发表机构
CQSI, School of Computer Science, University of Technology Sydney（悉尼科技大学计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明给定与基态有重叠的引导态时，判定2-局域量子比特哈密顿量的Berry相位是BQP完全的，并扩展到加权海森堡相互作用及二维晶格，利用Schrieffer-Wolff变换建立1-局域与2-局域间的复杂性转变。
AI 中文摘要
我们证明，当给定一个引导态的经典描述，并承诺该引导态与系统的基态有重叠时，判定参数化的2-局域量子比特哈密顿量的Berry相位是BQP完全的。我们的结果扩展到具有加权海森堡相互作用的系统，并限制在二维正方形或三角形晶格几何上。我们开发的技术利用了Schrieffer-Wolff变换，该变换通常用于局域哈密顿量问题的微扰小工具约简构造，我们将其扩展到参数化的哈密顿量族。我们证明存在一组参数化的模拟器哈密顿量，其Berry相位能很好地逼近参数化目标族的Berry相位。利用Oliveira和Terhal以及Schuch和Verstraete的微扰小工具约简框架，我们将论证适应于参数化相互作用，并证明所得模拟中的误差界可以被控制。此外，我们提供了一个明确的证明，即1-局域哈密顿量族的Berry相位可以被高效地计算到逆多项式精度。这建立了1-局域和2-局域哈密顿量族之间的复杂性转变。
英文摘要
We prove that deciding the Berry phase for parameterised 2-local qubit Hamiltonians is BQP-complete when presented with a classical description of a guiding state, promised to overlap with the ground state of the system. Our results extend to systems with weighted Heisenberg interactions and when restricted to a 2D square or triangular lattice geometry. The techniques we develop leverage the Schrieffer--Wolff transformation, typically used in the construction of perturbative gadget reductions for local Hamiltonian problems, extending it to parameterised families of Hamiltonians. We demonstrate that there exists a choice of parameterised simulator Hamiltonians whose Berry phase well-approximates that of a parameterised target family. Using the perturbative gadget reduction framework of Oliveira and Terhal and of Schuch and Verstraete, we adapt the arguments to parameterised interactions and demonstrate the error bounds in the resulting simulation can be controlled. Additionally, we provide an explicit proof that families of 1-local Hamiltonians have a Berry phase that can be efficiently computed to inverse-polynomial precision. This establishes a complexity transition between 1-local and 2-local Hamiltonian families.
Comments41 + 30 pages; 8 figures. v1: comments welcome
