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.
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 O(ε−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 O(ε−3) to O(ε−2log2(1/ε)), 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.
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.
Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work, we present qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. In qSWIFT, the required number of gates for a given precision is independent of the number of terms in the Hamiltonian, while the systematic error is exponentially reduced with regard to the order parameter. In this respect, our qSWIFT is a higher-order counterpart of the previously proposed quantum stochastic drift protocol (qDRIFT), the number of gates in which scales linearly with the inverse of the precision required. We construct the qSWIFT channel and establish a rigorous bound for the systematic error quantified by the diamond norm. qSWIFT provides an algorithm to estimate given physical quantities by using a system with one ancilla qubit, which is as simple as other product-formula-based approaches such as regular Trotter-Suzuki decompositions and qDRIFT. Our numerical experiment reveals that the required number of gates in qSWIFT is significantly reduced compared to qDRIFT. In particular, the advantage is significant for problems where high precision is required; e.g., to achieve a systematic relative propagation error of 10−6 , the required number of gates in third-order qSWIFT is 1000 times smaller than that of qDRIFT. Published by the American Physical Society 2024
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.
Hamiltonian simulation is a central application of quantum computing. Randomized Hamiltonian simulation approximates the target dynamics by sampling quantum circuits and often allows simpler circuit implementations. We develop a framework for randomized Hamiltonian simulation in which Hamiltonian terms are sampled with arbitrary probabilities and the corresponding short-time evolutions are implemented sequentially. The leading channel error relative to ideal time evolution is governed by the variance of the sampled generators. Minimizing the variance-based upper bound on the worst-case error recovers qDRIFT, a leading randomized Hamiltonian simulation algorithm that samples terms in proportion to their operator norms. For typical input states, however, this sampling distribution need not be optimal. Minimizing Haar-averaged error bounds instead yields sampling probabilities proportional to the Hilbert-Schmidt norms of the terms. This choice can provide smaller average-error bounds than conventional qDRIFT. For Hamiltonians expressed as sums of Pauli strings, we combine this framework with commuting Pauli grouping to obtain smaller error bounds than conventional qDRIFT at fixed Rz depth. Numerical benchmarks for the Sachdev-Ye-Kitaev model show error-bound improvement factors consistent with linear scaling in the number of qubits. For the 108-qubit FeMoco Hamiltonian, the error bounds are reduced by a factor of about 5.65. These results establish average-error optimization as a practical design principle for randomized Hamiltonian simulation.
Quantum simulation has wide applications in quantum chemistry and physics.Recently, scientists have begun exploring the use of randomized methods for accelerating quantum simulation.Among them, a simple and powerful technique, called QDRIFT, is known to generate random product formulas for which the average quantum channel approximates the ideal evolution.QDRIFT achieves a gate count that does not explicitly depend on the number of terms in the Hamiltonian, which contrasts with Suzuki formulas.This work aims to understand the origin of this speedup by comprehensively analyzing a single realization of the random product formula produced by QDRIFT.The main results prove that a typical realization of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error.The gate complexity is already independent of the number of terms in the Hamiltonian, but it depends on the system size and the sum of the interaction strengths in the Hamiltonian.Remarkably, the same random evolution starting from an arbitrary, but fixed, input state yields a much shorter circuit suitable for that input state.In contrast, in deterministic settings, such an improvement usually requires initial state knowledge.The proofs depend on concentration inequalities for vector and matrix martingales, and the framework is applicable to other randomized product formulas.Our bounds are saturated by certain commuting Hamiltonians.
Quantum simulation has wide applications in quantum chemistry and physics. Recently, scientists have begun exploring the use of randomized methods for accelerating quantum simulation. Among them, a simple and powerful technique, called qDRIFT, is known to generate random product formulas for which the average quantum channel approximates the ideal evolution. This work provides a comprehensive analysis of a single realization of the random product formula produced by qDRIFT. The main results prove that a typical realization of the randomized product formula approximates the ideal unitary evolution up to a small diamond-norm error. The gate complexity is independent of the number of terms in the Hamiltonian, but it depends on the system size and the sum of the interaction strengths in the Hamiltonian. Remarkably, the same random evolution starting from an arbitrary, but fixed, input state yields a much shorter circuit suitable for that input state. If the observable is also fixed, the same random evolution provides an even shorter product formula. The proofs depend on concentration inequalities for vector and matrix martingales. Numerical experiments verify the theoretical predictions.