Estimation of Quantum Fisher Information via Stein's Identity in Variational Quantum Algorithms

Mourad Halla

doi:10.22331/q-2025-07-21-1798

Estimation of Quantum Fisher Information via Stein’s Identity in Variational Quantum Algorithms

Mourad Halla

Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany

Published:	2025-07-21, volume 9, page 1798
Editor:	Mohan Sarovar
Eprint:	arXiv:2502.17231v2
Doi:	https://doi.org/10.22331/q-2025-07-21-1798
Citation:	Quantum 9, 1798 (2025).

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Abstract

The Quantum Fisher Information Matrix (QFIM) plays a crucial role in quantum optimization algorithms such as Variational Quantum Imaginary Time Evolution and Quantum Natural Gradient Descent. However, computing the full QFIM incurs a quadratic computational cost of $O(d^2)$ with respect to the number of parameters $d$, limiting its scalability for high-dimensional quantum systems. To address this limitation, stochastic methods such as the Simultaneous Perturbation Stochastic Approximation (SPSA) have been employed to reduce computational complexity to a constant (Quantum 5, 567 (2021)). In this work, we propose an alternative estimation framework based on Stein's identity that also achieves constant computational complexity. Furthermore, our method reduces the quantum resources required for QFIM estimation compared to the SPSA approach. We provide numerical examples using the transverse-field Ising model and the lattice Schwinger model to demonstrate the feasibility of applying our method to realistic quantum systems.

Featured image: Comparison of energy-error convergence versus the number of quantum circuits for QNSPSA and proposed QNSTEIN optimizers in a 12-qubit transverse-field Ising model.

Popular summary

Quantum computing holds significant promise for solving problems beyond classical computers' capabilities. One crucial component in optimizing quantum algorithms is the Quantum Fisher Information Matrix (QFIM), essential for efficiently navigating the complex landscape of quantum states.

The QFIM acts as a metric, helping algorithms understand how quantum states change when their parameters are slightly varied. It provides insights into the curvature of the quantum state space, guiding optimizations more effectively than traditional methods. Typically, calculating the QFIM has a high computational cost, especially as quantum systems grow in size.

To address this, a new method employing Stein’s identity has been proposed, significantly reducing computational complexity. Unlike existing approaches such as the Simultaneous Perturbation Stochastic Approximation (SPSA), the Stein-based method requires fewer quantum resources while preserving essential parameter correlations.

Numerical experiments, including applications to the Transverse Field Ising Model and the Schwinger Model, illustrate that this new method can achieve faster convergence and reduced quantum resource consumption. This advancement makes variational quantum algorithms more practical for realistic quantum systems, potentially impacting fields like quantum chemistry, materials science, and high-energy physics.

► BibTeX data

@article{Halla2025estimationofquantum,
  doi = {10.22331/q-2025-07-21-1798},
  url = {https://doi.org/10.22331/q-2025-07-21-1798},
  title = {Estimation of {Q}uantum {F}isher {I}nformation via {S}tein's {I}dentity in {V}ariational {Q}uantum {A}lgorithms},
  author = {Halla, Mourad},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1798},
  month = jul,
  year = {2025}
}

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Cited by

[1] Yuan-Hao Wang and Da-Jian Zhang, "Superiority of Krylov shadow tomography in estimating quantum Fisher information: from bounds to exactness", npj Quantum Information 12 1, 74 (2026).

[2] Michele Minervini, Dhrumil Patel, and Mark M. Wilde, "Quantum natural gradient with thermal-state initialization", Physical Review A 112 2, 022424 (2025).

[3] André J. Ferreira-Martins, Renato M. S. Farias, Giancarlo Camilo, Thiago O. Maciel, Allan Tosta, Ruge Lin, Abdulla Alhajri, Tobias Haug, and Leandro Aolita, "Quantum optimization with exact geodesic transport", arXiv:2506.17395, (2025).

[4] Mourad Halla, "Modified conjugate quantum natural gradient", EPJ Quantum Technology 12 1, 123 (2025).

[5] Francesco Scala, Giacomo Guarnieri, and Aurelien Lucchi, "Noise-Induced Equalization in quantum learning models", arXiv:2511.09428, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 17:05:36) and SAO/NASA ADS (last updated successfully 2026-08-08 17:05:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

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