arXiv 2409.041612024-09quant-ph

哈密顿动力学均值与引导局部哈密顿问题的经典算法

Classical Algorithms for Hamiltonian Dynamics Mean Value and Guided Local Hamiltonian Problem

Yusen Wu, Yukun Zhang, Chuan Wang, Xiao Yuan

备注：: Improve the classical simulation algorithm and rewrite the manuscript

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AI总结：

本文提出一种经典算法，可高效模拟任意局部量子系统的短时动力学并估计全局泡利可观测量期望值，同时给出引导局部哈密顿问题的经典去量子化算法，揭示了常数时间量子动力学的经典可解性，细化了量子优势的边界。

摘要：

量子动力学和基态的高效模拟是物理学的核心挑战，也是量子优势的关键前沿。虽然一维系统中的短时演化通常可以用经典方法模拟，但将其扩展到更高维度仍然困难。在此，我们提出一种高效的经典算法，用于模拟任意局部量子系统的短时动力学。对于任意局部哈密顿量 $H$ 和常数演化时间 $t$，我们的方法能够以高精度和指数级小的失败概率估计形式为 $\langleψ|e^{iHt}Oe^{-iHt}|ψ\rangle$ 的期望值，其中 $O$ 为全局泡利可观测量，$|ψ\rangle$ 为稳定子态。此外，我们给出了一个定制量子算法的经典去量子化版本，该算法能够高效地将引导局部哈密顿（GLH）问题求解到常数加性误差——此前该问题被认为在经典上困难，因此被视为量子计算优势的有力候选。这些结果揭示了常数时间量子动力学中出人意料的经典可解性，以及哈密顿动力学均值与 GLH 问题之间的根本联系。我们的工作细化了经典与量子计算能力之间的边界，为量子优势可能真正出现的区域确定了更清晰的判据。

英文摘要：

The efficient simulation of quantum dynamics and ground states is a central challenge in physics and a key frontier for quantum advantage. While short-time evolution in one-dimensional systems can often be simulated classically, extending this to higher dimensions remains difficult. Here, we introduce an efficient classical algorithm for simulating the short-time dynamics of arbitrary local quantum systems. For any local Hamiltonian $H$ and constant evolution time $t$, our method estimates expectation values of the form $\langleψ|e^{iHt}Oe^{-iHt}|ψ\rangle$ for global Pauli observables $O$ and stabilizer states $|ψ\rangle$, with high precision and exponentially small failure probability. Furthermore, we present a classical dequantization of a tailored quantum algorithm that efficiently solves the guided local Hamiltonian (GLH) problem to constant additive error - previously considered classically hard and hence a promising candidate for quantum computational advantage. These results reveal unexpected classical tractability in constant-time quantum dynamics and fundamental connections between Hamiltonian dynamics mean value and the GLH problem. Our work refines the boundary between classical and quantum computational power, identifying sharper criteria for regimes where quantum advantage may genuinely emerge.

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