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arXiv:2604.26263

qSHIFT: An Adaptive Sampling Protocol for Higher-Order Quantum Simulation

Sangjin Lee, Sangkook Cho

quant-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations
v1 · Apr 29, 2026v2 · Apr 30, 2026v3 · Sep 2, 2026
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Abstract

Quantum simulation is a cornerstone application for quantum computing, yet standard methods face a trade-off between circuit depth and accuracy: Trotterization depth scales with the number of Hamiltonian terms $L$, while sampling-based qDRIFT is restricted to $O(t^2)$ error scaling. Here, We introduce qSHIFT, an adaptive sampling protocol that overcomes these limitations. By adaptively updating sampling distributions, qSHIFT maintains $L$-independent gate complexity while achieving an improved error scaling of $O(t^{1+r})$ for an adjustable parameter $r$. This performance is enabled by a classical subroutine solving $L^r$ linear equations per sampling round. Numerical demonstrations confirm the $O(t^{1+r})$ scaling, showcasing qSHIFT as a resource-efficient framework for high-precision quantum simulation. Furthermore, the protocol's reduced circuit depth enhances its compatibility with physical error mitigation, making it a promising candidate for implementation on near-term quantum devices. In addition to its role as a standalone algorithm, qSHIFT can provide a high-precision foundation for modular quantum frameworks such as qSWIFT or Krylov quantum diagonalization.

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7+14 pages, 1 figure

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Simulating many-body quantum systems is a promising task for quantum computers. However, the depth of most algorithms, such as product formulas, scales with the number of terms in the Hamiltonian, and can therefore be challenging to implement on near-term, as well as early fault-tolerant quantum devices. An efficient solution is given by the stochastic compilation protocol known as qDrift, which builds random product formulas by sampling from the Hamiltonian according to the coefficients. In this work, we unify the qDrift protocol with importance sampling, allowing us to sample from arbitrary probability distributions, while controlling both the bias, as well as the statistical fluctuations. We show that the simulation cost can be reduced while achieving the same accuracy, by considering the individual simulation cost during the sampling stage. Moreover, we incorporate recent work on composite channel and compute rigorous bounds on the bias and variance, showing how to choose the number of samples, experiments, and time steps for a given target accuracy. These results lead to a more efficient implementation of the qDrift protocol, both with and without the use of composite channels. Theoretical results are confirmed by numerical simulations performed on a lattice nuclear effective field theory.
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 protocols are procedures that incorporate probabilistic choices during their execution and they play a central role in quantum algorithms, spanning Hamiltonian simulation, noise mitigation, and measurement tasks. In practical implementations, the dominant cost of such protocols typically arises from circuit execution and measurement, and depends on hardware-specific resources such as gate counts, circuit depth, runtime, or dissipated energy. We introduce a general framework for applying classical importance sampling to randomized quantum protocols. Given a cost function for running quantum circuits, the proposed approach minimizes a net-cost figure of merit that jointly captures the computational expense per circuit and the estimator variance. We further extend the framework to scenarios where the quantum computation is subject to errors arising either from algorithmic approximations or from physical noise, proving that importance sampling preserves estimator bias despite altering the sampling distribution, and to settings with error-detection schemes, where we characterize the resulting changes in the optimal sampling strategy and achievable net-cost reduction. Representative applications include the Qdrift protocol, dephasing channels, mixed-states simulation, composite observables estimation, classical shadows, and probabilistic error cancellation. Overall, our results establish a principled approach for reducing the computational resources required by randomized quantum protocols through classical sampling optimization.
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