Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes

2024-09-03

arXiv.org

AI Summary

This work addresses the problem of preserving the effective distance of weight-reduced quantum low-density parity-check (qLDPC) codes under syndrome extraction circuits employing a single ancilla qubit. To analyze fault-tolerant robustness, we integrate tools from homological algebra, stabilizer code theory, and circuit-level fault propagation modeling, augmented by distance-balancing techniques. Our key contribution is the first rigorous proof that single-ancilla circuits can nearly fully preserve the effective distance of weight-reduced qLDPC codes. Specifically, for high-dimensional hypergraph product (HGP) codes, we prove the absence of destructive hook errors. This establishes a provable lower bound on the effective distance of weight-reduced qLDPC codes under practical circuit constraints, providing the first formal guarantee of hook-free fault tolerance for high-dimensional HGP codes in single-ancilla architectures. The result significantly advances the engineering feasibility of qLDPC codes for scalable quantum error correction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Natural Language Processing: Safety and Robustness
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.

Problem

• Analyze effective distance of weight-reduced qLDPC codes.
• Preserve effective distance with single-ancilla syndrome extraction.
• Ensure no hook errors in higher-dimensional HGP codes.

Innovation

• Weight-reduced quantum LDPC codes
• Single-ancilla syndrome extraction
• Distance balancing technique