We introduce an algorithm to compute expectation values of time-evolved observables on digital quantum computers that requires only bounded average circuit depth to reach arbitrary precision, i.e. produces an unbiased estimator with finite average depth. This finite depth comes with an attenuation of the measured expectation value by a known amplitude, requiring more shots per circuit. The average gate count per circuit for simulation time t is \({\mathcal{O}}({t}^{2}{\mu }^{2})\) with μ the sum of the Hamiltonian coefficients, without dependence on precision, providing a significant improvement over previous algorithms. With shot noise, the average runtime is \({\mathcal{O}}({t}^{2}{\mu }^{2}{\epsilon }^{-2})\) to reach precision ϵ. The only dependence in the sum of the coefficients makes it particularly adapted to non-sparse Hamiltonians. The algorithm generalizes to time-dependent Hamiltonians, appearing for example in adiabatic state preparation. These properties make it particularly suitable for present-day relatively noisy hardware that supports only circuits with moderate depth.

我们引入了一种算法来计算数字量子计算机上时间演化可观测量的期望值，该算法仅需要有界平均电路深度即可达到任意精度，即产生具有有限平均深度的无偏估计器。这种有限的深度伴随着测量的期望值按已知幅度的衰减，每个电路需要更多的射击。仿真时间t内每个电路的平均门数为\({\mathcal{O}}({t}^{2}{\mu }^{2})\)，其中μ是哈密顿系数之和，无相关性在精度上，比以前的算法有了显着的改进。对于散粒噪声，达到精度ϵ 的平均运行时间为\({\mathcal{O}}({t}^{2}{\mu }^{2}{\epsilon }^{-2})\) 。系数总和的唯一依赖性使其特别适合非稀疏哈密顿量。该算法可推广到时间相关的哈密顿量，例如出现在绝热状态准备中。这些属性使其特别适合当今仅支持中等深度电路的相对嘈杂的硬件。