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arXiv:2502.17231 · Quantum

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

Mourad Halla

Published Jul 21, 2025• 3 citations
arXivHTMLPDF

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(d2) 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.

DOI
10.22331/q-2025-07-21-1798

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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(d2) 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.
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.
The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with d parameters, however, is computationally expensive and generally requires O ( d 2 ) function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.
Preconditioning with the quantum Fisher information matrix (QFIM) is a popular approach in quantum variational algorithms. Yet the QFIM is costly to obtain directly, usually requiring more state preparation than its classical counterpart: the classical Fisher information matrix (CFIM). It is known that averaging the classical Fisher information matrix over Haar-random measurement bases yields EU∼μH[FU(θ)]=12Q(θ)\mathbb{E}_{U\simμ_H}[F^U(\boldsymbolθ)] = \frac{1}{2}Q(\boldsymbolθ)EU∼μH​​[FU(θ)]=21​Q(θ) for pure states in CN\mathbb{C}^NCN. In this paper, we review this identity by revealing its connection to covariant measurement in quantum metrology. Furthermore, we go beyond this and obtain the exact variance of CFIM (O(N−1)O(N^{-1})O(N−1)), estimate its moment, and establish non-asymptotic concentration bounds (exp⁡(−Θ(N)t2)\exp(-Θ(N)t^2)exp(−Θ(N)t2)), demonstrating that using few random measurement bases is sufficient to approximate the QFIM accurately in high-dimensional settings. This work establishes a solid theoretical foundation for efficient quantum natural gradient methods via randomized measurements.
Abstract The Quantum Fisher Information Matrix (QFIM) quantifies how sensitive parameterized quantum states are to changes in their parameters. Recently, it has been used to improve variational quantum algorithm optimization through geometry-aware techniques. However, estimating the QFIM—particularly its off-block-diagonal elements—requires substantial resources. To address this, we introduce a novel protocol that efficiently computes these elements for commuting-block variational circuits. Our approach reduces the number of quantum state preparations from O(m2)O(m^2)O(m2) O ( m 2 ) to O(L2)O(L^2)O(L2) O ( L 2 ) , where m is the number of parameters and L the number of circuit layers. This also lowers classical measurement and post-processing requirements, improving computational efficiency.
In this study, we investigated the Variational Quantum Eigensolver (VQE) application for the Ising model as a testbed model, in which we thoroughly delved into several optimizers, both classical and quantum, and analyzed the extent to which each of these methods would offer a benefit. We then investigated a new combinatorial optimization scheme, termed QN-SPSA+PSR, in which the Fubini-Study metric is approximated within the Quantum Natural Gradient (QN) framework, with its inner gradient estimated by the Simultaneous Perturbation Stochastic Approximation (SPSA), while the outer gradient of the cost function is evaluated exactly by the Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving the stability of QN-SPSA-based and convergence speed per parameter update while maintaining low computational consumption. Our results provide a potential performance improvement in the VQAs' optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.
The study of variational quantum algorithms (VQCs) has received significant attention from the quantum computing community in recent years. These hybrid algorithms, utilizing both classical and quantum components, are well-suited for noisy intermediate-scale quantum devices. Though estimating exact gradients using the parameter-shift rule to optimize the VQCs is realizable in NISQ devices, they do not scale well for larger problem sizes. The computational complexity, in terms of the number of circuit evaluations required for gradient estimation by the parameter-shift rule, scales linearly with the number of parameters in VQCs. On the other hand, techniques that approximate the gradients of the VQCs, such as the simultaneous perturbation stochastic approximation (SPSA), do not scale with the number of parameters but struggle with instability and often attain suboptimal solutions. In this work, we introduce a novel gradient estimation approach called Guided-SPSA, which meaningfully combines the parameter-shift rule and SPSA-based gradient approximation. The Guided-SPSA results in a 15% to 25% reduction in the number of circuit evaluations required during training for a similar or better optimality of the solution found compared to the parameter-shift rule. The Guided-SPSA outperforms standard SPSA in all scenarios and outperforms the parameter-shift rule in scenarios such as suboptimal initialization of the parameters. We demonstrate numerically the performance of Guided-SPSA on different paradigms of quantum machine learning, such as regression, classification, and reinforcement learning.