# Towards local testability for quantum coding

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 $\Omega(\log^{-2} (N))$ beating the current state-of-the-art of $\log^{-3} (N)$ 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 $1/\log(N)$, 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.
Document type :
Preprints, Working Papers, ...

https://hal.inria.fr/hal-02432360
Contributor : Anthony Leverrier <>
Submitted on : Wednesday, January 8, 2020 - 1:12:08 PM
Last modification on : Friday, April 9, 2021 - 3:38:49 PM

### Identifiers

• HAL Id : hal-02432360, version 1
• ARXIV : 1911.03069

### Citation

Anthony Leverrier, Vivien Londe, Gilles Zémor. Towards local testability for quantum coding. 2020. ⟨hal-02432360⟩

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