Hamiltonian Simulation via Stochastic Zassenhaus Expansions

Quantum algorithms for Hamiltonian simulation promise to enable not only the study of quantum dynamics but also to serve as key subroutines in applications across chemistry, materials science, and optimization. We introduce a class of ancilla-free quantum algorithms for Hamiltonian simulation called the Stochastic Zassenhaus Expansions (SZEs). These algorithms map nested Zassenhaus formulas onto quantum gates and then employ randomized sampling to minimize circuit depths. Unlike Suzuki-Trotter product formulas, which grow exponentially long with approximation order, the nested commutator structures of SZEs enable high-order formulas for many systems of interest. For a 10-qubit transverse-field Ising model, we construct an 11th-order SZE with 42x fewer CNOTs than the standard 10th-order product formula. Further, we empirically demonstrate regimes where SZEs reduce trace distance errors by many orders of magnitude compared to leading algorithms.