Mathematical and numerical study of exponential product formulas and application to quantum dynamics

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, $H=A+αB$ , where $α$ is small. In the standard access model, where one can implement exponentials of $A$ and $B$ separately, our randomized formulas achieve $\mathcal O(α^2)$ error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an $Ω(α)$ lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of $A+αB_\ell$ 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 $\mathcal O(α^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $Ω(α^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.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, $H=A+αB$, where $α$ is small. In the standard access model, where one can implement exponentials of $A$ and $B$ separately, our randomized formulas achieve $\mathcal O(α^2)$ error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an $Ω(α)$ lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of $A+αB_\ell$ 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 $\mathcal O(α^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $Ω(α^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.

quant-pharXiv:2608.07720Aug 11, 20260 citations

Exponential product formulas and quantum analysis

Masuo Suzuki

東京大学 eBooksJan 1, 19961 citation

Error bounds on exponential product formulas for Schrödinger operators

Atsushi Doumeki, Takashi Ichinose, Hideo Tamura

Journal of the Mathematical Society of JapanApr 1, 199826 citations

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

Mauro Bologna

This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine the two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate closed formulas that have several applications in solving problems ranging from linear differential equations and quantum mechanics to non-linear differential equations.This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine the two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate closed formulas that have several applications in solving problems ranging from linear differential equations and quantum mechanics to non-linear differential equations.

arXiv:1804.01125Progress of Theoretical and Experimental PhysicsMar 28, 20192 citations

H=A+αB

, where

α

is small. In the standard access model, where one can implement exponentials of

A

and

B

separately, our randomized formulas achieve

\mathcal O(α^2)

error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an

Ω(α)

lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of

A+αB_\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(α^3)

error scaling with only constant-factor expected gate overhead. We also establish an

Ω(α^2)

lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.