Quantum Fisher information matrix via its classical counterpart from random measurements

• URL: http://arxiv.org/abs/2509.08196v2
• Date: Sun, 19 Oct 2025 00:47:29 GMT
• Title: Quantum Fisher information matrix via its classical counterpart from random measurements
• Authors: Jianfeng Lu, Kecen Sha,
• Abstract summary: Preconditioning with the quantum Fisher information matrix (QFIM) is a popular approach in quantum variational algorithms.<n>We show that averaging the classical Fisher information matrix over Haar-random measurement bases yields $mathbbE_Usimmu_H[FU(boldsymboltheta)] = frac12Q(boldsymboltheta)$ for pure states in $mathbbCN$.
• Score: 5.726854405157353
• License: http://creativecommons.org/licenses/by/4.0/
• 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). By revealing its relation to covariant measurement in quantum metrology, we show that averaging the classical Fisher information matrix over Haar-random measurement bases yields $\mathbb{E}_{U\sim\mu_H}[F^U(\boldsymbol{\theta})] = \frac{1}{2}Q(\boldsymbol{\theta})$ for pure states in $\mathbb{C}^N$. Furthermore, we obtain the variance of CFIM ($O(N^{-1})$) and establish non-asymptotic concentration bounds ($\exp(-\Theta(N)t^2)$), demonstrating that using few random measurement bases is sufficient to approximate the QFIM accurately, especially in high-dimensional settings. This work establishes a solid theoretical foundation for efficient quantum natural gradient methods via randomized measurements.

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