Resource-Optimal Importance Sampling for Randomized Quantum Algorithms

TL;DR

This paper presents a framework that optimizes resource usage in randomized quantum algorithms by applying importance sampling to reduce costs like circuit depth and energy, even in noisy or error-prone settings.

Contribution

It introduces a general importance sampling method for quantum protocols that minimizes combined cost and variance, extending to noisy and error-detection scenarios.

Findings

Reduces resource costs in quantum algorithms

Maintains estimator bias despite sampling changes

Applicable to various protocols like Qdrift and classical shadows

Abstract

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.