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

Error bounds for the truncated Baker--Campbell--Hausdorff and Zassenhaus formulas in unitary problems

A. Arnal, F. Casas, J. L. Ruiz-Benito

The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.

math-phCross-listedarXiv:2607.07692Jul 9, 2026

Hybrid exponential product formulas for unbounded operators with possible applications to Monte Carlo simulations

Masuo Suzuki

Physics Letters AJun 1, 1995146 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.