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arXiv:2103.01194 · Journal of Scientific Computing

Structure-Preserving Numerical Schemes for Lindblad Equations

Yu Hua Cao, Jianfeng Lu

Published Dec 7, 2024• 9 citations
arXivHTMLPDF
DOI
10.1007/s10915-024-02707-x

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Optimal control of open quantum systems governed by the Lindblad master equation requires the repeated solution of forward and adjoint evolution equations. For large Hilbert spaces, these computations demand numerical methods that are structure-preserving, accurate, and computationally efficient, while remaining compatible with nonsmooth optimization techniques. In this work, a unified framework for optimal control of Lindblad equations is presented. The continuous optimality system is derived from the Pontryagin maximum principle and solved by a sequential quadratic Hamiltonian (SQH) method. Its numerical realization is based on second-order exponential midpoint propagators for the forward and adjoint Lindblad equations with time-dependent Hamiltonians. Full-rank schemes preserving the Hermitian and positive-semidefinite structure of the density matrix are developed together with low-rank formulations that substantially reduce storage requirements and computational cost. Rigorous error estimates are established for the full- and low-rank forward and adjoint propagators. Numerical experiments confirm the predicted convergence rates, demonstrate the effectiveness of the low-rank approximations, and illustrate the performance of the proposed FREM-SQH and LREM-SQH algorithms for optimal control problems with smooth and nonsmooth control costs.
The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. This equation is also a fundamental building block to design optimal control functions. In this paper we develop full- and low-rank exponential midpoint integrators for solving both the forward and adjoint Lindblad equations. These schemes are applicable to optimize-then-discretize approaches for optimal control of open quantum systems. We show that the proposed schemes preserve positivity and trace unconditionally. Furthermore, convergence of these numerical schemes is proved theoretically and verified numerically.