Fawzi Hamza, Fawzi Omar, Scalet Samuel O
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom.
Univ Lyon, Inria, ENS Lyon, UCBL, LIP, Lyon, France.
Nat Commun. 2024 Aug 27;15(1):7394. doi: 10.1038/s41467-024-51592-3.
Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision ε, local observables can be approximated by a convex optimization program of quasi-polynomial size in 1/ε.
预测量子多体系统平衡态下的可观测量是一个核心问题，但也是出了名的难题。在物理上相关的热力学极限下，这一任务的某些数学表述甚至已被证明会导致不可判定的问题。由于缺乏针对该极限的经过认证的收敛界限，使用为有限系统设计的算法进行有限尺寸标度通常会失败。在这项工作中，我们设计了经过认证的算法，用于计算局部量子哈密顿量平衡态下可观测量的期望值，包括零温和正温情况。重要的是，我们的算法输出这些值的严格下限和上限。这使我们能够表明，局部可观测量的期望值可以在有限时间内近似，这与相关的不可判定性结果形成对比。当哈密顿量在二维晶格上可对易时，我们证明了高温下层级的快速收敛，结果是对于所需精度ε，局部可观测量可以由一个关于1/ε的拟多项式规模的凸优化程序近似。
