随机乘积公式中的集中性
Concentration for random product formulas
AI总结：
本文研究了qDRIFT方法中随机乘积公式的集中性，证明其在小的钻石范数误差下近似理想单位元演算，且门复杂度不依赖哈密顿量项数但依赖系统规模和相互作用强度之和。
摘要：
量子模拟在量子化学和物理中有广泛应用。最近，科学家开始探索使用随机方法加速量子模拟。其中，一种简单而强大的技术称为qDRIFT，能够生成随机乘积公式，其平均量子通道近似理想演算。qDRIFT实现的门数不显式依赖于哈密顿量项数，这与Suzuki公式形成对比。本文旨在通过综合分析qDRIFT生成的单个随机乘积公式实现，理解这种加速的来源。主要结果证明，随机乘积公式的一个典型实现能在小的钻石范数误差下近似理想单位元演算。门复杂度已不依赖哈密顿量项数，但依赖于系统规模和哈密顿量中相互作用强度之和。值得注意的是，相同的随机演算从任意但固定的输入状态出发，会产生更短的电路适用于该输入状态。相比之下，在确定性设置中，此类改进通常需要初始状态知识。证明依赖于向量和矩阵鞅的集中不等式，该框架适用于其他随机乘积公式。我们的界限由某些交换哈密顿量饱和。
英文摘要：
Quantum simulation has wide applications in quantum chemistry and physics. Recently, scientists have begun exploring the use of randomized methods for accelerating quantum simulation. Among them, a simple and powerful technique, called qDRIFT, is known to generate random product formulas for which the average quantum channel approximates the ideal evolution. qDRIFT achieves a gate count that does not explicitly depend on the number of terms in the Hamiltonian, which contrasts with Suzuki formulas. This work aims to understand the origin of this speed-up by comprehensively analyzing a single realization of the random product formula produced by qDRIFT. The main results prove that a typical realization of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error. The gate complexity is already independent of the number of terms in the Hamiltonian, but it depends on the system size and the sum of the interaction strengths in the Hamiltonian. Remarkably, the same random evolution starting from an arbitrary, but fixed, input state yields a much shorter circuit suitable for that input state. In contrast, in deterministic settings, such an improvement usually requires initial state knowledge. The proofs depend on concentration inequalities for vector and matrix martingales, and the framework is applicable to other randomized product formulas. Our bounds are saturated by certain commuting Hamiltonians.
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