Quantum simulation, the simulation of quantum processes on quantum computers, suggests a path forward for the efficient simulation of problems in condensed-matter physics, quantum chemistry, and materials science. While the majority of quantum simulation algorithms are deterministic, a recent surge of ideas has shown that randomization can greatly benefit algorithmic performance. In this work, we introduce a scheme for quantum simulation that unites the advantages of randomized compiling on the one hand and higher-order multiproduct formulas, as they are used for example in linear-combination-of-unitaries (LCU) algorithms or quantum error mitigation, on the other hand. In doing so, we propose a framework of randomized sampling that is expected to be useful for programmable quantum simulators and present two new multi-product formula algorithms tailored to it. Our framework reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by the implementation of multi-product formulas using standard LCU methods, rendering it especially useful for early quantum computers used to estimate the dynamics of quantum systems instead of performing full-fledged quantum phase estimation. Our algorithms achieve a simulation error that shrinks exponentially with the circuit depth. To corroborate their functioning, we prove rigorous performance bounds as well as the concentration of the randomized sampling procedure. We demonstrate the functioning of the approach for several physically meaningful examples of Hamiltonians, including fermionic systems and the Sachdev–Ye–Kitaev model, for which the method provides a favorable scaling in the effort.

中文翻译：

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哈密​​顿模拟的多乘积公式随机化

量子模拟，即在量子计算机上对量子过程的模拟，为高效模拟凝聚态物理、量子化学和材料科学中的问题提供了一条前进的道路。虽然大多数量子模拟算法都是确定性的，但最近涌现的大量想法表明，随机化可以极大地提高算法性能。在这项工作中，我们介绍了一种量子模拟方案，它结合了一方面随机编译和高阶多乘积公式的优点，因为它们用于例如线性组合单位 (LCU) 算法或量子误差另一方面，缓解。在这样做，我们提出了一个随机采样框架，预计该框架对可编程量子模拟器有用，并提出了两种为其量身定制的新的多乘积公式算法。我们的框架通过规避使用标准 LCU 方法实现多乘积公式所需的不经意幅度放大的需要来减少电路深度，使其对于用于估计量子系统动力学而不是执行成熟的早期量子计算机特别有用量子相位估计。我们的算法实现了一个模拟误差，该误差随着电路深度呈指数级缩小。为了证实它们的功能，我们证明了严格的性能界限以及随机抽样程序的集中度。