Quantum message-passing algorithm for optimal and efficient decoding

TL;DR

This paper advances quantum decoding algorithms by proving BPQM's optimality for tree-structured codes, formalizing its description, addressing implementation challenges, and proposing an efficient approximate message-passing variant with potential advantages over classical decoders.

Contribution

It provides a formal, detailed description of BPQM, proves its optimality for tree codes, identifies implementation flaws, and introduces an efficient approximate algorithm and a method for extending to cyclic graphs.

Findings

BPQM is proven to be optimal for binary linear codes with tree Tanner graphs.

The paper identifies a flaw in the original BPQM implementation affecting circuit size.

An approximate message-passing algorithm with polynomial complexity is proposed.

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 $\mathcal{O}(\text{poly } n, \text{polylog } \frac{1}{\epsilon})$, where $n$ is the code length and $\epsilon$ 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.