Quasi-probability decompositions (QPDs) have proven essential in many quantum algorithms and protocols -- one replaces a ``difficult'' quantum circuit with an ensemble of ``easier'' circuit variants whose weighted outcomes reproduce any target observable. This, however, inevitably yields an increased configuration variance beyond Born-rule shot noise. We develop a broad framework for accounting for and reducing this variance and prove that stratified sampling -- under ideal proportional allocation -- results in an unbiased estimator with a variance that is never worse than naïve sampling (with equality only in degenerate cases).
Furthermore, we provide a classical dynamic programme to enable stratification on arbitrary product-form QPDs. Numerical simulations of typical QPDs, such as Probabilistic Error Cancellation (PEC) and Probabilistic Angle Interpolation (PAI), demonstrate constant-factor reductions in overall variance (up to $\sim 60$--$80\%$ in an oracle model) and robust $\sim 10\%$ savings in the pessimistic single-shot regime. Our results can be applied immediately to reduce the net sampling cost of practically relevant QPDs that are commonly used in near term and early fault-tolerant algorithms without requiring additional quantum resources.
Quasi-probability decompositions (QPDs) have proven essential in many quantum algorithms and protocols -- one replaces a difficult'' quantum circuit with an ensemble of easier'' circuit variants whose weighted outcomes reproduce any target observable. This, however, inevitably yields an increased configuration variance beyond Born-rule shot noise. We develop a broad framework for accounting for and reducing this variance and prove that stratified sampling -- under ideal proportional allocation -- results in an unbiased estimator with a variance that is never worse than naïve sampling (with equality only in degenerate cases).
Furthermore, we provide a classical dynamic programme to enable stratification on arbitrary product-form QPDs. Numerical simulations of typical QPDs, such as Probabilistic Error Cancellation (PEC) and Probabilistic Angle Interpolation (PAI), demonstrate constant-factor reductions in overall variance (up to ∼60--80% in an oracle model) and robust ∼10% savings in the pessimistic single-shot regime. Our results can be applied immediately to reduce the net sampling cost of practically relevant QPDs that are commonly used in near term and early fault-tolerant algorithms without requiring additional quantum resources.
Quasiprobabilistic decompositions (QPDs) play a key role in maximizing the utility of near-term quantum hardware. For example, Probabilistic Error Cancellation (PEC) (an error mitigation technique) and circuit cutting (which enables large quantum computations to be performed on quantum hardware with a limited number of qubits) both involve QPDs. Computations based on QPDs typically incur large sampling overheads that grow exponentially with the number of error-terms mitigated or number of circuit-cuts employed, limiting their practical feasibility. In this work, we adapt the control variates variance reduction technique from the statistics literature in order to reduce the sampling overhead in QPD-based computations. We demonstrate our method using simulation experiments that mimic a realistic PEC scenario. In our experiments, we observed a more than 50% reduction in the number of samples needed to achieve a given precision, in more than 50% of the PEC-based estimations performed in the study when using our approach. We discuss how future research on constructing good control variates can lead to even stronger sampling overhead reduction.
Near-term quantum devices are limited by noise and hardware constraints, motivating algorithmic approaches that trade circuit complexity for increased sampling overhead. Quasi-probability decompositions (QPDs), for example, allow replacing non-local operations by multiple circuits with local operations, but the associated sampling overhead generally scales exponentially and limits their practicality. In this work, we introduce a reweighting strategy for QPDs for circuits with the same variational structure across parameter settings, reusing samples and thereby reducing the sampling overhead. We first demonstrate this approach by estimating fidelities between parameterized quantum states, a key primitive in variational time evolution and quantum kernel methods. Importantly, this setup allows controlling the exponential QPD sampling overhead while preserving the structure of the state-encoding ansatz. We then apply the method to estimate the real part of the quantum geometric tensor using the simultaneous perturbation stochastic approximation and find that, in the presence of realistic hardware noise, our method outperforms other standard estimation techniques. These results highlight the potential of reweighting strategies to extend the applicability of QPD-based methods in variational quantum algorithms.
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.
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.
Randomised quantum simulation algorithms such as qDRIFT reduce circuit depth by replacing deterministic product formulas with stochastic compilations, but achieving high accuracy can require many steps. In vanilla qDRIFT, Hamiltonian terms are sampled with probabilities proportional to their coefficient magnitudes. We propose StateDRIFT, a variant of qDRIFT which optimises its sampling distribution to estimate time-evolved quantities using knowledge of the initial state and target observable, with a provable truncation bound that keeps qDRIFT's guarantees. The distribution at each depth is chosen by solving an optimisation problem targeting bias and/or variance. StateDRIFT can significantly reduce the bias and variance of estimators, and hence circuit depth, which we test on short-time dynamics of lattice models and molecules under Jordan-Wigner up to 18 qubits. For example, with BeH2 StateDRIFT can reduce the number of two-qubit gates required to achieve a given accuracy from 11,201 to 983. StateDRIFT involves measuring observables and commutators on the initial state, so we propose two practical ways to estimate these: classical surrogates such as tensor networks, and classical shadows, which recover most or all of the saving from exact oracles in the systems tested.
Quasiprobabilistic decompositions (QPDs) play a key role in maximizing the utility of near-term quantum hardware. For example, Probabilistic Error Cancellation (PEC) (an error mitigation technique) and circuit cutting (which enables large quantum computations to be performed on quantum hardware with a limited number of qubits) both involve QPDs. Computations based on QPDs typically incur large sampling overheads that grow exponentially, e.g., with the number of error-terms mitigated or the number of circuit-cuts employed, limiting their practical feasibility. In this work, we adapt the control variates variance reduction technique from the statistics literature in order to reduce the sampling overhead in QPD-based computations. We demonstrate our method, dubbed CV4Quantum, using simulation experiments that mimic a realistic PEC scenario. In more than 50% of the PEC-based estimations performed in the study, we observed a more than 50% reduction in the number of samples needed to achieve a given precision when using our technique. We discuss how future research on constructing good control variates can lead to even stronger sampling overhead reduction.