发表机构
Monash University; Universitas Kristen Immanuel; Hon-Hai Research Institute（莫纳什大学; 伊曼纽尔基督大学; 鸿海研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明无限温度谱隙可传递至任意有限温度下CKG量子吉布斯采样器，并构造单纯复形耦合算子，为量子拓扑数据分析中的态制备提供条件收敛保证。
AI 中文摘要
通过耗散动力学制备吉布斯态需要控制所选哈密顿量 $H$ 和耦合算子的收敛性。然而，高效实现需要从动力学生成元的谱隙获得收敛保证，这仍然是一个关键挑战。近期研究通过限制动力学并假设哈密顿量及其相互作用的结构来解决这一挑战。对于 Chen-Kastoryano-Gilyen (CKG) 构造和任意有限维哈密顿量，我们证明了一个显式比较，将无限温度生成元的谱隙估计转化为 CKG 量子吉布斯采样器在每个有限正逆温度 $\beta$ 下谱隙的下界。这使得单个无限温度谱隙估计能够跨哈密顿量和温度认证收敛。这样的估计通常可从经典谱隙估计（如经典随机游走谱隙）得知。对于作为哈密顿量的单纯复形的霍奇拉普拉斯算子，我们构造了耦合算子，其无限温度生成元具有与经典单纯下-上行走相同的谱隙。行走的正谱隙和有界谱宽 $W=\lambda_{\max}(H)-\lambda_{\min}(H)$ 给出在每个固定温度下与单纯形数量无关的生成元谱隙。然后我们建立了采样器在多项式演化时间内近似制备零能量（调和）态的条件。这为热方法中量子拓扑数据分析的态制备提供了条件收敛保证。
英文摘要
Preparing Gibbs states through dissipative dynamics requires controlling convergence for the chosen Hamiltonian $H$ and coupling operators. However, efficient implementation requires convergence guarantees from the spectral gap of the generator of the dynamics, which remains a crucial challenge. Recent results address the challenge by restricting the dynamics with structural assumptions about the Hamiltonian and its interactions. For the Chen-Kastoryano-Gilyen (CKG) construction and arbitrary finite-dimensional Hamiltonians, we prove an explicit comparison that converts a spectral gap estimate for the infinite-temperature generator into a lower bound on the spectral gap of the CKG quantum Gibbs sampler at every finite positive inverse temperature $β$. It allows a single spectral gap estimate at infinite temperature to certify convergence across Hamiltonians and temperatures. Such an estimate is often known from a classical spectral gap estimate, such as the classical random walk spectral gap. For Hodge Laplacians of simplicial complexes as Hamiltonians, we construct coupling operators whose generator at infinite temperature has the same spectral gap as a classical simplicial down-up walk. A positive spectral gap of the walk and bounded spectral width $W=λ_{\max}(H)-λ_{\min}(H)$ give a generator spectral gap independent of the number of simplices at every fixed temperature. We then establish conditions under which the sampler approximately prepares zero-energy (harmonic) states in polynomial evolution time. This provides a conditional convergence guarantee for the state preparation in the thermal approach to quantum topological data analysis.
