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arXiv:2604.07448

When is randomization advantageous in quantum simulation?

Francesco Paganelli, Michele Grossi, Andrea Giachero, Thomas E. O'Brien, Oriel Kiss

quant-phSubmitted Apr 8, 2026• Announced Apr 10, 2026• 0 citations
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Abstract

We study the regimes in which Hamiltonian simulation benefits from randomization. We introduce a sparse-QSVT construction based on composite stochastic decompositions, where dominant terms are treated deterministically and smaller contributions are sampled stochastically. Crucially, we analyze how stochastic and approximation errors propagate through block-encoding and QSVT procedures. To benchmark this approach, we construct ensembles of random Hamiltonians with controlled coefficient dispersion, locality, and number of terms, designed to favor randomization, and therefore providing an upper bound on its practical advantage. For Hamiltonians with many terms and highly inhomogeneous coefficient distributions, randomized methods reduce gate counts by up to an order of magnitude. However, this advantage is confined to moderate-precision regimes: as the target error decreases, deterministic methods become more efficient, with a crossover near $\varepsilon \sim 10^{-3}$. Although this regime partially overlaps with quantum chemistry Hamiltonians, realistic systems exhibit additional structure, such as commutation patterns, not captured by our model, which are expected to further favor deterministic approaches.

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We study the regimes in which Hamiltonian simulation benefits from randomization. We introduce a sparse-QSVT construction based on composite stochastic decompositions, where dominant terms are treated deterministically and smaller contributions are sampled stochastically. Crucially, we analyze how stochastic and approximation errors propagate through block-encoding and QSVT procedures. To benchmark this approach, we construct ensembles of random Hamiltonians with controlled coefficient dispersion, locality, and number of terms, designed to favor randomization, and therefore providing an upper bound on its practical advantage. For Hamiltonians with many terms and highly inhomogeneous coefficient distributions, randomized methods reduce gate counts by up to an order of magnitude. However, this advantage is confined to moderate-precision regimes: as the target error decreases, deterministic methods become more efficient, with a crossover near ε∼10−3\varepsilon \sim 10^{-3}ε∼10−3. Although this regime partially overlaps with quantum chemistry Hamiltonians, realistic systems exhibit additional structure, such as commutation patterns, not captured by our model, which are expected to further favor deterministic approaches.
Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.
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 RzR_zRz​ 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.
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/ε1/ε1/ε. We introduce Pathwise Random Hamiltonian Simulation (PRHS), which extends qDrift to arbitrary order by subdividing each time step into MMM 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 MMM. Optimizing jointly over the number of slices MMM and the number of independent blocks NNN 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/ε1/ε1/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.
Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.
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
For Hamiltonian H=∑jhjH = \sum_j h_jH=∑j​hj​, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms ∥hj∥\|h_j\|∥hj​∥, time ttt, and trace-distance error εεε. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case ∥hj∥\|h_j\|∥hj​∥ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in 1/ε1/ε1/ε, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.