Structure-Aware Variance Reduction for Unbiased Randomized Hamiltonian Simulation
Abstract
Randomized Hamiltonian simulation methods are often governed by a trade-off between systematic bias and sampling overhead. We study how classical variance-reduction techniques can be applied to such methods without changing their mean channel, and therefore without introducing additional bias. As a motivating unbiased estimator, we formulate continuous time-evolution probabilistic angle interpolation (continuous TE-PAI), a quasiprobabilistic random-circuit protocol whose remaining Monte Carlo error is purely statistical. Continuous TE-PAI removes Trotter discretization error with finite-depth random circuits, whereas deterministic Trotterization does so only in the infinite-depth limit. Further, in tensor-network simulations, we demonstrate that discretization error can cause an unphysical exponential growth in the bond dimension required for Trotterized simulations, whereas comparable-depth continuous TE-PAI circuits avoid this growth. We then show that the variance of randomized product-formula-based estimators admits a canonical decomposition into a classical counting component and a quantum ordering component such that the dominant simulation overhead results from the non-commutative parts of the Hamiltonian dynamics. Motivated by this decomposition, we achieve an ≈70% error-reduction using the counting-component for small systems whereas our tensor-network simulations of n=30 spin-chain dynamics use coarser statistics tailored to the observable and estimator attaining a negligible bias and a reduction of ≈ 80% leading to ≈91% and ≈96% sampling-cost reductions, respectively.
Related
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- Transient subtraction method: An alternative approach for variance reduction in dissipative particle dynamics
BibTeX
@misc{dai2026structureaware,
title = {{Structure-Aware Variance Reduction for Unbiased Randomized Hamiltonian Simulation}},
author = {Joshua W. Dai and Fredrik Hasselgren and Chusei Kiumi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.23544},
eprint = {2606.23544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23544},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Structure-Aware Variance Reduction for Unbiased Randomized Hamiltonian Simulation
AU - Joshua W. Dai
AU - Fredrik Hasselgren
AU - Chusei Kiumi
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2606.23544
UR - https://arxiv.org/abs/2606.23544
AB - Randomized Hamiltonian simulation methods are often governed by a trade-off between systematic bias and sampling overhead. We study how classical variance-reduction techniques can be applied to such methods without changing their mean channel, and therefore without introducing additional bias. As a motivating unbiased estimator, we formulate continuous time-evolution probabilistic angle interpolation (continuous TE-PAI), a quasiprobabilistic random-circuit protocol whose remaining Monte Carlo error is purely statistical. Continuous TE-PAI removes Trotter discretization error with finite-depth random circuits, whereas deterministic Trotterization does so only in the infinite-depth limit. Further, in tensor-network simulations, we demonstrate that discretization error can cause an unphysical exponential growth in the bond dimension required for Trotterized simulations, whereas comparable-depth continuous TE-PAI circuits avoid this growth. We then show that the variance of randomized product-formula-based estimators admits a canonical decomposition into a classical counting component and a quantum ordering component such that the dominant simulation overhead results from the non-commutative parts of the Hamiltonian dynamics. Motivated by this decomposition, we achieve an ≈70% error-reduction using the counting-component for small systems whereas our tensor-network simulations of n=30 spin-chain dynamics use coarser statistics tailored to the observable and estimator attaining a negligible bias and a reduction of ≈ 80% leading to ≈91% and ≈96% sampling-cost reductions, respectively.
ER -