优秀量子局部可测试码上的横向非Clifford门

中文摘要 AI 辅助

我们通过将[ arXiv:2604.01874 ]的门框架应用于[ arXiv:2609.20780 ]最近的优秀qLTC构造，在量子低密度奇偶校验码和量子局部可测试码上同时实现了具有渐近最优参数的非平凡横向逻辑多控制-$Z$门。为此，我们使用覆盖空间方法在有限算术立方复形上构造非零杯积配对。这不同于之前的近似好码构造，后者的基空间是超图积。我们将配对表示为Moore行列式乘积中的系数，并通过二分多重图特化证明多项式非消失。然后我们构造覆盖空间，以获得一个渐近族优秀qLTC，在其上拉回配对诱导出所需的非平凡横向逻辑作用。

英文摘要

We achieve linearly many parallelizable and addressable transversal logical multi-controlled-Z gates with asymptotically optimal parameters simultaneously on good quantum low-density parity check codes and quantum locally testable codes, by applying the gate framework of [arXiv:2604.01874] to good qLTC constructions. To establish the nontriviality of the logical operation, we express the cup product pairing as a coefficient in a product of Moore determinants and prove that this coefficient is nonzero. Moreover, we use the symmetries of the covering spaces to construct large spaces of cocycles. By multiplying the initial cocycles by suitable elements of these spaces, we obtain linearly many independent pairings, each selected by a corresponding cycle constructed using Poincaré duality. This yields linearly many parallelizable and addressable multi-controlled-Z gates and enables asymptotically constant-overhead magic state distillation with good qLDPC codes.

发表机构

• Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）

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