Quantum Fisher information from tensor network integration of Lyapunov equation
arXiv:2506.11330 · doi:10.1103/xjln-7ddy
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.
References in corpus (44)
- Advances in Quantum Metrology
- Area laws for the entanglement entropy - a review
- Efficient classical simulation of slightly entangled quantum computations
- Efficient simulation of one-dimensional quantum many-body systems
- Matrix Product Density Operators: Simulation of finite-T and dissipative systems
- The ITensor Software Library for Tensor Network Calculations
- On the Improvement of Frequency Stardards with Quantum Entanglement
- General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology
- Time-dependent variational principle for quantum lattices
- Quantum metrology from a quantum information science perspective
- Unifying time evolution and optimization with matrix product states
- The elusive Heisenberg limit in quantum enhanced metrology
- Generalized uncertainty relations: Theory, examples, and Lorentz invariance
- Tensor networks for complex quantum systems
- Quantum Fisher Information Flow in Non-Markovian Processes of Open Systems
- Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
- Universality of Entropy Scaling in 1D Gap-less Models
- Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
- Time-evolving a matrix product state with long-ranged interactions
- Quantum criticality as a resource for quantum estimation
- The ultimate precision limits for noisy frequency estimation
- Quantum Fisher information for density matrices with arbitrary ranks
- Discontinuities of the quantum Fisher information and the Bures metric
- Simple expression for the quantum Fisher information matrix
- Canonical Energy is Quantum Fisher Information
- Density Matrix Renormalization Group in the Heisenberg Picture
- Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response
- Variational Quantum Algorithm for Estimating the Quantum Fisher Information
- Complexity of thermal states in quantum spin chains
- Quantum metrology for non-Markovian processes
- Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
- Quantum Fisher information and symmetric logarithmic derivative via anti-commutators
- Exact matrix product solutions in the Heisenberg picture of an open quantum spin chain
- On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank
- General expressions for the quantum Fisher information matrix with applications to discrete quantum imaging
- Tensor-Network Approach for Quantum Metrology in Many-Body Quantum Systems
- Probe optimization for quantum metrology via closed-loop learning control
- Improved scaling of Time-Evolving Block-Decimation algorithm through Reduced-Rank Randomized Singular Value Decomposition
- Determining the Continuous Family of Quantum Fisher Information from Linear Response Theory
- Probabilistic low-rank factorization accelerates tensor network simulations of critical quantum many-body ground states
- Uhlmann fidelities from tensor networks
- Krylov shadow tomography: Efficient estimation of quantum Fisher information
- TNQMetro: Tensor-network based package for efficient quantum metrology computations
- Quantum Natural Gradient optimizer on noisy platforms: QAOA as a case study