Towards local testability for quantum coding

TL;DR

This paper introduces hemicubic quantum codes with improved local testability and efficient decoding, extending to generalized codes that could lead to highly scalable quantum locally testable codes if soundness is improved.

Contribution

The paper presents a new family of quantum codes with enhanced local testability and decoding algorithms, and extends these codes via classical code quotients for potential scalability.

Findings

Hemicubic codes encode one logical qubit into exponentially many physical qubits.

Hemicubic codes achieve local testability with soundness  rac{1}{\u221a{ ext{log}(N)}}.

Decoding algorithms correct errors up to the minimum distance with polylogarithmic factors.

Abstract

We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the $p$-faces of the $n$-cube (for $n>p$) and stabilizer constraints with faces of dimension $(p\pm1)$. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into $N = 2^{n-p-1} \tbinom{n}{p}$ physical qubits and displays local testability with a soundness of $\Omega(1/\log(N))$ beating the current state-of-the-art of $1/\log^{2}(N)$ due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. We then extend this code family by considering the quotient of the $n$-cube by arbitrary linear classical codes of length $n$. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary $n$-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.