We introduce multivariate multicycle (MM) codes, a new family of quantum error-correcting codes (QECCs) that unifies bivariate bicycle, multivariate bicycle, abelian two-block group algebra, generalized bicycle, trivariate tricycle, and toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-$\textit{t+1}$ chain complexes with $\textit{$t \ge 4$}$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[144, 12, 12]]$, $[[216, 12, 14]]$, $[[288, 12, 16]]$, $[[324, 12, 20]]$, $[[432, 12, 27]]$, $[[486, 24, 12]]$, $[[630, 70, 9]]$, and $[[648, 18, 23]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.

翻译：我们提出多变量多循环（MM）码，这是一类新型量子纠错码（QECC），它统一了双变量自行车码、多变量自行车码、阿贝尔两区块群代数码、广义自行车码、三变量三轮车码以及环面码。MM码是Calderbank-Shor-Steane（CSS）码，由长度为$\textit{t+1}$的链复形定义，其中$\textit{$t \ge 4$}$。这类码的主要优势在于它们具有元校验（metachecks）和高约束性，能够实现完全单次解码（complete single-shot decoding）。我们提出一个框架，通过使用Koszul复形促进长链复形的构造。特别是，在我们的方法中，获取显式边界映射（奇偶校验矩阵和元校验矩阵）尤为直接。这种简单但高度通用的码参数化使得我们能够高效进行数值搜索，识别出多个展示这些能力的高码率、高码距MM码候选。新得到参数为$[[n,k,d]]$的码示例包括$[[96, 12, 8]]$、$[[144, 12, 12]]$、$[[216, 12, 14]]$、$[[288, 12, 16]]$、$[[324, 12, 20]]$、$[[432, 12, 27]]$、$[[486, 24, 12]]$、$[[630, 70, 9]]$和$[[648, 18, 23]]$。值得注意的是，我们的码实现的约束性曲线超越了所有已知具有实际块长度的可单次解码量子CSS码。此外，我们的码也是首个显式实例化的压缩5维至9维高维量子纠错码，其校验权重显著低于近期小型量子Tanner码实例。