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Resource-Optimal Importance Sampling for Randomized Quantum Algorithms

Davide Cugini, Touheed Anwar Atif, Yigit Subasi

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.

Resource-Optimal Importance Sampling for Randomized Quantum Algorithms

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.
Paper Structure (29 sections, 6 theorems, 247 equations, 3 figures, 1 table)

This paper contains 29 sections, 6 theorems, 247 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Optimal IS
  3. Composite Quantum Channels
  4. IS for imperfect quantum channels
  5. IS with error detection
  6. ZeroFill($L$)
  7. Discard($L$)
  8. Algorithm performance
  9. A Case Study: IID Parameter Model under Poisson Error Statistics
  10. Applications
  11. Qdrift
  12. He2 example
  13. Comparison with Related Approaches
  14. Dephasing channels
  15. Mixed States and Composite Observables
  16. ...and 14 more sections

Key Result

Theorem 1

Let $p(\underline{\theta})$ be a probability distribution, $c(\underline{\theta})$ a cost function, and $q(\underline{\theta})$ an arbitrary IS distribution. Then, with equality if and only if

Figures (3)

  • Figure 1:
  • Figure 2: Structure of a single circuit layer. The first stage consists of single-qubit gates applied to each qubit, followed by the randomized Pauli operations required to statistically cancel the noise and the localized single-qubit depolarizing channels. Similarly, the second stage analogously applies two-qubit gates, after which randomized Pauli operations are performed and the $2$-local depolarizing channels act.
  • Figure 3: Net cost ratio $NC_{q^*}/NC_p$ for a single layer with the structure in Figure \ref{['fig:layer_structure']}, as a function of the number of qubits $n$ and the depolarizing-noise parameter $\varepsilon$.

Theorems & Definitions (13)

  • Theorem 1: Optimal Sampling Distribution
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • Theorem 4
  • proof
  • ...and 3 more