Pathwise Random Hamiltonian Simulation
Abstract
Randomized product formulas such as qDrift offer a resource-efficient alternative to deterministic Trotter--Suzuki decompositions for Hamiltonian simulation, removing their polynomial dependence on the number of Hamiltonian terms. qDrift, however, is intrinsically limited to first order in the evolution time, so its query complexity remains linear in the inverse of the target accuracy, 1/ε. We introduce Pathwise Random Hamiltonian Simulation (PRHS), which extends qDrift to arbitrary order by subdividing each time step into M correlated slices, each evolving under a term sampled from a quasi-probability distribution that we construct in closed form and prove unique, with a bias decaying factorially in M. Optimizing jointly over the number of slices M and the number of independent blocks N interpolates between the standard qDrift protocol at long times and a high-precision regime where the query cost grows slower than any power of 1/ε, without requiring ancillary qubits. Numerical simulations of five molecular Hamiltonians confirm this advantage, with PRHS achieving accuracies two to four orders of magnitude beyond qDrift at equal query cost.
Related
- High-order Magnus expansion for Hamiltonian simulation
- Optimal Linear Combination of Hamiltonian Simulation for Dissipative Dynamics
- Efficient Quantum Simulation of Linearized Vlasov--Poisson Dynamics Using Trotter and THRIFT Hamiltonian Simulation Methods
- Observable-targeted variational quantum simulation of Hamiltonian dynamics
- Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation
BibTeX
@misc{cugini2026pathwise,
title = {{Pathwise Random Hamiltonian Simulation}},
author = {Davide Cugini},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29756},
eprint = {2608.29756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29756},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pathwise Random Hamiltonian Simulation
AU - Davide Cugini
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29756
UR - https://arxiv.org/abs/2608.29756
AB - Randomized product formulas such as qDrift offer a resource-efficient alternative to deterministic Trotter--Suzuki decompositions for Hamiltonian simulation, removing their polynomial dependence on the number of Hamiltonian terms. qDrift, however, is intrinsically limited to first order in the evolution time, so its query complexity remains linear in the inverse of the target accuracy, 1/ε. We introduce Pathwise Random Hamiltonian Simulation (PRHS), which extends qDrift to arbitrary order by subdividing each time step into M correlated slices, each evolving under a term sampled from a quasi-probability distribution that we construct in closed form and prove unique, with a bias decaying factorially in M. Optimizing jointly over the number of slices M and the number of independent blocks N interpolates between the standard qDrift protocol at long times and a high-precision regime where the query cost grows slower than any power of 1/ε, without requiring ancillary qubits. Numerical simulations of five molecular Hamiltonians confirm this advantage, with PRHS achieving accuracies two to four orders of magnitude beyond qDrift at equal query cost.
ER -