Quantum Fisher information matrix via its classical counterpart from random measurements

TL;DR

The paper establishes a rigorous link between the quantum Fisher information matrix $Q(\boldsymbol{\theta})$ and the classical Fisher information matrices $F^U(\boldsymbol{\theta})$ obtained from measurements in Haar-random bases. It proves that for pure states, the expected CFIM satisfies $\mathbb{E}_{U}[F^U(\boldsymbol{\theta})]=\tfrac{1}{2}Q(\boldsymbol{\theta})$ and derives the exact variance $\mathrm{Var}_{U}[F^U(\boldsymbol{\theta})] = \tfrac{1}{8N}\big( Q_{ii}Q_{jj}+Q_{ij}^2+\tilde{Q}_{ij}^2 \big)$, with $\tilde{Q}$ from the imaginary part of the QGT. The authors further establish concentration bounds on $F^U(\boldsymbol{\theta})$ around its mean, showing exponential tails in $N$ and providing high-probability spectral bounds that ensure few random bases suffice to approximate the QFIM in high dimensions. This yields a solid theoretical foundation for efficient quantum natural gradient methods based on randomized measurements. The work connects quantum metrology and quantum information geometry, enabling practical estimation of geometric preconditioners via randomized CFIM 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). 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 $\mathbb{E}_{U\simμ_H}[F^U(\boldsymbolθ)] = \frac{1}{2}Q(\boldsymbolθ)$ for pure states in $\mathbb{C}^N$. Furthermore, we obtain the variance of CFIM ($O(N^{-1})$) and establish non-asymptotic concentration bounds ($\exp(-Θ(N)t^2)$), 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.

Figures (4)

• Figure 1: Relative Frobenius norm errors of different dimensions $N$.
• Figure 2: Histograms of the QFIM estimation error.
• Figure 3: Empirical distribution functions and exponential upper-bound curves $\exp(-cNt^2)$ for different values of $N$. For each $N$, the constant $c$ is the largest estimated value such that the empirical tail distribution (up to the $99.99$-th percentile) lies entirely below the theoretical tail distribution curve. The specific estimation procedure is given in Figure \ref{['fig:anaylsis']}.
• Figure 4: Tail distribution analysis for $N=80$. The top-left figure shows the empirical tail probability (complementary CDF) of the relative Frobenius norm error. The top-right figure plots the logarithm of the tail probability against $t^2$ and provides linear regression. Then we find the best $c$ based on the slope of the regression. The bottom-left figure depicts the best theoretical curve and the bottom-right figure plots the ratio between the empirical and the upper-bound.

Theorems & Definitions (36)

• Definition 1
• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5: Maximum norm
• Theorem 6: Frobenius norm
• Theorem 7: Eigenvalue control
• Definition 8
• Lemma 1
• Lemma 2