发表机构
Inria; LIX, École Polytechnique; CPHT, CNRS; Institut Polytechnique de Paris（法国国家信息与自动化研究所; 巴黎综合理工学院LIX实验室; 法国国家科学研究中心CPHT中心; 巴黎理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对费米子量子计算中并行组合交换律失效的问题，提出对称预幺半范畴语义，并引入pronaps扩展图解记号，为费米子ZW演算的多个片段给出规范形式和完备性证明，连接了电路语义、图解重写与行列式代数。
AI 中文摘要
基于局域费米子模式（LFM）的量子计算，其纯态空间通过Jordan-Wigner表示与量子比特态空间同构，但其组合结构存在微妙差异。实际上，规定如何嵌入费米子系统的代数形式主义支撑了一种并行组合的概念，对于这种并行组合，通常幺半范畴的交换律会失效。我们对此现象给出范畴论解释。从费米子门应用的CAR代数语义出发，我们证明LFM过程构成一个对称预幺半范畴，其中心恰好是偶数、保宇称的子范畴。同样的过程也可以改用普通张量积和一种仅在偶数映射上自然的费米子预对称性来组织。为了在后一种表述中进行图解推理，我们引入pronaps，这是对prop概念的一种放宽，其中置换不一定是自然的，并利用它们来组织与物理动机门集上的费米子电路研究相关的费米子ZW演算的片段层次结构。然后我们将可扩展图解记号扩展到pronaps。在此设定中，定义了受SZX和GSA启发的语法糖，其特殊性在于矩阵箭头编码子式和行列式，而图三角形编码主子阵的普法夫ian。这些构造为我们的EoW、FoW、EW和FW片段的表示提供了优雅的规范形式和完备性证明。特别是，FW表示给出了matchgate片段的一种新规范形式，不同于同一范畴的平面-W（pW）表示的rWGS-X规范形式。由此产生的框架连接了费米子电路语义、具有可扩展记号的图解重写以及行列式和普法夫ian的代数。
英文摘要
Local fermionic mode (LFM) based quantum computation has pure state spaces that are isomorphic, via the Jordan-Wigner representation, to qubit state spaces, but its compositional structure is subtly different. Indeed, the algebraic formalism specifying how to embed fermionic systems underlies a notion of parallel composition, for which, in general, the usual interchange law of monoidal categories fails. We give a categorical account of this phenomenon. Starting from the CAR-algebraic semantics of fermionic gate application, we show that LFM processes form a symmetric premonoidal category whose centre is precisely the even, parity-preserving subcategory. The same processes can alternatively be organized with the ordinary tensor product and a fermionic presymmetry that is natural exactly on even maps. To reason diagrammatically in this latter presentation, we introduce pronaps, a relaxation of the notion of prop, in which permutations are not-necessarily-natural, and use them to organize a hierarchy of fragments of the fermionic ZW calculus relevant to the study of fermionic circuits on physically motivated gate-sets. We then extend scalable diagrammatic notation to pronaps. In this setting, syntactic sugars inspired by SZX and GSA are defined, with the specificity that matrix arrows encode minors and determinants, while graph triangles encode pfaffians of principal submatrices. These constructions yield elegant normal forms and completeness proofs for our presentations of the EoW, FoW, EW, and FW fragments. In particular, the FW presentation gives a new normal form for the matchgate fragment, distinct from the rWGS-X normal form of the planar-W (pW) presentation of the same category. The resulting framework connects fermionic circuit semantics, diagrammatic rewriting with scalable notations, and the algebra of determinants and pfaffians.
