Operator-aware shadow importance sampling for accurate fidelity estimation

TL;DR

The paper tackles the problem of efficiently estimating the fidelity $ ext{tr}( ho O)$ between an unknown quantum state $ ho$ and a fixed target $O$. It introduces operator-aware shadow importance sampling (OASIS) built on informationally overcomplete POVMs (IOC-POVMs) to optimize measurement distributions and minimize estimator variance, yielding unbiased fidelities. It presents two classes: OASIS-GT for general targets via a linear-programming-optimized IOC-POVM expansion, and OASIS-ST for structured targets (GHZ and W) with scalable, grouping-based estimators that avoid exponential memory. Empirical results show state-of-the-art performance for Haar-random states with local Pauli measurements and strong accuracy for GHZ and W targets, highlighting both improved fidelity estimation and better scalability for structured states.

Abstract

Estimating the fidelity between an unknown quantum state and a fixed target is a fundamental task in quantum information science. Direct fidelity estimation (DFE) enables this without full tomography by sampling observables according to a target-dependent distribution. However, existing approaches face notable trade-offs. Grouping-based DFE achieves strong accuracy for small systems but suffers from exponential scaling, and its applicability is restricted to Pauli measurements. In contrast, classical-shadow-based DFE offers scalability but yields lower accuracy on structured states. In this work, we address these limitations by developing two classes of operator-aware shadow importance sampling algorithms using informationally overcomplete positive operator-valued measures. Instantiated with local Pauli measurements, our algorithm improves upon the grouping-based algorithms for Haar-random states. For structured states such as the GHZ and W states, our algorithm also eliminates the exponential memory requirements of previous grouping-based methods. Numerical experiments confirm that our methods achieve state-of-the-art performance across Haar-random, GHZ, and W targets.

Figures (5)

• Figure 1: General framework of DFE. The goal is to estimate $\text{tr}(\rho O)$ for a target state $O$. Given many copies of the unknown state $\rho$, random measurements are performed according to a distribution optimized for each $O$. After collecting the measurement statistics, the estimation algorithm produces an estimate of the fidelity.
• Figure 1: OASIS-GT.
• Figure 2: OASIS-ST for the GHZ state.
• Figure 3: OASIS-ST for the W state.
• Figure 4: Sorted insertion crawford2021efficient.