Structure-Aware Variance Reduction for Unbiased Randomized Hamiltonian Simulation
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 $\approx70\%$ 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 $\approx 80\%$ leading to $\approx91\%$ and $\approx96\%$ sampling-cost reductions, respectively.