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

TL;DR

This paper investigates the effective distance of weight-reduced quantum LDPC codes, demonstrating that certain fault-tolerant syndrome extraction methods preserve their error-correcting capabilities, especially in higher-dimensional HGP codes.

Contribution

It proves that weight reduction techniques and distance balancing preserve effective distance, ensuring fault tolerance in quantum LDPC codes, including higher-dimensional HGP codes.

Findings

Single-ancilla syndrome extraction largely preserves effective distance.

Distance balancing technique maintains effective distance.

Higher-dimensional HGP codes have no troublesome hook errors with these methods.

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.