Tradeoff Constructions for Quantum Locally Testable Codes

TL;DR

This paper introduces three new constructions for quantum locally testable codes (qLTCs) that improve their parameters, including soundness amplification and distance amplification, enabling the creation of codes with previously unattainable properties.

Contribution

The paper presents novel methods for constructing and enhancing qLTCs, including soundness and distance amplification techniques, expanding the range of achievable code parameters.

Findings

Soundness amplification increases qLTC soundness to a constant.

Distance amplification yields qLTCs with linear distance.

Applications produce codes with previously unknown parameters.

Abstract

In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings' weight reduction construction for qLDPC codes arXiv:2102.10030 to give a weight reduction procedure for qLTCs. Secondly, we describe a novel `soundness amplification' procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications to prove the existence of codes in previously unknown parameter regimes. In particular, applications of these operations to the hypersphere product code arXiv:1608.05089 and the hemicubic code arXiv:1911.03069 yield many previously unknown parameters. Additionally, soundness amplification can be used to produce the first asymptotically good testable quantum code (rather than locally testable) - that being one with linear distance and dimension, as well as constant soundness. Lastly, applications of all three results are described to an upcoming work.