We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $kd^2/n = 8.40$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$, $kd^2/n = 2.67$). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$. On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$. We additionally present two self-dual codes with weight-8 stabilizers: $[[54,14,5]]$ ($kd^2/n = 6.48$) and $[[128,20,8]]$ ($kd^2/n = 10.0$). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

翻译：我们引入了六个独立的三元循环自行车（ITB）码，将Bravyi等人的二元循环自行车框架扩展至三个循环维度。利用三维环面上的非对称多项式对，我们构造了四个码，包括一个$[[140,6,14]]$码，其参数$kd^2/n = 8.40$。在码容量设置下，$[[140,6,14]]$码实现了$8.0\%$的伪阈值且$kd^2/n = 8.40$，超过了Voss等人最佳多元循环自行车码的性能（$7.9\%$，$kd^2/n = 2.67$）。在电路级去极化噪声下，$[[140,6,14]]$码的伪阈值达到$0.59\%$，$[[84,6,10]]$码达到$0.53\%$。在SI1000超导噪声模型下，当$p = 0.20\%$时，$[[140,6,14]]$码每轮每可观测量的错误率为$5.6 \times 10^{-5}$。此外，我们还提出了两个具有权重8稳定子的自对偶码：$[[54,14,5]]$（$kd^2/n = 6.48$）和$[[128,20,8]]$（$kd^2/n = 10.0$）。这些结果扩展了代数量子LDPC码的设计空间，并表明第三循环维度为实际容错实现提供了有竞争力的候选方案。