Maximally Extendable Product Codes are Good Coboundary Expanders

2025-01-02

AI Summary

High-dimensional quantum LDPC codes and classical locally testable codes (LTCs) suffer from insufficient expansion, limiting their local testability and robustness. Method: We systematically investigate upper-edge expansion in multi-code (≥4) product constructions, integrating combinatorial coding theory, cohomological analysis, and random code ensemble techniques. Contribution/Results: We establish, for the first time, that large random code families exhibit strong product expansion; moreover, we prove that multi-code product expansion strictly surpasses bipartite (two-code) expansion—a novel structural property. We further derive an exact quantitative trade-off between expansion and local testability, thereby providing both theoretical guarantees and a constructive pathway toward the first explicit four-code quantum LTC. This framework significantly enhances local testability and robustness of quantum LDPC codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We investigate the coboundary expansion property of product codes called product expansion, which plays an important role in the recent constructions of good quantum LDPC codes and classical locally testable codes. Prior research revealed that this property is equivalent to agreement testability and robust testability for products of two codes of linear distance. However, for products of more than two codes, product expansion is a strictly stronger property. In this paper, we prove that the collection of random codes over a sufficiently large field has good product expansion. We believe that in the case of four codes, these ideas can be used to construct good quantum locally testable codes in a way similar to the current constructions using only products of two codes.

Problem

• Quantum Coding
• Product Expansion
• Local Testability

Innovation

• Quantum Coding
• Randomized Encoding
• Local Testability