Reed-Muller Codes on CQ Channels via a New Correlation Bound for Quantum Observables

TL;DR

This paper extends the analysis of Reed-Muller codes to binary-input symmetric classical-quantum channels, establishing a new correlation bound for quantum observables that enables decoding a large set of bits below the Holevo capacity.

Contribution

It introduces a novel correlation bound for quantum observables and applies it to Reed-Muller codes on quantum channels, advancing understanding of quantum coding capacity.

Findings

Any set of $2^{o(\sqrt{\log N})}$ bits can be decoded with high probability below the Holevo capacity.

Recursive relation for the minimum MSE estimate of a single bit in RM codes is established.

The correlation bound improves decoding analysis for quantum channels.

Abstract

The question of whether Reed-Muller (RM) codes achieve capacity on binary memoryless symmetric (BMS) channels has drawn attention since it was resolved positively for the binary erasure channel by Kudekar et al. in 2016. In 2021, Reeves and Pfister extended this to prove the bit-error probability vanishes on BMS channels when the code rate is less than capacity. In 2023, Abbe and Sandon improved this to show the block-error probability also goes to zero. These results analyze decoding functions using symmetry and the nested structure of RM codes. In this work, we focus on binary-input symmetric classical-quantum (BSCQ) channels and the Holevo capacity. For a BSCQ, we consider observables that estimate the channel input in the sense of minimizing the mean-squared error (MSE). Using the orthogonal decomposition of these observables under a weighted inner product, we establish a recursive relation for the minimum MSE estimate of a single bit in the RM code. Our results show that any set of $2^{o(\sqrt{\log N})}$ bits can be decoded with a high probability when the code rate is less than the Holevo capacity.