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arXiv:2501.19228 · Quantum

Sampling Groups of Pauli Operators to Enhance Direct Fidelity Estimation

Júlia Barberà-Rodríguez, Mariana Navarro, Leonardo Zambrano

Published Jul 3, 2025• 3 citations
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Abstract

Direct fidelity estimation is a protocol that estimates the fidelity between an experimental quantum state and a target pure state. By measuring the expectation values of Pauli operators selected through importance sampling, the method is exponentially faster than full quantum state tomography. We propose an enhanced direct fidelity estimation protocol that uses fewer copies of the experimental state by grouping Pauli operators before the sampling process. We derive analytical bounds on the measurement cost and estimator variance, showing improvements over the standard method. Numerical simulations validate our approach, demonstrating that for 8-qubit Haar-random states, our method achieves a one-third reduction in the required number of copies and reduces variance by an order of magnitude using only local measurements. These results underscore the potential of our protocol to enhance the efficiency of fidelity estimation in current quantum devices.

DOI
10.22331/q-2025-07-03-1784

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Direct fidelity estimation is a protocol that estimates the fidelity between an experimental quantum state and a target pure state. By measuring the expectation values of Pauli operators selected through importance sampling, the method is exponentially faster than full quantum state tomography. We propose an enhanced direct fidelity estimation protocol that uses fewer copies of the experimental state by grouping Pauli operators before the sampling process. We derive analytical bounds on the measurement cost and estimator variance, showing improvements over the standard method. Numerical simulations validate our approach, demonstrating that for 8-qubit Haar-random states, our method achieves a one-third reduction in the required number of copies and reduces variance by an order of magnitude using only local measurements. These results underscore the potential of our protocol to enhance the efficiency of fidelity estimation in current quantum devices.
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.
Direct fidelity estimation provides a way to estimate the fidelity between an experimentally prepared state and a desired pure target state without performing full tomography. Two influential formulations were introduced in 2011 by Flammia and Liu and by da Silva, Landon-Cardinal, and Poulin. In these protocols, the total estimation error is controlled through two distinct probabilistic steps: first, the fidelity is approximated using randomly sampled Pauli observables; second, each sampled expectation value is estimated from finitely many measurement outcomes. In this work we show that additional structural information about the noise can substantially sharpen the corresponding resource bounds. In particular, for some canonical channels the effective number of sampled Pauli settings can be reduced, leading to lower measurement cost both in the general pure-state setting and in the case of a stabilizer state. These results illustrate a broader point: worst-case confidence bounds in direct fidelity estimation can be significantly conservative when experimentally relevant structure is ignored. As a technical ingredient, we also revisit the allocation of the total accuracy and confidence budgets between the two probabilistic steps. Reformulating the analysis in terms of separate error parameters yields a constrained optimization problem whose solution lowers the average number of measurements in the general pure-state setting. Numerical simulations based on quantum circuits implemented in Qiskit illustrate both the improvement obtained under structured-noise assumptions and the conservativeness of the original worst-case bounds.
Reliable characterization of quantum states and quantum processes is central to the development of scalable quantum computing. Standard characterization techniques such as full quantum state tomography become rapidly impractical as system size increases. In this work we present two software-oriented fidelity estimation protocols that avoid full tomographic reconstruction while remaining experimentally relevant for noisy intermediate-scale quantum devices. The first protocol is direct fidelity estimation (DFE), where the fidelity between a target state and an experimentally prepared state is estimated from importance-sampled Pauli measurements. The second protocol is a survival-probability-based method that reconstructs fidelity from experimentally accessible moments of a noisy evolution and its approximate inverse, and is intrinsically robust against state-preparation-and-measurement (SPAM) errors. We discuss the theoretical basis of both approaches, analyze their noise sensitivity, and present numerical evidence on representative multiqubit states under depolarizing, dephasing, and amplitude-damping channels. For DFE we show that attenuation-aware correction improves robustness against measurement noise. For the survival-probability approach we show that an L2L_2L2​-optimized expansion substantially outperforms naive Taylor truncation in high-noise regimes.
Direct fidelity estimation (DFE) is a famous tool for estimating the fidelity with a target pure state. However, such a method generally requires exponentially many sampling copies due to the large magic of the target state. This work proposes a sample- and gate-efficient fidelity estimation algorithm that is affordable within feasible quantum devices. We show that the fidelity estimation with pure states close to the structure of phase states, for which sample-efficient DFE is limited by their strong entanglement and magic, can be done by using O(poly(n))\mathcal{O}({\rm poly}(n))O(poly(n)) sampling copies, with a single nnn-qubit fan-out gate. As the target state becomes a phase state, the sampling complexity reaches O(1)\mathcal{O}(1)O(1). Such a drastic improvement stems from a crucial step in our scheme, the so-called phase stripping, which can significantly reduce the target-state magic. Furthermore, we convert a complex diagonal gate resource, which is needed to design a phase-stripping-adapted algorithm, into nonlinear classical post-processing of Pauli measurements so that we only require a single fan-out gate. Additionally, as another variant using the nonlinear post-processing, we propose a nonlinear extension of the conventional DFE scheme. Here, the sampling reduction compared to DFE is also guaranteed, while preserving the Pauli measurement as the only circuit resource. We expect our work to contribute to establishing noise-resilient quantum algorithms by enabling a significant reduction in sampling overhead for fidelity estimation under the restricted gate resources, and ultimately to clarifying a fundamental gap between the resource overhead required to understand complex physical properties and that required to generate them.
A constant number of random Clifford measurements allows the classical shadow protocol to perform direct fidelity estimation (DFE) with high precision. However, estimating properties of an unknown quantum state is expected to be more feasible with random Pauli measurements than with random Clifford measurements in the near future. Inspired by the importance sampling technique applied to sampling Pauli measurements for DFE, we show that similar strategies can be derived from classical shadows. Specifically, we describe efficient methods using only local Pauli measurements to perform DFE with Greenberger-Horne-Zeilinger (GHZ), W , and Dicke states, establishing tighter bounds (by factors of 14.22 and 16 for GHZ and W /Dicke, respectively) on the number of measurements required for desired precision. These protocols are derived by adjusting the distribution of observables. Notably, they require no preprocessing steps other than the sampling algorithms.
Fault-tolerant quantum computation hinges on the requirement for low physical error rates. Reaching below threshold regime requires the accounting of circuit dependent noise, that is inherent to the execution context in which a quantum gate is usually embedded. Direct fidelity estimation is a technique that offers natural context preservation as it solely requires the insertion of local Pauli preparation and measurement fiducials around the window of interest. However, each sampled input-output Pauli pair demands its own preparation and measurement setting, an overhead that grows rapidly once the target gate is no longer Clifford. We introduce joint fiducial grouping, which partitions Pauli pairs into sets with commuting input and output operators, allowing several Pauli-transfer coefficients to be estimated within the same preparation-measurement setting. We derive an unbiased grouped estimator and finite-sample guarantees showing that grouping always reduces the number of distinct input-output settings and can also reduce the required channel uses when the target weight is concentrated within compatible groups. We characterize these gains for the parametric two-qubit gate fSim(θ,φ)\mathrm{fSim}(θ,\varphi)fSim(θ,φ), and use the grouped estimator as a context-sensitive reward for reinforcement-learning-based gate calibration. Our results provide a practical route to lower-overhead, context-preserving fidelity estimation for continuously parameterized quantum gates.