Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from the random sampling of the Hamiltonian terms and the operator splitting error. Leveraging on the bias/variance trade-off, the error can be minimized by balancing the two. The concentration bound provides an estimate on the number of gates. The estimates are verified by using numerical experiments on classical computers.

中文翻译：

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用于量子哈密顿模拟的部分随机 Trotter 算法

鉴于哈密顿量，幺正算子的计算一直是许多量子算法的核心。受现有确定性和随机方法的启发，我们提出了一种混合方法，其中在每个时间步评估具有大振幅的哈密顿量，而其余项则随机评估。获得均方误差的界限以及浓度界限。均方误差由方差项和偏差项组成，分别来自哈密顿项的随机抽样和算子分裂误差。利用偏差/方差权衡，可以通过平衡两者来最小化误差。浓度界限提供了对门数的估计。通过在经典计算机上使用数值实验来验证估计。