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Quantum message-passing algorithm for optimal and efficient decoding

Christophe Piveteau, Joseph M. Renes

TL;DR

The paper develops a rigorous quantum analogue of belief propagation for decoding classical data over pure-state CQ channels. It first formalizes BPQM, proves bitwise and blockwise optimality on tree Tanner graphs, and then introduces a true quantum message-passing BPQM that carries discretized angle information to achieve polynomial quantum circuit complexity. To handle graphs with cycles, it proposes an approximate cloning approach that unrolls the code’s Tanner graph into a tree, yielding strong numerical gains over classical decoders in tested examples. The work provides detailed circuit constructions, discretization error bounds, and extensive numerical results, offering a path toward scalable quantum decoding and laying groundwork for applying BPQM to more general code families and quantum channels.

Abstract

Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel [Renes, NJP 19 072001 (2017)]. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy et al. [npj Quantum Information 7 97 (2021)] observed that BPQM implements the optimal decoder on a small example code. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code dimension. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has quantum circuit complexity $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}ε)$, where $n$ is the code length and $ε$ is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder.

Quantum message-passing algorithm for optimal and efficient decoding

TL;DR

The paper develops a rigorous quantum analogue of belief propagation for decoding classical data over pure-state CQ channels. It first formalizes BPQM, proves bitwise and blockwise optimality on tree Tanner graphs, and then introduces a true quantum message-passing BPQM that carries discretized angle information to achieve polynomial quantum circuit complexity. To handle graphs with cycles, it proposes an approximate cloning approach that unrolls the code’s Tanner graph into a tree, yielding strong numerical gains over classical decoders in tested examples. The work provides detailed circuit constructions, discretization error bounds, and extensive numerical results, offering a path toward scalable quantum decoding and laying groundwork for applying BPQM to more general code families and quantum channels.

Abstract

Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel [Renes, NJP 19 072001 (2017)]. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy et al. [npj Quantum Information 7 97 (2021)] observed that BPQM implements the optimal decoder on a small example code. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code dimension. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has quantum circuit complexity , where is the code length and is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder.

Paper Structure

This paper contains 52 sections, 10 theorems, 144 equations, 22 figures, 1 table.

Table of Contents

  1. Introduction
  2. Description of BPQM
  3. Block optimality
  4. Message-passing BPQM
  5. Codes with cycles
  6. Other notions of quantum belief propagation
  7. Preliminaries
  8. Factor graphs and classical belief propagation decoding
  9. Commmunication over a pure state classical-quantum channel
  10. Representing states and channels using Forney-style factor graphs
  11. Description of BPQM
  12. Single-bit decoding
  13. Message passing graph
  14. Check and branch lists
  15. Quantum circuit
  16. ...and 37 more sections

Key Result

Theorem 4.1

The BPQM algorithm performs block optimal decoding.

Figures (22)

  • Figure 1: Tanner graph and message-passing graph (MPG) for a five-bit code. The MPG is centered on $X_1$.
  • Figure 2: Unrolling the Tanner graph of a code $\mathcal{C}$ to generate a new code $\mathcal{C}'$ and computation tree of $\mathcal{C}$. (a) The Tanner graph of a 6-bit code $\mathcal{C}$. (b) Tanner graph of the 9-bit code $\mathcal{C}'$ defined by unrolling the original Tanner graph around $X_1$ to depth 2. (c) $h=2$ computation tree of $\mathcal{C}$, obtained by identifying the variables of $\mathcal{C}'$ with those of $\mathcal{C}$.
  • Figure 3: (a) Forney-style representation of the Tanner graph of the 5-bit code. (b) The graph on top is an example of a (standard) factor graph containing a variable $X_4$ that is connected to more than two factor nodes. The corresponding Forney-style representation of the factor graph, depicted on the bottom, correspondingly contains an equality node. (c) (Standard) factor graph characterizing the joint distribution of the codeword bits of the 5-bit code given that some channel output $\vec{y}$ was observed.
  • Figure 4: Factor graph representing the distribution of a uniformly random codeword of the 5-bit code.
  • Figure 5: (a) Factor graph describing the classical channel from $X_1$ to $X_1,X_2,X_3,X_4,X_5$ for the 5-bit code. (b) Factor graph describing the CQ channel from $X_1$ to the CQ channel outputs $Y_1,Y_2,Y_3,Y_4,Y_5$ for the 5-bit code.
  • ...and 17 more figures

Theorems & Definitions (15)

  • Theorem 4.1
  • Lemma 4.2: Optimality of the PGM ban_optimum_1997sasaki_quantum_1998eldar_2001
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • Lemma 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Lemma B.1
  • proof
  • ...and 5 more