Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates without overhead. As examples, we construct $[[98,6,12]]$ and $[[162, 8, 12]]$ BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which might be of independent interest.

翻译：量子低密度奇偶校验（qLDPC）码为实现具有恒定开销的可扩展容错量子计算提供了一条前景广阔的路径。最新进展表明，即使使用近期硬件，qLDPC码也能超越表面码的量子存储能力。如何为这类码实现容错的逻辑门操作仍是一个开放性问题。本文提出了具有增强对称性的高码率双变量自行车（BB）码新实例。这些码具备显式的逻辑算符优良基（类似于环面码），并能无开销地支持折叠横向克利福德门。作为示例，我们构造了$[[98,6,12]]$和$[[162, 8, 12]]$的BB码，这些码可实现有趣的容错克利福德门。我们的工作还为量子双块码与群代数码中逻辑算符显式基与折叠横向门的数学基础建立了框架，这可能具有独立的学术价值。