AI 中文总结
研究费米子到量子比特编码，基于加权超图和耦合空间表示构建几何框架，通过BK和XBK编码进行分析，发现几何普适类与谱组织，应用于多体模型，捕捉编码哈密顿量结构演化，确立超图几何为新框架。
AI 中文摘要
传统上，精确的费米子到量子比特的变换被视为将多体哈密顿量转换为量子模拟的量子比特表示的算法工具。本文表明，它们还定义了内在的几何表示，其结构编码了超越谱等价的物理有意义的信息。我们基于加权超图和由Bravyi-Kitaev（BK）和Xia-Bian-Kais（XBK）编码构建的耦合空间表示开发了一个几何框架。在BK表示中，我们引入了一个几何可观测量，它比较了动力学和相互作用超图的代数连通性，推导出其对相互作用强度的精确解析依赖性，并发现了两个几何普适类以及源自编码二叉树架构的精确谱组织。互补的XBK表示通过耦合空间中的概率测度描述了编码哈密顿量的演化，其中最优传输独立于谱分析量化了相互作用驱动的重组。对哈伯德模型、无自旋tV模型、单杂质安德森模型和Kitaev模型的应用表明，这些基于连通性和传输的几何描述一致地捕捉了不同多体系统类中编码量子哈密顿量的结构演化。我们的结果将超图几何确立为理解费米子到量子比特编码的新框架，揭示了它们不仅作为计算映射，而且作为量子多体哈密顿量的几何表示。
英文摘要
Exact fermion to qubit transformations are conventionally regarded as algorithmic tools that translate many-body Hamiltonians into qubit representations for quantum simulation. Here we show that they also define intrinsic geometric representations whose structure encodes physically meaningful information beyond spectral equivalence. We develop a geometric framework based on weighted hypergraphs and coupling space representations constructed from the Bravyi--Kitaev (BK) and Xia--Bian--Kais (XBK) encodings. Within the BK representation, we introduce a geometric observable that compares the algebraic connectivities of the kinetic and interaction hypergraphs, derive its exact analytical dependence on interaction strength, and uncover two geometric universality classes together with an exact spectral organization originating from the binary tree architecture of the encoding. The complementary XBK representation describes the evolution of encoded Hamiltonians through probability measures in coupling space, where optimal transport quantifies interaction-driven reorganization independently of the spectral analysis. Applications to the Hubbard, spinless tV , single impurity Anderson, and Kitaev models demonstrate that these connectivity and transport based geometric descriptions consistently capture the structural evolution of encoded quantum Hamiltonians across distinct classes of many-body systems. Our results establish hypergraph geometry as a new framework for understanding fermion-to-qubit encodings,revealing that they serve not only as computational mappings but also as geometric representations of quantum many-body Hamiltonians.
