Sampling Groups of Pauli Operators to Enhance Direct Fidelity Estimation

Júlia Barberà-Rodríguez; Mariana Navarro; Leonardo Zambrano

doi:10.22331/q-2025-07-03-1784

Sampling Groups of Pauli Operators to Enhance Direct Fidelity Estimation

Júlia Barberà-Rodríguez¹, Mariana Navarro^1,2, and Leonardo Zambrano¹

¹ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
²LuxQuanta Technologies S.L., Mediterranean Technology Park. Carrer d’Esteve Terradas, 1, Office 206, 08860 Castelldefels, Barcelona, Spain

Published:	2025-07-03, volume 9, page 1784
Editor:	Jiangwei Shang
Eprint:	arXiv:2501.19228v2
Doi:	https://doi.org/10.22331/q-2025-07-03-1784
Citation:	Quantum 9, 1784 (2025).

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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.

Featured image: Scheme of our fidelity estimation protocol with grouping. Pauli strings that decompose a pure quantum state are grouped into mutually commuting sets, which can be measured simultaneously. By applying importance sampling over these groups, the number of measurements required to estimate the fidelity between a pure and an experimental quantum state is reduced, resulting in a more efficient estimation.

Popular summary

As quantum computers grow in size and complexity, we need better ways to verify that they are producing the correct results. A key benchmark is fidelity estimation, which measures how close a quantum state prepared in the lab is to a desired, ideal state. One method for this task is Direct Fidelity Estimation (DFE), an approach that estimates the fidelity using randomized local measurements. However, DFE might require a large number of measurements as the system size increases.
Here we introduce an improved version of DFE that reduces the experimental cost. The key idea behind our work is to group compatible measurements; those that can be performed simultaneously. By organizing the measurements into such groups, we extract more information from each experiment. This reduces both the number of measurements and the variance of the fidelity estimator.

► BibTeX data

@article{BarberaRodriguez2025samplinggroupsof,
  doi = {10.22331/q-2025-07-03-1784},
  url = {https://doi.org/10.22331/q-2025-07-03-1784},
  title = {Sampling {G}roups of {P}auli {O}perators to {E}nhance {D}irect {F}idelity {E}stimation},
  author = {Barber{\`{a}}-Rodr{\'{i}}guez, J{\'{u}}lia and Navarro, Mariana and Zambrano, Leonardo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1784},
  month = jul,
  year = {2025}
}

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Cited by

[1] Hyunho Cha, Sangwoo Hong, and Jungwoo Lee, "Operator-aware shadow importance sampling for accurate fidelity estimation", Physical Review A 113 4, 042602 (2026).

[2] Leonardo Zambrano, Teodor Parella-Dilmé, Antonio Acín, and Donato Farina, "Certification of quantum properties with imperfect measurements", Quantum Science and Technology 11 3, 035011 (2026).

[3] J Knörzer, X Liu, B F Schiffer, and J Tura, "Distributed quantum information processing: a review of recent progress", Reports on Progress in Physics 89 7, 074401 (2026).

[4] Tobias Micklitz, "Simulation-free fidelity estimation of universal quantum processors via output order statistics of chaotic circuits", Physical Review Applied 26 2, 024016 (2026).

[5] Dirk Oliver Theis, "Verifying a stabilizer state with few observables but many shots", arXiv:2412.16690, (2024).

[6] Sami Abdul Sater and Harold Ollivier, "Composable Verification in the Circuit-Model via Magic-Blindness", arXiv:2601.07111, (2026).

[7] Tobias Micklitz, "Simulation-Free Fidelity Estimation via Quantum Output Order Statistics", arXiv:2510.13026, (2025).

[8] Netanel Barel, Lee Peleg, Yotam Kadish, Amit Ben Kish, and Yotam Shapira, "Optimizing resource bounds in direct fidelity estimation", arXiv:2606.16336, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 15:57:34) and SAO/NASA ADS (last updated successfully 2026-08-13 15:57:35). The list may be incomplete as not all publishers provide suitable and complete citation data.

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