Optimizing the Tradeoff Between Trotter Error and Gate Error in Three- and Four-Wave Plasma Problems

Simulations on near-term quantum hardware are limited by hardware error, from gate infidelity and decoherence, and by algorithmic error introduced by approximations, such as the Trotter-Suzuki expansion. Using compilation techniques and an optimal Trotter step size, the algorithmic error incurred by the Trotter-Suzuki expansion, referred to as the "Trotter error," can be mitigated, and the simulation depth can be improved. In this paper, we explore the tradeoff between Trotter error and gate error in pursuit of the optimal Trotter step size. In particular, we simulate the three-wave and four-wave interaction Hamiltonian, describing a nonlinear optical process, on quantum hardware using a single compiled gate, which we repeatedly apply in a series of Trotterized steps to reach a desired simulation period. We evaluate expectation values of occupation numbers to assess the quality of simulations and use these results to evaluate an optimal Trotter step size. These results serve to facilitate the plasma community's interest and investment in quantum simulations by demonstrating successful simulation of nonlinear dynamics using product formulas and Trotter expansions to simulate the three-wave and four-wave unitary of interest.