Superiority of Krylov shadow tomography in estimating quantum Fisher information: From bounds to exactness

Abstract

Estimating the quantum Fisher information (QFI) is a crucial yet challenging task with widespread applications across quantum science and technologies. The recently proposed Krylov shadow tomography (KST) opens a new avenue for this task by introducing a series of Krylov bounds on the QFI. In this work, we address the practical applicability of the KST, unveiling that the Krylov bounds of low orders already enable efficient and accurate estimation of the QFI. We show that the Krylov bounds converge to the QFI exponentially fast with increasing order and can surpass the state-of-the-art polynomial lower bounds known to date. Moreover, we show that certain low-order Krylov bound can already match the QFI exactly for low-rank states prevalent in practical settings. Such exact match is beyond the reach of polynomial lower bounds proposed previously. These theoretical findings, solidified by extensive numerical simulations, demonstrate practical advantages over existing polynomial approaches, holding promise for fully unlocking the effectiveness of QFI-based applications.

Figures (6)

• Figure 1: Basic idea of the Krylov shadow tomography. We construct a nested sequence of Krylov subspaces $\mathcal{K}_n$ and associate each of them with the Krylov bound defined by $B_n^{(\mathsf{Kry})}=\norm{L_n}_\rho^2$, where $L_n\in\mathcal{K}_n$ is chosen to be as close as possible to the SLD, represented by the symbol $L$ in the plot. As $\mathcal{K}_n$ continuously expands, $B_n^{(\mathsf{Kry})}$ approximates $F_Q$ increasingly better and eventually matches it exactly when $n=n^*$.
• Figure 2: Relative gap $\mathcal{E}_n^{(\mathsf{Kry})}$ as a function of $n$. The four subplots correspond to (a) $N=6$, (b) $N=8$, (c) $N=10$, and (d) $N=12$. We generate 100 relative gaps $\mathcal{E}_n^{(\mathsf{Kry})}$ for a fixed $n$ in each subplot by randomly choosing $100$ full-rank states. The dashed line in each subplot is the average of $100$ relative gaps.
• Figure 3: Numerical results for $\mathcal{E}_n^{(\mathsf{Kry})}$ and $\mathcal{E}_{2n-1}^{(\mathsf{Tay})}$ as functions of the order $n$. These two relative gaps are represented by the red solid line with dot markers and the blue solid line with square markers, respectively. The data points in this plot are associated with a randomly generated full-rank quantum state of $N=8$ qubits. The orange dashed line represents the theoretical upper bound on $\mathcal{E}_n^{(\mathsf{Kry})}$ predicted in Theorem \ref{['ThmExpDecay']}.
• Figure 4: Numerical simulations for estimating different bounds. The estimates of the three Krylov bounds, $\hat{B}_n^{(\mathsf{Kry})}$, $n=1,2,3$, are obtained by numerically simulating the KST for a randomly generated rank-$2$ state of $N=6$ qubits. Different solid lines correspond to different estimates of the bounds, as indicated in the legend. We also show the estimate of the Taylor bound $\hat{B}_5^{(\mathsf{Tay})}$ for comparison. All estimates are plotted as functions of the number $M$ of classical shadows required. The dashed lines correspond to the exact values of the respective bounds.
• Figure 5: Numerical simulations for demonstrating the exact match between the highest Krylov bound and the QFI. We examine rank-2 states of $N=6$ qubits, for which Theorem \ref{['ThmSmallNStar']} guarantees that the highest Krylov bound $B_3^{(\mathsf{Kry})}$ can match exactly with the QFI. The estimate $\hat{B}_3^{(\mathsf{Kry})}$ is obtained via numerically simulating the KST with $M=7\times 10^6$ classical shadows. The plot displays 100 data points, each representing a pair $(\hat{B}_3^{(\mathsf{Kry})},F_Q)$ corresponding to a randomly generated rank-2 state. The dashed line indicates the diagonal.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3