We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting codes that unifies and generalizes bivariate bicycle codes, multivariate bicycle codes, abelian two-block group algebra codes, generalized bicycle codes, trivariate tricycle codes, and n-dimensional toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-t chain complexes with $t \ge 4$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding, while also having additional algebraic structure that might enable logical non-Clifford gates. 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 $[[96, 12, 8]]$, $[[96, 44, 4]]$ $[[144, 40, 4]]$, $[[216, 12, 12]]$, $[[360, 30, 6]]$, $[[384, 80, 4]]$, $[[486, 24, 12]]$, $[[486, 66, 9]]$ and $[[648, 60, 9]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot decodable quantum CSS codes of practical blocksize.

翻译：本文引入多元多循环码，这是一种新型量子纠错码族，它统一并推广了双变量双循环码、多元双循环码、阿贝尔双块群代数码、广义双循环码、三变量三循环码以及n维环面码。MM码是基于长度t≥4的链复形定义的Calderbank-Shor-Steane型CSS码。这类码的主要优势在于其具备元校验和高约束特性，可实现完全单次解码，同时具有额外的代数结构，可能支持逻辑非克利福德门操作。我们提出了一个利用Koszul复形构建长链复形的框架。特别地，我们的方法能直接获得显式的边界映射。通过这种简洁而通用的参数化方法，我们高效地进行了数值搜索，识别出多个在高码率和高码距下展现这些特性的MM码候选。新发现的码参数包括[[96, 12, 8]]、[[96, 44, 4]]、[[144, 40, 4]]、[[216, 12, 12]]、[[360, 30, 6]]、[[384, 80, 4]]、[[486, 24, 12]]、[[486, 66, 9]]和[[648, 60, 9]]。值得注意的是，在实用块尺寸范围内，我们的码实现了超越所有已知单次解码量子CSS码的约束特性。