Reed-Muller Codes for Quantum Pauli and Multiple Access Channels

2025-06-10

AI Summary

This work extends Reed–Muller (RM) codes to two distinct settings: the multiple-access channel (MAC) and the quantum Pauli channel—particularly addressing additive correlated noise in multi-user quantum communication. Method: We introduce a novel quantum MAC (Q-MAC) model and derive, for the first time, the exact achievable rate region for RM codes over this model. Leveraging generalized coset-bending analysis, polarization theory, and algebraic coding techniques, we establish capacity achievability of quantum RM codes across a broad range of Pauli noise parameters. Contribution/Results: Our results break the traditional limitation of RM code optimality—previously confined to single-user classical channels—by demonstrating universal rate-optimality in both multi-user and quantum regimes. The work unifies the performance limits of classical RM codes over correlated MACs and the robustness of quantum RM codes over Pauli channels, thereby forging theoretical bridges across classical/quantum and single-user/multi-user coding paradigms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Multiagent Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Extend Reed-Muller codes to multiple-access and quantum Pauli channels
• Derive achievable rate region for RM codes on Q-MACs
• Show quantum RM codes' universality across channel parameters

Innovation

• Extend Reed-Muller codes to quantum Pauli channels
• Derive achievable rate region for Q-MACs
• Show universality of quantum RM codes