We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check codes and $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum locally testable codes with soundness $\tildeΘ(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates for the first time. Remarkably, our proofs are almost entirely algebraic-topological, showing that such presumably intricate logical gates naturally arise as a fundamental topological phenomenon. We develop a general framework for constructing a rich new family of homological invariant forms which we call ''cupcap gates'' that induce transversal logical multi-controlled-$Z$ and, building on insights from [Li et al., arXiv:2603.25831], covering space methods to certify their nontriviality. The claimed almost-good code results follow immediately as examples.

翻译：我们展示了在[\![N,Θ(N),˜Θ(N)]\!]量子低密度奇偶校验码和具有˜Θ(1)健全性的[\![N,Θ(N),˜Θ(N)]\!]量子局部可检验码上，存在非平凡的横向逻辑多控制-Z门，首次将近乎最优的码参数与容错非Clifford门相结合。值得注意的是，我们的证明几乎完全基于代数拓扑方法，表明这些看似复杂的逻辑门自然表现为一种基本拓扑现象。我们发展了一个通用框架，用于构造一个丰富的同调不变形式新族，称之为"杯积门"，该门可诱导横向逻辑多控制-Z门，并基于[Li等人，arXiv:2603.25831]的洞见，利用覆盖空间方法证明其非平凡性。所声称的近乎最优码结果可立即作为实例得到验证。