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arXiv:2505.09818 · Quantum Information Processing

Efficient protocol to estimate the Quantum Fisher Information Matrix for commuting-block circuits

Rafael Gómez-Lurbe

Published Aug 7, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $$O(m^2)$$ O ( m 2 ) to $$O(L^2)$$ 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.

DOI
10.1007/s11128-026-05304-z

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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.
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.
Mixed-state quantum geometry plays a fundamental role in quantum metrology, multiparameter estimation, and the characterization of quantum many-body systems. Yet, unlike its pure-state counterpart, accessing mixed-state geometry requires solving an ill-conditioned Lyapunov equation for the symmetric logarithmic derivative (SLD), posing a major challenge for resource-efficient quantum computation. Here we address the challenge and develop a novel quantum algorithmic framework for efficiently accessing the quantum Fisher information matrix (QFIM) and quantum geometric tensor (QGT) of mixed states. Our key insight is to uncover a previously unexplored connection between the positive semidefinite structure of density matrices and exponentially convergent Gauss-Laguerre quadrature, enabling an efficient integral-based construction of the SLD. A rigorous complexity analysis further establishes the resource efficiency of our algorithm, demonstrating logarithmic precision dependence and double-logarithmic ancillary-qubit overhead at fixed conditioning. Building on this construction, we achieve near-Heisenberg precision scaling for diagonal QFI estimation without quantum phase estimation and construct unified block encodings of the full QFIM and QGT under a bounded spectral-gap assumption. A complementary regularized algorithm extends efficient diagonal estimation to exponentially small spectral gaps under a bounded-response condition. Our results establish a resource-efficient route to coherent mixed-state quantum geometry, enabling its direct integration into quantum linear-algebra algorithms without classical reconstruction and opening new possibilities for quantum-enhanced multiparameter metrology.
We consider metric-aware quantum algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object.For example, the recently introduced quantum natural gradient approach uses the Fisher matrix as a metric tensor to correct the gradient vector for the codependence of the circuit parameters.We rigorously characterize and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries.Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.
Abstract Quantum Fisher information matrix (QFIM) is a fundamental quantity in quantum physics, which closely links to diverse fields such as quantum metrology, phase transitions, entanglement witness, and quantum speed limit. It is crucial in quantum parameter estimation, central to the ultimate Cramér-Rao bound. Recently, the evaluation of QFIM using quantum circuit algorithms has been proposed for systems with multiplicative parameters Hamiltonian. However, systems with generic Hamiltonians still lack these proposed schemes. This work introduces a quantum-circuit-based approach for evaluating QFIM with generic Hamiltonians. We present a time-dependent stochastic parameter-shift rule for the derivatives of evolved quantum states, whereby the QFIM can be obtained. The scheme can be executed in universal quantum computers under the family of parameterized gates. In magnetic field estimations, we demonstrate the consistency between the results obtained from the stochastic parameter-shift rule and the exact results, while the results obtained from a standard parameter-shift rule slightly deviate from the exact ones. Our work sheds new light on studying QFIM with generic Hamiltonians using quantum circuit algorithms.
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.