



01557nas a2200169   4500008004100000245004600041210004600087260001500133490000800148520110300156100001301259700001401272700002401286700001701310700002301327856003701350       2022                        eng d00aHamiltonian simulation with random inputs0 aHamiltonian simulation with random inputs  c12/30/20220 v1293 a<p>The algorithmic error of digital quantum simulations is usually explored in terms of the spectral norm distance between the actual and ideal evolution operators. In practice, this worst-case error analysis may be unnecessarily pessimistic. To address this, we develop a theory of average-case performance of Hamiltonian simulation with random initial states. We relate the average-case error to the Frobenius norm of the multiplicative error and give upper bounds for the product formula (PF) and truncated Taylor series methods. As applications, we estimate average-case error for digital Hamiltonian simulation of general lattice Hamiltonians and k-local Hamiltonians. In particular, for the nearest-neighbor Heisenberg chain with n spins, the error is quadratically reduced from O(n) in the worst case to O(n&minus;&minus;&radic;) on average for both the PF method and the Taylor series method. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models. We also apply our results to error analysis in the simulation of quantum scrambling.</p>1 aZhao, Qi1 aZhou, You1 aShaw, Alexander, F.1 aLi, Tongyang1 aChilds, Andrew, M.  uhttps://arxiv.org/abs/2111.04773