General Distance Balancing for Quantum Locally Testable Codes

TL;DR

This paper establishes a lower bound on the soundness of quantum locally testable codes using distance balancing, enabling the use of classical codes with independent checks to construct quantum codes with improved parameters.

Contribution

It introduces a generalized distance balancing method that applies to any classical code with independent checks, expanding the toolkit for constructing quantum locally testable codes.

Findings

Lower bound on quantum code soundness derived

Ability to use classical LDPC codes for quantum code construction

New quantum LTCs with improved parameters obtained

Abstract

In this paper, we prove a lower bound on the soundness of quantum locally testable codes under the distance balancing construction of Evra et al. arXiv:2004.07935 [quant-ph]. Our technical contribution is that the new soundness of the quantum code is at least the old soundness divided by the classical code length (up to a constant factor). This allows us to use any classical code with independent checks when distance balancing, where previously only the repetition code had been considered for qLTCs. By using a good classical LDPC code, we are able to grow the dimension of the hypersphere product codes arXiv:1608.05089 [quant-ph] and the hemicubic codes arXiv:1911.03069 [quant-ph] while maintaining their distance and locality, but at the expense of soundness. From this, and also by distance balancing a chain complex of Cross et al. arXiv:2209.11405 [cs.IT], we obtain quantum locally testable codes of new parameters.