Error-Mitigated Hamiltonian Simulation: Complexity Analysis and Optimization for Near-Term and Early-Fault-Tolerant Quantum Computers

Abstract

Simulating real-time dynamics under a Hamiltonian is a central goal of quantum information science. While numerous Hamiltonian-simulation quantum algorithms have been proposed, the effects of physical noise have rarely been incorporated into their performance analysis, despite the non-negligible noise levels of quantum devices. We present an end-to-end complexity analysis of noisy Hamiltonian simulation combined with quantum error mitigation (QEM) to answer how many circuit runs are required to reach a given target accuracy. Because the QEM sampling overhead grows exponentially with the circuit depth while the algorithmic error decreases with it, the circuit depth becomes an optimization variable, and we derive an analytic depth-selection rule for two algorithm families. For the order-$k$ Suzuki--Trotter formula, the optimized cost exhibits a critical error $\epsilon_c$, below which the required number of circuit runs grows exponentially. QEM improves the noise dependence of $\epsilon_c$ from sublinear to $k$th-power scaling, an exponent improvement by a factor of $k+1$. For randomized-LCU-based simulation, optimizing the repetition number yields a square-root improvement in the simulation-time dependence of the sampling-overhead exponent over the standard parameter choice. We further quantify the noise-characterization cost required for error mitigation via gate set tomography and the recently proposed space-time noise inversion method, showing that the latter can significantly reduce this cost.