Clifford 层级的上同调刻画
A Cohomological Characterization of the Clifford Hierarchy
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中文总结 AI 辅助
本文通过证明酉算子的量子导数构成非阿贝尔1-上循环，给出Clifford层级的递归上同调刻画，并应用于第三层级，证明了所有二量子比特和三量子比特门均为半Clifford。
中文摘要 AI 辅助
Clifford 层级在容错量子计算中扮演核心角色，但其更高层级仅被部分理解。近期关于量子高阶傅里叶分析的工作利用量子导数刻画了 Clifford 层级中的成员资格，但未提供各个层级的显式描述。在本工作中，我们通过证明酉算子的量子导数集合构成非阿贝尔 1-上循环，且反之每个这样的 1-上循环都源于一个酉算子，来解决此问题。这产生了 Clifford 层级的递归上同调刻画。随后，我们将此框架专门应用于第三层级，并将上循环数据分解为辛、仿射和相位分量，每个分量均具有上同调解释。作为应用，我们给出了第三层级中每个双量子比特门是半 Clifford 的新证明，并首次证明了第三层级中每个三量子比特门是半 Clifford。
英文摘要
The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian \(1\)-cocycle, and conversely that every such \(1\)-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.
发表机构
- University of Maryland（马里兰大学）
- University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。
