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.
翻译:我们引入了超模代码, 这是一种量子代码的组合, 通过将qubit与美元立方美元( $> p$) 的美元面相挂钩, 以及以维度表面( p\ pm1) $( p\ p1) 来稳定器的制约值。 我们利用这个本地测试能力来设计一种有效的解码算法, 将一个逻辑的qubit的任意差错改成 N= 2 ⁇ n- p-1}\ tbinom}\\\\\\\\\\\\\ p} 美元物理夸比特, 并显示本地测试能力与美元长度的任意直线式古典代码相比。 我们建立了这些通用的超标度( 1/\ log ( N) 美元) 的参数, 以1/\ log2 ( N) 美元击打当前因黑白( $) 而导致的直线代码。 我们利用这种本地测试来设计一种有效的解算算法, 纠正一个比最小距离小于最小距离的任意直方码的任意直方码的任意直方码。 。 我们确定这些直方代码的常值的代码的常值, 可能显示直方码, 直方码的常值为直达的直达的直方码。