Reed-Muller Codes for Quantum Pauli and Multiple Access Channels

TL;DR

This paper extends Reed-Muller codes analysis to quantum Pauli and multiple access channels, deriving achievable rate regions and demonstrating their universal, rate-optimal performance across various quantum noise models.

Contribution

It introduces a unified approach to analyze Reed-Muller codes for quantum and multiple-access channels, establishing their rate-optimality and universality in quantum Pauli noise environments.

Findings

Derived achievable rate region for Q-MACs with correlated noise

Established a connection between classical and quantum Reed-Muller codes

Demonstrated rate-optimal performance of quantum RM codes across channel parameters

Abstract

Reed-Muller (RM) codes have undergone significant analytical advancements over the past decade, particularly for binary memoryless symmetric (BMS) channels. We extend the scope of RM codes development and analysis to multiple-access channels (MACs) and quantum Pauli channels, leveraging a unified approach. Specifically, we first derive the achievable rate region for RM codes on so-called Q-MACs, a class of MACs with additive correlated noise. This is achieved via a generalization of the bending and boosting arguments defined in arXiv:2304.02509. We then put forward a connection between the rate region of these QMACs and quantum RM codes designed for Pauli noise channels. This connection highlights a universality property of quantum RM codes, demonstrating their rate-optimal performance across a range of channel parameters, rather than for a single Pauli channel.