AI 中文总结
该研究开发了一种资源高效的量子算法，利用对易子缩放和经典后处理随机乘积公式电路，实现更低电路深度，可用于矩阵函数性质估计等任务，数值实验验证了其性能优势。
AI 中文摘要
我们开发了用于估计厄米矩阵的一般矩阵函数性质的量子算法，可应用于相位估计、格林函数评估以及时间演化态的测量分布估计。所得方法在矩阵参数上展现出与乘积公式中常见的对易子缩放特性，在其他参数下具有更低的电路深度，且仅需1个辅助量子比特。我们的核心基础是对随机选取的乘积公式电路进行经典后处理，数学上对应于理查森外推的近似。在该框架内，我们引入了一种用于近似量子态测量分布的协议，其适用范围超出了标准可观测量估计。我们还为实际相关系统（包括具有k局域相互作用、长尾矩阵系综和守恒量的系统）提供了更严格的门复杂度界。最后，数值实验证实，在特定参数范围内，我们的方法能实现比标准乘积公式显著更浅的电路深度，并凸显了其启发式应用的潜力。
英文摘要
We develop quantum algorithms for estimating properties of general matrix functions of Hermitian matrices, with applications to phase estimation, Green's function evaluation, and estimating measurement distributions of time-evolved states. The resulting methods exhibit commutator scaling in matrix parameters similar to that usually found for product formulae, lower circuit depth in other parameters, and require only a single ancillary qubit. Our central primitive consists of classically postprocessing randomly chosen product formulae circuits, which mathematically corresponds to an approximation of a Richardson extrapolation. Within our framework, we introduce a protocol for approximating the measurement distributions of quantum states, extending beyond standard observable estimation. We also provide tightened gate complexity bounds for practically relevant systems, including those with k-local interactions, long-tailed matrix ensembles, and conserved quantities. Finally, numerical experiments confirm that our method can achieve significantly shallower circuit depths than standard product formulae in certain parameter regimes, and highlight the potential of their heuristic application.
Comments17+56 pages, 3 tables, 6 figures
