Low rank approximation for the numerical simulation of high dimensional Lindblad and Riccati equations

Abstract

A systematic numerical approach to approximate high dimensional Lindblad equations is described. It is based on a deterministic rank m approximation of the density operator, the rank m being the only parameter to adjust. From a known initial value, this rank m approximation gives at each time-step an estimate of the largest m eigen-values with their eigen-vectors of the density operator. A numerical scheme is proposed. Its numerical efficiency in the case of a rank 12 approximation is demonstrated for oscillation revivals of 50 atoms interacting resonantly with a slightly damped coherent quantized field of 200 photons. The approach may be employed for other similar equations. We in particularly show how to adapt such low-rank approximation for Riccati differential equations appearing in Kalman filtering.

19 pages, 7 figures

References in corpus (4)

• Derivatives of Multilinear Functions of Matrices
• Quantum projection filter for a highly nonlinear model in cavity QED
• Derivation of Maxwell-Bloch-type equations by projection of quantum models
• Entanglement and decoherence of N atoms and a mesoscopic field in a cavity

Cited by in corpus (17)

• Algebraic solution of the Lindblad equation for a collection of multilevel systems coupled to independent environments
• Optimal control in large open quantum systems: the case of transmon readout and reset
• Adaptive variational low-rank dynamics for open quantum systems
• Continuous-time dynamics and error scaling of noisy highly-entangling quantum circuits
• Stochastic dynamical low-rank approximation method
• Structure-preserving numerical schemes for Lindblad equations
• Convergence and adiabatic elimination for a driven dissipative quantum harmonic oscillator
• Adaptive low-rank approximation and denoised Monte-Carlo approach for high-dimensional Lindblad equations
• Tensor-network-based variational Monte Carlo approach to the non-equilibrium steady state of open quantum systems
• Parallel transport dynamics for mixed quantum states with applications to time-dependent density functional theory
• Low-Rank Variational Quantum Algorithm for the Dynamics of Open Quantum Systems
• Continuous Measurement and Stochastic Methods in Quantum Optical Systems
• Symplectic split-operator method for the time-dependent unitary Tavis-Cummings model
• Positivity Preserving Density Matrix Minimization at Finite Temperatures via Square Root
• Full- and low-rank exponential Euler integrators for the Lindblad equation
• Low-rank optimal control of quantum devices
• Dynamically Optimal Unraveling Schemes for Simulating Lindblad Equations