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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> with respect to the number of parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>, 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.