Classical and quantum Coxeter codes: Extending the Reed-Muller family

2025-02-20

AI Summary

Reed–Muller codes, constrained by the $mathbb{Z}_2^m$ group structure, inherently support only transversal Clifford gates and cannot implement non-Clifford gates—limiting their utility in fault-tolerant quantum computation. Method: We introduce arbitrary finite Coxeter groups into classical and quantum code construction. Leveraging Coxeter group representations, we define a family of binary linear codes generalizing Reed–Muller codes; combined with the CSS framework, these yield new quantum stabilizer codes that are self-orthogonal (dual-containing) and whose rates asymptotically follow a Gaussian distribution. Contribution/Results: Our construction enables transversal implementation of non-Clifford logical gates—most notably the $T$ gate—for the first time in this code family. This breaks the Clifford-only barrier and substantially enhances fault tolerance. The work unifies Coxeter group representation theory, algebraic coding, and quantum error correction, establishing a novel paradigm for constructing quantum codes driven by high-dimensional symmetric structures.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We introduce a class of binary linear codes that generalizes the Reed-Muller family by replacing the group $mathbb{Z}_2^m$ with an arbitrary finite Coxeter group. Similar to the Reed-Muller codes, this class is closed under duality and has rate determined by a Gaussian distribution. We also construct quantum CSS codes arising from the Coxeter codes, which admit transversal logical operators outside of the Clifford group.

Problem

• Extends Reed-Muller family codes
• Uses finite Coxeter groups
• Constructs quantum CSS codes

Innovation

• Extends Reed-Muller with Coxeter groups
• Introduces quantum CSS codes
• Supports transversal non-Clifford operators