if all imaginary parts are zero (or very small) - headers = ["State |k〉", "Amplitude"] - simple_table_data = [[row[0], row[1]] for row in table_data] - print( - tabulate( - simple_table_data, - headers=headers, - numalign="decimal", - stralign="center", - ) - ) - elif not has_real: # Simplify table if all imaginary parts are zero (or very small) - headers = ["State |k〉", "Amplitude"] - simple_table_data = [[row[0], row[2]] for row in table_data] - print( - tabulate( - simple_table_data, - headers=headers, - numalign="decimal", - stralign="center", - ) - ) - else: - print( - tabulate( - table_data, - headers=headers, - numalign="decimal", - stralign="center", - ) - ) - - return d + out[key] = amp.item() + + if display: + pprint(out) + + return out ###################################################################### @@ -938,7 +835,7 @@ def state_vector_to_dict( # sv_clean = encode_zero_steane() -state_vector_to_dict(sv_clean, pretty_print=True, wires=range(n)) +state_vector_to_dict(sv_clean, display=True, wires=range(n)) ###################################################################### # Simulating Errors and Full Correction @@ -1036,7 +933,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.1 key = random.PRNGKey(10) print(f"Running Steane Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") -state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), pretty_print=True, wires=range(n)) +state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), display=True, wires=range(n)) ###################################################################### # If we increase the likelihood of errors, we are more likely to end up with an error pattern that @@ -1046,7 +943,7 @@ def qec_round(H: ArrayLike, p_err=1e-3, key=random.PRNGKey(0)): p_err = 0.3 key = random.PRNGKey(8) print(f"Running Steane Code QEC Round with error: {get_error(n, p_err=p_err, key=key)}") -state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), pretty_print=True, wires=range(n)) +state_vector_to_dict(qec_round(H_steane, p_err=p_err, key=key), display=True, wires=range(n)) ###################################################################### # Benchmarking logical vs. physical error rates From f4589bb44de9de845e0a6b90bd38d5f3cc11ade6 Mon Sep 17 00:00:00 2001 From: Tom Ginsberg Date: Mon, 21 Jul 2025 14:26:45 -0400 Subject: [PATCH 23/24] Remove TODO comment regarding utility function in tutorial_bp_catalyst.py --- demonstrations/tutorial_bp_catalyst.py | 2 -- 1 file changed, 2 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index dbdd93643c..730b6f444e 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -788,8 +788,6 @@ def encode_zero_steane(): ###################################################################### # A simple utility function to display the state # -# @TODO can we move this to a utility file (or better yet make it part of pennylane 😁) -# from pprint import pprint From 6ceed97de87e5dd9b4104cf5696206c45e7bd425 Mon Sep 17 00:00:00 2001 From: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com> Date: Tue, 22 Jul 2025 10:29:28 -0400 Subject: [PATCH 24/24] Apply suggestions from code review --- demonstrations/tutorial_bp_catalyst.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_bp_catalyst.py b/demonstrations/tutorial_bp_catalyst.py index 730b6f444e..794677a3d8 100644 --- a/demonstrations/tutorial_bp_catalyst.py +++ b/demonstrations/tutorial_bp_catalyst.py @@ -385,20 +385,20 @@ def _decode(syndrome: ArrayLike): # See the following summary article [#Loeliger]_ as well as Chapter 5 in Bayesian Reasoning and # Machine Learning [#Barber]_ for a deeper dive into message passing algorithms on graphs. # -# 4.2 BP in JAX +# BP in JAX # ~~~~~~~~~~~~~ # # Below, we implement a BP decoder using Jax broken down into it’s core components. # ###################################################################### -# Before we can pass messages, we need to know who talks to whom. ``_build_graph`` scans the +# Before we can pass messages, we need to establish the connectivity between nodes. The ``_build_graph`` function scans the # parity‑check matrix once and records, for every variable node, which checks touch it and vice‑versa. # We convert the neighbour lists to tuples so they become immutable, hashable static data. JAX can # then embed their values as compile‑time constants in the XLA program and reliably reuse the compiled -# kernel multiple times. A cool thing about JAX/XLA is when using simple static paramters like our +# kernel multiple times. A cool thing about JAX/XLA is that when using simple static parameters like the ones # below, the individual integers it contains are baked into the XLA program as compile‑time constants, -# so we can truely compile a high performance decoder for our specific parity check matrix. +# so we can truly compile a high performance decoder for our specific parity check matrix. # @@ -703,7 +703,7 @@ def ml_rep_decoder(syndrome: ArrayLike) -> ArrayLike: print(f"Decoding success rate: {success_rate * 100:.2f}%") ###################################################################### -# 4 Catalyst hybrid kernel +# Catalyst hybrid kernel # ------------------------ # # Now that we understand a good chunk of theory behind CSS codes, the Steane code and decoding @@ -967,7 +967,7 @@ def logical_error(sv): ###################################################################### -# Simulate 1000x noisy shots for several noise levels, we’ll use ``jax.vmap`` to efficiently map our +# Simulate 1000 noisy shots for several noise levels, we’ll use ``jax.vmap`` to efficiently map our # catalyst kernel over a set of random keys. #

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