标题： 广义环面码在扭曲环面上的量子纠错应用 显示英文标题

标题： Generalized toric codes on twisted tori for quantum error correction

摘要： Kitaev toric码被广泛认为是容错量子计算纠错领域中最领先的候选者之一。然而，直接增加其逻辑维度的方法，例如格子手术或引入穿孔，往往会导致不可接受的资源开销。在这项工作中，我们引入了一种基于环论的方法，用于高效分析二维拓扑CSS码，从而能够在扭环上探索具有更大逻辑维度的广义toric码。利用Gröbner基，我们将稳定子伴随简化，以高效识别任意子激发及其几何周期性，即使是在扭曲周期边界条件下也能实现。由于码的性质由任意子决定，这种方法使我们能够直接计算逻辑维度，而无需构建大型奇偶校验矩阵。我们的方法为寻找新的量子纠错码以及通过Laurent多项式环展示其潜在拓扑秩序提供了一个统一的方法。此框架自然适用于双变量自行车码。 例如，我们在扭曲环面上构造了参数为$[[ n, k, d ]]$的最优权值-6广义环面码$n \leq 400$，得到了诸如$[[120,8,12]]$、$[[186,10,14]]$、$[[210,10,16]]$、$[[248, 10, 18]]$、$[[254, 14, 16]]$、$[[294, 10, 20]]$、$[[310, 10, \leq 22]]$和$[[340, 16, 18]]$等新码。 此外，我们通过在由基矢量 $(0,30)$ 和 $(6,6)$ 定义的扭环面上使用 $(3,3)$-双变量自行车码，提出了一种新的 $[[360, 12, \leq 24]]$ 量子码实现方法，相对于之前的构造，提高了稳定子的局域性。 这些结果凸显了拓扑有序性视角在推进量子低密度奇偶校验（LDPC）码的设计和理论理解方面的力量。 显示英文摘要

摘要： The Kitaev toric code is widely considered one of the leading candidates for error correction in fault-tolerant quantum computation. However, direct methods to increase its logical dimensions, such as lattice surgery or introducing punctures, often incur prohibitive overheads. In this work, we introduce a ring-theoretic approach for efficiently analyzing topological CSS codes in two dimensions, enabling the exploration of generalized toric codes with larger logical dimensions on twisted tori. Using Gr\"obner bases, we simplify stabilizer syndromes to efficiently identify anyon excitations and their geometric periodicities, even under twisted periodic boundary conditions. Since the properties of the codes are determined by the anyons, this approach allows us to directly compute the logical dimensions without constructing large parity-check matrices. Our approach provides a unified method for finding new quantum error-correcting codes and exhibiting their underlying topological orders via the Laurent polynomial ring. This framework naturally applies to bivariate bicycle codes. For example, we construct optimal weight-6 generalized toric codes on twisted tori with parameters $[[ n, k, d ]]$ for $n \leq 400$, yielding novel codes such as $[[120,8,12]]$, $[[186,10,14]]$, $[[210,10,16]]$, $[[248, 10, 18]]$, $[[254, 14, 16]]$, $[[294, 10, 20]]$, $[[310, 10, \leq 22]]$, and $[[340, 16, 18]]$. Moreover, we present a new realization of the $[[360, 12, \leq 24]]$ quantum code using the $(3,3)$-bivariate bicycle code on a twisted torus defined by the basis vectors $(0,30)$ and $(6,6)$, improving stabilizer locality relative to the previous construction. These results highlight the power of the topological order perspective in advancing the design and theoretical understanding of quantum low-density parity-check (LDPC) codes.