Quantum Locally Testable Code with Constant Soundness

TL;DR

This paper introduces two novel constructions of quantum locally testable codes with constant soundness, advancing the development of quantum error correction with scalable parameters and potential implications for the NLTS theorem.

Contribution

The paper presents two new methods for constructing quantum locally testable codes with constant soundness and scalable parameters, using check product and hypergraph product techniques.

Findings

Constructed QLTCs with constant soundness and constant rate.

Achieved QLTCs with scalable distance and locality.

Provided insights towards high soundness and distance in quantum codes.

Abstract

In this paper, we present two constructions of quantum locally testable codes (QLTC) with constant soundness. In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.