发表机构
The University of Tokyo; University of Oxford; The University of Osaka; Kyoto University（东京大学; 牛津大学; 大阪大学; 京都大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出近乎最优的量子算法估计二维哈密顿量族的陈数，并证明匹配的量子下界及BQP完备性，展示了量子计算在拓扑相表征中的优势。
AI 中文摘要
物质的拓扑相由超越常规局域序参量的量子态全局性质所表征。陈数是这一表征中的核心不变量，它将量子态的几何性质与稳健的物理可观测量联系起来，使其确定成为理解量子物质的关键。我们构造了一个近乎最优的量子算法，用于估计定义在二维参数环面上的、具有唯一且带能隙基态的光滑哈密顿量族（包括相互作用多体系统）的陈数。该算法传输并重用在一个参考点制备的两个基态副本，并使用广义量子信号处理来相干累积几何相位。它实现了哈密顿量预言机演化时间 $\tilde{O}(L_xL_y/\Delta_{\min}^3)$，其中 $L_x,L_y$ 界定了哈密顿量的参数导数，$\Delta_{\min}$ 是已知的谱隙下界。我们证明了匹配的最坏情况预言机下界，即使在陈数被承诺为零或一的情况下也成立，从而确立了在多项式对数因子意义下的最优性。对于具有逆多项式能隙和提供的引导态的局域哈密顿量族，精确陈数计算属于 $\mathsf{FBQP}$ 且是 $\mathsf{BQP}$-困难的，而仅判断陈数是否为零或一已经是 $\mathsf{BQP}$-完全的。这些结果确立了量子计算机估计量化拓扑不变量的效率，并为表征拓扑相中的量子优势提供了证据。
英文摘要
Topological phases of matter are characterized by global properties of quantum states beyond conventional local order parameters. The Chern number is a central invariant in this characterization, linking the geometry of quantum states to robust physical observables and making its determination key to understanding quantum matter. We construct a nearly optimal quantum algorithm for estimating the Chern number of a smooth Hamiltonian family over a two-dimensional parameter torus with a unique, gapped ground state, including interacting many-body systems. The algorithm transports and reuses two ground-state copies prepared at a single reference point and uses generalized quantum signal processing to coherently accumulate geometric phases. It achieves a Hamiltonian-oracle evolution time of $\tilde{O}(L_xL_y/Δ_{\min}^3)$, where $L_x,L_y$ bound the parameter derivatives of the Hamiltonian and $Δ_{\min}$ is a known spectral-gap lower bound. We prove a matching worst-case oracle lower bound, even when the Chern number is promised to be zero or one, establishing optimality up to polylogarithmic factors. For local Hamiltonian families with an inverse-polynomial gap and a supplied guiding state, exact Chern-number computation is in $\mathsf{FBQP}$ and is $\mathsf{BQP}$-hard, and merely deciding whether the Chern number is zero or one is already $\mathsf{BQP}$-complete. These results establish how efficiently quantum computers can estimate a quantized topological invariant and provide evidence for quantum advantage in characterizing topological phases.
