Quantum Fisher information from tensor network integration of Lyapunov equation

TL;DR

This paper introduces a tensor-network–based numerical method to compute the quantum Fisher information (QFI) and symmetric logarithmic derivative (SLD) via Lyapunov integral representations, avoiding explicit diagonalization of large density matrices. By formulating L_theta = 2 ∫_0^∞ e^{-ρ_theta x} dot{ρ}_theta e^{-ρ_theta x} dx and F_theta = 2 ∫_0^∞ tr[dot{ρ}_theta e^{-ρ_theta x} dot{ρ}_theta e^{-ρ_theta x}] dx, and implementing these with matrix product operators through imaginary-time evolution, the authors enable scalable QFI calculations for many-body thermal states. They analyze convergence, provide bounds depending on spectral properties and entropy, and benchmark the approach on a thermal TFIM probe under unitary encoding, comparing integration with variational TN methods and exploring adaptive truncation and encoding strategies. The method broadens practical access to QFI computations in large quantum systems, with potential extensions to noisy/metrological scenarios and broader Lyapunov-equation problems. This advances quantum metrology by delivering a scalable, diagonalization-free route to quantify ultimate parameter-estimation precision in complex many-body states.

Abstract

The Quantum Fisher Information (QFI) is a geometric measure of state deformation calculated along the trajectory parameterizing an ensemble of quantum states. It serves as a key concept in quantum metrology, where it is linked to the fundamental limit on the precision of the parameter that we estimate. However, the QFI is notoriously difficult to calculate due to its non-linear mathematical form. For mixed states, standard numerical procedures based on eigendecomposition quickly become impractical with increasing system size. To overcome this limitation, we introduce a novel numerical approach based on Lyapunov integrals that combines the concept of symmetric logarithmic derivative and tensor networks. Importantly, this approach requires only the elementary matrix product states algorithm for time-evolution, opening a perspective for broad usage and application to many-body systems. We discuss the advantages and limitations of our methodology through an illustrative example in quantum metrology, where the thermal state of the transverse-field Ising model is used to measure magnetic field amplitude.

Figures (8)

• Figure 1: Typical procedure for quantum metrology. The generally entangled input probe (white) is prepared in arbitrary quantum protocol. The probe is exposed to the signal generated by the unitary $U_{\theta}$ that encodes the parameter ${\theta}$ (blue). After the encoding the probe is measured with the positive operator-valued measure $\{\Pi_j\}$ (red). The measurement data are analyzed with statistical processing (yellow) to estimate the parameter value ${\theta}_{\rm est}$ and its variance $\delta^2{\theta}$.
• Figure 2: Phase estimation with thermal states of the TFIM. Benchmark of the QFI obtained from truncated integration up to ${X}$ (crosses and circles) compared to the reference value (dashed). For ${X}=100$ (dots) includes additional term from the fit of the residual integral obtained from the final $D[{\Delta{x}}]=10$ range. (a) Convergence in terms of system size at ${g}=2J$ for various system size and truncation cutoff ${X}$. (b) The benchmark of the QFI across the phase transition. The convergence time increases around the critical point ${g}=J$ where the gap to the excited state becomes small. Data obtained for encoding parameter ${\theta}=J$, inverse temperature $\beta=4J^{-1}$ and bond dimension $D_{{\rho}_{{\theta}}}\leq32$ for $N=16$ and $D_{{\rho}_{{\theta}}}\leq64$ otherwise. The integration obtained from the interpolation fit to the QFI integrand that was evaluated numerically with tensor networks. The QFI integrand is calculated with the MPO ansatz of the bond dimension $D_{\mathcal{B}_{\theta}}\leq64$. Reference data obtained with TNQMETRO chabuda_tnqmetro_2022 using $D_{\mathcal{L}_{\theta}}=16$.
• Figure 3: Convergence of the QFI integral. Convergence of our method for the TFIM across the phase diagram with an example for the disordered phase ${g}=2J$ in (a), the critical point ${g}=J$ (b) and the ordered phase ${g}=0$ in (c). Relative error in Eq. \ref{['eq:qfi_integral']} (solid lines) is plotted together with the low-temperature bound in Eq. \ref{['eq:boundLotTemp']} (dashed) and the worst-case bound in Eq. \ref{['eq:worstbound']} (dotted). Data obtained for the thermal state of the TFIM in Eq. \ref{['eq:model']} with $N=6$ for $\beta=4J^{-1}$ by exact integration of the QFI in Eq. \ref{['eq:qfi_integral']}.
• Figure 4: Variations for unitary encoding. Comparison of the bond dimension obtained under the cutoff on Schmidt values ${\rm SV tol.}=10^{-9}$ and maximal bond dimension $D\leq64$. The figure compares the standard QFI integration introduced in Sec. \ref{['sec:implementation']} (blue) and integration of the encoding in Sec. \ref{['sec:unitary']} (red). The integrand operators ${\mathcal{B}_{\theta}}({X})$ in Eq. \ref{['eq:Bs']} and ${\mathcal{A}_{\theta}}({X})$ in Eq. \ref{['eq:As']} (solid lines) are compared to the integrated operators ${\mathcal{L}_{\theta}}({X})$ in Eq. \ref{['eq:BT0']} and ${\mathcal{K}_{\theta}}({X})$ in Eq. \ref{['eq:AT0']} (dashed). Data obtained for $N=16$, ${g}=2J$, $\beta=4J^{-1}$, bond dimensions $D_{{\rho}_{{\theta}}}\leq32$ truncation cutoff ${\rm SV tol.}=10^{-9}$ and integration step ${\Delta{x}}=0.1$.
• Figure 5: Cost of single update. Comparison of the computational time for a single update obtained for the integration approach with the integrand ${\mathcal{B}_{\theta}}$, see Sec. \ref{['sec:implementation']} (blue), ${\mathcal{A}_{\theta}}$, see Sec. \ref{['sec:unitary']} (red), the variational approaches in the original implementation TNQMETRO chabuda_tnqmetro_2022 (brown), and based on the Krylov method, see App. \ref{['sec:varia']} (black). The computational time is the smallest for our variational method implemented in TN4QM where Krylov methods are used reduce dimension of the problem we solve for local update. Then, there is the ${\mathcal{B}_{\theta}}({x})$-based integration that takes simple overlaps in Eq. \ref{['eq:qfi_integrand']} to evaluate the integrand. After that, the ${\mathcal{A}_{\theta}}({x})$-based integration is mode costly due to higher order overlaps in Eq. \ref{['eq:B4A']}. Finally, the TNQMETRO is generally the most expensive because the preudoinverse is applied directly on effective operators that has a large dimension. Data obtained using TNQMETRO chabuda_tnqmetro_2022 and TN4QM tn4qm for $N=16$, ${g}=2J$, $\beta=4J^{-1}$, bond dimensions $D_{{\rho}_{{\theta}}}\leq32$ and integrand $D_{\mathcal{B}_{\theta}}\leq64$, integration step ${\Delta{x}}=0.1$. Computer time obtained on a single CPU core.