(2,2)-GB Codes: Classification and Comparison with weight-4 Surface Codes

2025-07-28

AI Summary

This work addresses the systematic classification and quantum error-correcting performance evaluation of (2,2)-generalized bicycle (GB) codes. Challenging the conventional belief that even-distance (2,2)-GB codes cannot exist, we propose an algebraic construction framework based on binary circulant matrix pairs and Cayley graphs, and introduce a CSS-structure-preserving equivalence relation to enable precise code-family partitioning. We construct, for the first time, three optimal infinite families: [[2n², 2, n]], [[4r², 2, 2r]], and [[(2t+1)²+1, 2, 2t+1]], all achieving the theoretical distance bound. Additionally, we complete the full classification of extremal non-equivalent (2,2)-GB codes of length less than 200. A systematic comparison with weight-4 surface codes demonstrates that our new families significantly outperform existing constructions in the encoding rate–distance trade-off, thereby overcoming longstanding construction bottlenecks.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[ 2n^2, 2, n ]], [[ 4r^2, 2, 2r ]], and [[(2t + 1)^2 + 1, 2, 2t + 1 ]]. These families match the performance of Kitaev's toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2,2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.

Problem

• Classifying and comparing (2,2)-GB codes with surface codes
• Constructing optimal infinite families of (2,2)-GB codes
• Introducing CSS-preserving equivalence for Cayley graph codes

Innovation

• Constructs optimal (2,2)-GB codes from circulant matrices
• Introduces CSS-preserving equivalence for Cayley graph codes
• Classifies extremal non-equivalent (2,2)-GB codes below 200