Title: High-order Cayley based methods for quantum optimal control problems

Abstract: Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motion defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance.

This thesis concerns the construction, implementation, and numerical analysis of a novel Cayley-based, commutator-free multistep scheme for general differential equations evolving on quadratic Lie groups, with an emphasis on applications in quantum optimal control. The method builds on Cayley commutator-free schemes recently introduced, but, in contrast, is not restricted to linear equations; it can, for example, handle dynamics governed by nonlinear Schrödinger equations.

We conduct numerical experiments to compare accuracy and computational performance with established Lie group integrators. We also discuss applications to quantum optimal control problems whose dynamics are governed by the nonlinear Schrödinger (Gross–Pitaevskii) equation, highlighting a design advantage afforded by the new method in this setting.

Title: Variation integrators for stochatic Hamiltonian systems

Abstract: We develop variational integrators for stochastic Hamiltonian systems. Stochastic Hamilton’s equations are the stochastic extension of Hamilton’s equations, which can be derived from a stochastic version of the phase space principle with an additional stochastic term expressed as a Stratonovich integral of stochastic Hamiltonians. Variational integrators can be constructed from the discretization of the stochastic variational principle. We construct such integrators for systems whose configuration space is a Lie group, and also for systems with advected parameters. We focus on the midpoint method, which provides the most basic strongly convergent integrator, and explore the geometry-preserving properties of the resulting integrator. Strong convergence is proven for the special case of stochastic rigid body equations. Various applications to both finite-dimensional and infinite-dimensional systems are also presented.