Towards local testability for quantum coding

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