Independent Trivariate Bicycle Codes

TL;DR

This paper introduces six new independent trivariate bicycle codes extending previous frameworks, demonstrating improved thresholds and parameters for quantum error correction, with potential for practical fault-tolerant quantum computing.

Contribution

The paper presents six novel trivariate bicycle codes, including a high-performance $[[140,6,14]]$ code, expanding the design space of quantum LDPC codes with competitive parameters.

Findings

The $[[140,6,14]]$ code exceeds previous multivariate bicycle codes in pseudothresholds.

Circuit-level depolarizing noise thresholds reach 0.59%.

Codes perform well on superconducting noise models.

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 $kd^2/n = 8.40$. 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.