Randomized product formulas beyond optimal deterministic scaling

Abstract

Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems on quantum computers. Here we introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales, $H=A+\alpha B$, where $\alpha$ is small. In the standard access model, where one can implement exponentials of $A$ and $B$ separately, our randomized formulas achieve $\mathcal{O}({\alpha }^2)$ error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an $\Omega (\alpha )$ lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of $A+\alpha B_{\ell }$ for $B={\sum }_{\ell }{B}_{\ell }$, our randomized formula, based on Trotter Heuristic Resource Improved Formulas for Time-dynamics (THRIFT)~[J. L. Bosse et al., Nat. Commun. 16, 2673 (2025)], achieves $\mathcal{O}({\alpha }^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $\Omega ({\alpha }^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.