Independent Trivariate Bicycle Codes

Abstract

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 $k{d}^2/n=8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $k{d}^2/n=8.40$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$, $k{d}^2/n=2.67$). 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: $[[54,14,5]]$ ($k{d}^2/n=6.48$) and $[[128,20,8]]$ ($k{d}^2/n=10.0$). 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.