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Efficient Fidelity Estimation with Few Local Pauli Measurements

Mingyu Sun, Gabriel Waite, Michael Bremner, Christopher Ferrie

TL;DR

This work develops a scalable fidelity-estimation framework that uses a polynomial number of non-adaptive, local Pauli measurements to assess how closely a prepared state matches multiple target states. By analyzing the estimator’s bias through the mixing time $τ$ of a Markov chain induced by the target distribution and introducing the $k$-Generalized Local Escape Property ($k$-GLEP), the authors extend the method beyond Haar-random states to $ ext{ε}$-approximate state $t$-designs, low-depth circuits, and various physically relevant states, including certain mixed states. They provide an empirical check to diagnose applicability and demonstrate, with theory and numerics, efficient simultaneous fidelity estimation for $M$ targets with overhead $O( ext{log }M)$, along with practical use cases in benchmarking, barren-plateau mitigation, and phase learning in gapped Hamiltonians. The approach offers a scalable, versatile tool for quantum-device benchmarking and tomography assistance, clarifying fundamental limits of learning from local measurements and enabling adaptive quantum algorithms in high dimensions.

Abstract

As quantum devices scale, quantifying how close an experimental state aligns with a target becomes both vital and challenging. Fidelity is the standard metric, but existing estimators either require full tomography or apply only to restricted state/measurement families. Huang, Preskill, and Soleimanifar (Nature Physics, 2025) introduced an efficient certification protocol for Haar-random states using only a polynomial number of non-adaptive, single-copy, local Pauli measurements. Here, we adopt the same data collection routine but recast it as a fidelity estimation protocol with rigorous performance guarantees and broaden its applicability. We analyze the bias in this estimator, linking its performance to the mixing time $τ$ of a Markov chain induced by the target state, and resolve the three open questions posed by Huang, Preskill, and Soleimanifar (Nature Physics, 2025). Our analysis extends beyond Haar-random states to state $t$-designs, states prepared by low-depth random circuits, physically relevant states and families of mixed states. We introduce a $k$-generalized local escape property that identifies when the fidelity estimation protocol is both efficient and accurate, and design a practical empirical test to verify its applicability for arbitrary states. This work enables scalable benchmarking, error characterization, and tomography assistance, supports adaptive quantum algorithms in high dimensions, and clarifies fundamental limits of learning from local measurements.

Efficient Fidelity Estimation with Few Local Pauli Measurements

TL;DR

This work develops a scalable fidelity-estimation framework that uses a polynomial number of non-adaptive, local Pauli measurements to assess how closely a prepared state matches multiple target states. By analyzing the estimator’s bias through the mixing time of a Markov chain induced by the target distribution and introducing the -Generalized Local Escape Property (-GLEP), the authors extend the method beyond Haar-random states to -approximate state -designs, low-depth circuits, and various physically relevant states, including certain mixed states. They provide an empirical check to diagnose applicability and demonstrate, with theory and numerics, efficient simultaneous fidelity estimation for targets with overhead , along with practical use cases in benchmarking, barren-plateau mitigation, and phase learning in gapped Hamiltonians. The approach offers a scalable, versatile tool for quantum-device benchmarking and tomography assistance, clarifying fundamental limits of learning from local measurements and enabling adaptive quantum algorithms in high dimensions.

Abstract

As quantum devices scale, quantifying how close an experimental state aligns with a target becomes both vital and challenging. Fidelity is the standard metric, but existing estimators either require full tomography or apply only to restricted state/measurement families. Huang, Preskill, and Soleimanifar (Nature Physics, 2025) introduced an efficient certification protocol for Haar-random states using only a polynomial number of non-adaptive, single-copy, local Pauli measurements. Here, we adopt the same data collection routine but recast it as a fidelity estimation protocol with rigorous performance guarantees and broaden its applicability. We analyze the bias in this estimator, linking its performance to the mixing time of a Markov chain induced by the target state, and resolve the three open questions posed by Huang, Preskill, and Soleimanifar (Nature Physics, 2025). Our analysis extends beyond Haar-random states to state -designs, states prepared by low-depth random circuits, physically relevant states and families of mixed states. We introduce a -generalized local escape property that identifies when the fidelity estimation protocol is both efficient and accurate, and design a practical empirical test to verify its applicability for arbitrary states. This work enables scalable benchmarking, error characterization, and tomography assistance, supports adaptive quantum algorithms in high dimensions, and clarifies fundamental limits of learning from local measurements.

Paper Structure

This paper contains 31 sections, 15 theorems, 86 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Prior work.
  3. Contributions.
  4. Outline.
  5. Main Results
  6. Fidelity Estimation and Performance Guarantees
  7. Fidelity Estimation Procedure
  8. Procedure.
  9. Analysis.
  10. --Generalized Local Escape Property
  11. Empirical Check
  12. Numerical Results
  13. Benchmarking Quantum Devices
  14. Cost-Function-Dependent Barren Plateaus Mitigation
  15. Learning Properties of Ground States of Gapped Hamiltonians
  16. ...and 16 more sections

Key Result

Theorem 2.1

Let $\rho$ be an unknown $n$-qubit state and $\{ \vert \psi_i \rangle \}_{i=1}^M$ be $M$ pure target states. For each $\vert \psi_i \rangle$, let $\tau_i$ denote the mixing time of the Markov chain induced by $\vert \psi_i \rangle$. Suppose all $\tau_i$ are polynomially bounded in $n$, then local Pa

Figures (5)

  • Figure 1: Performance comparison of XEB (blue), shadow fidelity (orange), and true fidelity (gray dashed) under white (top), coherent (middle), and dephasing (bottom) noise. Columns show IQP states, phase states (diagonal-$Z$), stabilizer (Clifford) states, and ground state of TFIM (TFIM). Shadow overlap closely follows true fidelity across all cases, while XEB fails for states that are not anticoncentrated and fails for dephasing noise.
  • Figure 2: Training of a $41$-qubit quantum autoencoder ($n_A=1$, $n_B=40$) optimized with SPSA using: global cost (red), local cost (blue), and shadow fidelity cost (green). The global cost (red) remains flat near unity, indicating barren plateau, while the local (blue) and shadow fidelity (green) costs decrease steadily, indicating barren plateau mitigation. Shadow fidelity cost (green) closely tracks the performance of local cost (blue).
  • Figure 3: Estimated fidelities obtained from Protocol \ref{['prot:fidelity']} for system size $n=12$. The lab state is the ground state of $H_{bXXZ}$ (left)/$H_{bXXZ}$ (right). Left: Bond-alternating XXZ model with anisotropy $\Delta=3.0$, varying the ratio $r=J'/J$. The system transitions from a trivial phase (blue) to a symmetry-broken phase (green), and finally to a symmetry-protected topological (SPT) phase (red). Right: $J_1$–$J_2$ antiferromagnetic Heisenberg chain, varying $\alpha =J_2/J_1$. A Berezinskii–Kosterlitz–Thouless (BKT) transition separates the gapless Luttinger liquid phase (blue) from the gapped dimerized phase (red) at the critical point $\alpha_c \approx 0.2411$ (dashed line). For both Hamiltonians, the protocol accurately tracks the true fidelity (gray dashed lines) across phases, both under the noiseless case (orange circles) and the white noise with strength $p=0.3$ (orange squares).
  • Figure 4: Estimated fidelities obtained from Protocol \ref{['prot:fidelity']} for system sizes $n=3, 6, 9, 12, 15$. The lab state is the ground state of the cluster Hamiltonian (cf. Equation \ref{['eq:1d-cl-Ham']}). Left: noiseless; right: white noise with strength $p=0.3$. Orange circles are the estimates against the true cluster ground state, which agree with the true fidelities (gray dashed lines). Green circles show fidelities estimated against five independent random product states, consistently yielding negligible values. The protocol correctly estimates the fidelity values and clearly distinguishes the highly entangled SPT cluster state from trivial product states.
  • Figure 5: Two paths (purple and orange) from $x=000$ to $y=111$ on the $3$-dimensional hypercube.

Theorems & Definitions (22)

  • Theorem 2.1: Sample Complexity of Fidelity Estimation
  • Theorem 2.2: Fidelity Estimation for Most Quantum States
  • Theorem 2.3: Mixed States Certification
  • Definition 1: Good Vertex
  • Definition 2: $k$-Generalized Local Escape Property
  • Lemma 3.1: Connectivity
  • Theorem A.1: Chebyshev inequality
  • Theorem A.2: Hölder’s inequality
  • proof : Proof of Lemma \ref{['graph diam']}
  • Proposition B.1
  • ...and 12 more