Numerical and Algebraic Studies for the Control of Quantum Systems

Alfio Borzı̀, Gabriele Ciaramella, M. Sprengel

Uwe Sander

In this thesis, two aspects of control theory, namely controllability and optimal control, are applied to quantum systems. The presented results are based on group theoretical techniques and numerical studies. By Lie-algebraic analysis, the controllability properties of systems with an arbitrary topology are described and related to the symmetries existing in these systems. We find that symmetry precludes full controllability. Our work investigates well-known control systems and gives rules for the design of new systems. Furthermore, theoretical and numerical concepts are instrumental to studying quantum channels: Their capacities are optimised using gradient flows on the unitary group in order to find counterexamples to a long-established additivity conjecture. The last part of this thesis presents and benchmarks a modular optimal control algorithm known as GRAPE. Numerical tests show how the interplay of its modules can be optimised for higher performance, and how the algorithm performs in comparison to a Krotov-type optimal control algorithm. It is found that GRAPE performs particularly well when aiming for high qualities.

Claude Le Bris, Yvon Maday, Gabriel Turinici

Re-Bing Wu, Gong Cheng

Robust control of quantum systems against parametric uncertainties is crucial for the practical implementation of quantum technologies. However, whether all gates are robustly realizable, i.e., can be precisely realized against errors, remain not fully understood. This paper investigates the identification of robustly realizable quantum gates from a Lie algebraic perspective. For single-parameter uncertainties, we derive an algebraic criterion that is straightforward to test. Numerical simulations involving single-qubit control systems demonstrate the validity of the proposed criterion.

Re-Bing Wu, Jing Zhang, Chunwen Li +2 more

Anastasia Aleksandrovna Myachkova, Alexander Nikolaevich Pechen

Abstract We analyze the controllability of three quantum systems that belong to a specific class of four-level quantum systems with twice degenerate highest excited energy level and with forbidden transition between the two remaining non-degenerate levels. For this purpose we perform numerical computation, construct the dynamical Lie algebras generated by all commutators of the free and interaction Hamiltonians, and show that two quantum systems are irreducible and controllable while the third system is reducible and hence uncontrollable. The reducibility and uncontrollability are proved by constructing a conserved Hermitian operator (physical quantity). The controllability is proved by constructing the dynamical Lie algebra and showing that it has maximal rank. These findings indicate that depending on the values of certain particular matrix entries of the interaction Hamiltonian, quantum systems in the class under consideration can be either uncontrollable or controllable.