Quantum Fisher information from tensor-network integration of the Lyapunov equation

The quantum Fisher information (QFI) is a geometric measure of state deformation calculated along the trajectory parametrizing 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 nonlinear 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 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.