Quantum Fisher information matrix via its classical counterpart from random measurements

Abstract

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 ${\mathbb{E}}_{U\sim \hat{I}{¼}_H}[F^U(\hat{\mathbf{I}}¸)]=\frac{1}{2}Q(\hat{\mathbf{I}}¸)$ for pure states in ${\mathbb{C}}^N$. 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 ($O(N^{-1})$), estimate its moment, and establish non-asymptotic concentration bounds ($\exp (-\hat{I}˜(N)t^2)$), 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.

v2 made connections with existing results in the literature of quantum metrology. v4 added moment estimation results, supplementary remarks and refined typos