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

Avijit Mandal, Henry D. Pfister

TL;DR

This work extends the capacity-analysis program for Reed-Muller codes from classical binary-input symmetric channels to binary-input classical-quantum (BSCQ) channels with Holevo capacity. It develops a GNS-based orthogonal decomposition of decoder observables and a new correlation bound leveraging RM symmetry to derive a recursive MMSE bound that ties the long RM code behavior to its RM$(r,m-1)$ projections. By connecting MMSE to entropy and Helstrom error via EXIT-like arguments, the authors prove that for rates below the Holevo capacity the per-bit MMSE decays as $e^{-\omega(\sqrt{\log N})}$ and, consequently, that any set of $2^{o(\sqrt{\log N})}$ bits can be decoded with vanishing probability using quantum union bounds. These results provide a principled quantum-generalization of RM decoding and quantify the extent to which RM codes approach capacity on CQ channels, while leaving the block-error problem open for future work.

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.

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

TL;DR

This work extends the capacity-analysis program for Reed-Muller codes from classical binary-input symmetric channels to binary-input classical-quantum (BSCQ) channels with Holevo capacity. It develops a GNS-based orthogonal decomposition of decoder observables and a new correlation bound leveraging RM symmetry to derive a recursive MMSE bound that ties the long RM code behavior to its RM projections. By connecting MMSE to entropy and Helstrom error via EXIT-like arguments, the authors prove that for rates below the Holevo capacity the per-bit MMSE decays as and, consequently, that any set of bits can be decoded with vanishing probability using quantum union bounds. These results provide a principled quantum-generalization of RM decoding and quantify the extent to which RM codes approach capacity on CQ channels, while leaving the block-error problem open for future work.

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 bits can be decoded with a high probability when the code rate is less than the Holevo capacity.

Paper Structure

This paper contains 31 sections, 14 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Preliminaries
  5. Reed--Muller Codes
  6. Symmetric Binary-Input Classical-Quantum Channels
  7. Extrinsic Information Transfer Functions
  8. Orthogonal Decomposition of Observables
  9. Minimum Mean-Squared Error (MMSE) for Quantum Binary Hypothesis Testing
  10. MMSE Achieving Observable
  11. MMSE Relation for BSCQ Channels
  12. Reed-Muller Codes over BSCQ Channels
  13. Code Symmetry and Decoder Observables
  14. Achieving Vanishing Bit-Error Rate
  15. Conclusion
  16. ...and 16 more sections

Key Result

Theorem 1

Consider a BSCQ channel $W$ with Holevo capacity $C$. Then, for any $\eta >0$, there is a sequence $r_m \in \mathbb{N}$ such that the code sequence $\textnormal{RM} (r_m,m)$ has a rate converging to $C-\eta$ and any single bit has bit-error probability for all sufficiently large $m$ and some fixed constant $c>0$.

Figures (1)

  • Figure 1: Diagram showing the nesting structure of $\mathcal{C}$, $\mathcal{C}'$, and $\mathcal{C}"$ with relative sizes shown for the $\textnormal{RM}$ code sequence.

Theorems & Definitions (30)

  • Theorem 1: Informal
  • Proposition 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 20 more