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

2025-02-06

AI Summary

This work investigates the capacity achievability of Reed–Muller (RM) codes over the binary-input symmetric classical-quantum (BSCQ) channel, specifically whether reliable estimation of a subset of bits is possible at rates below the Holevo capacity. We introduce a weighted inner-product-based orthogonal decomposition of quantum observables, establishing for the first time a recursive connection to the nested structure of RM codes. Integrating Holevo information theory with the minimum mean-square error (MMSE) estimation framework, we derive a novel correlation bound. We prove that, for any rate below the Holevo capacity, every bit subset of size $2^{o(sqrt{log N})}$ can be decoded with block-error probability tending to zero with high probability. This constitutes the first rigorous demonstration of capacity achievability for RM codes over the BSCQ channel.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Reed-Muller Codes capacity on BMS channels
• Decoding functions using RM codes symmetry
• Holevo capacity on binary-input symmetric CQ channels

Innovation

• Reed-Muller Codes decoding
• Quantum Observables correlation
• Holevo capacity analysis