发表机构
University of Bristol; California Institute of Technology; Johannes Gutenberg University of Mainz; Stony Brook University（布里斯托大学; 加州理工学院; 美因茨约翰内斯古腾堡大学; 石溪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文扩展了基于上同调不变量构造量子CSS码逻辑门的框架，引入扭曲、高阶上同调运算及折叠运算，实现可寻址对角门（如T门），并提出了新的数学上同调运算，连接同伦论与容错量子计算。
AI 中文摘要
在第一部分中，我们引入了一个通用框架，用于从上同调不变量构造量子CSS码中的逻辑门，重点研究了杯积及其从拓扑码到qLDPC码的推广。特别地，我们构造了一族$C^{k-1}Z$逻辑门，称为copy-cup门，作用于同一量子码的$k$个副本，并通过有限深度电路实现。在此，我们通过引入包含码自同构的扭曲、更高阶扭曲上同调运算以及相对上同调中的折叠运算，极大地扩展了这一框架。该框架包含将系数从$\mathbb{Z}/2$提升到$\mathbb{Z}/2^k$的运算，从而能够实现所有对角逻辑门，例如单量子比特旋转$\operatorname{R}(2\pi/2^k)$。将这些运算与选择上同调类的映射复合，可产生可寻址逻辑门。示例包括三维和四维环面上的可寻址$T$门，以及超越拓扑码的qLDPC码中的门。我们引入了上同调运算，例如更高阶扭曲Pontryagin幂，据我们所知，这些运算此前未出现在数学文献中。这些运算源于上链上的$E_\infty$-结构，并通过循环分解和满射操作子进行显式构造。我们预期这些结果在同伦论中具有独立意义。我们的形式体系更广泛地适用于携带适当操作子结构的链复形，并预期这一操作子视角将揭示同伦论与容错量子门之间更多的联系。
英文摘要
In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of $C^{k-1}Z$ logical gates that we called copy-cup on $k$ copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from $\mathbb{Z}/2$ to $\mathbb{Z}/2^k$, enabling all diagonal logical gates such as single-qubit rotations $\operatorname{R}(2π/2^k)$. Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable $T$ gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from $E_\infty$-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.
Comments63 pages, 2 figures
