MLMC-qDRIFT: Multilevel Variance Reduction for Randomized Quantum Hamiltonian Simulation
This work reduces the sampling overhead for randomized quantum Hamiltonian simulation, benefiting researchers in quantum computing who need to estimate observables with high precision.
The paper introduces a multilevel Monte Carlo framework for qDRIFT that reduces the total gate complexity for fixed-precision observable estimation from O(ε^{-3}) to O(ε^{-2} log^2(1/ε)), while preserving qDRIFT's lack of explicit dependence on the number of Hamiltonian terms.
Simulating quantum dynamics is one of the central applications of quantum computing. For Hamiltonians written as a sum of many terms, deterministic Trotter--Suzuki product formulas can require applying a large number of term-wise evolutions at each time step, leading to high circuit costs for large or dense systems. Randomized methods such as qDRIFT offer an alternative: each step samples only one Hamiltonian term, giving a circuit depth with no explicit dependence on the number of terms. However, when qDRIFT is used for observable estimation, high precision requires many independent random circuit realizations, resulting in a total gate complexity that scales as $\mathcal{O}(\varepsilon^{-3})$. We introduce a multilevel Monte Carlo framework for qDRIFT that reduces this sampling overhead. The method constructs a hierarchy of qDRIFT estimators with increasing circuit depths and couples adjacent levels by sharing their random Hamiltonian-term samples. This coupling makes the variance of the level differences decay with depth, allowing most samples to be taken on cheaper, coarse circuits and only a few on expensive, fine circuits. We prove that the resulting MLMC-qDRIFT estimator reduces the total gate complexity for fixed-precision observable estimation from the standard qDRIFT scaling $\mathcal{O}(\varepsilon^{-3})$ to $\mathcal{O}(\varepsilon^{-2}\log^2(1/\varepsilon))$, while preserving qDRIFT's lack of explicit dependence on the number of Hamiltonian terms. Numerical experiments for spin-chain dynamics confirm the predicted variance decay and demonstrate the practical gate-count savings of the multilevel construction.