arXiv:1004.3492v3 [quant-ph] 2 Mar 2014 November 19, 2018 9:24 WSPC/INSTRUCTION FILE 2010FS1-v4 A Closer Look at Quantum Control Landscapes & their Implication for Control Optimization Pierre de Fouquieres and Sophie G. Schirmer College of Science, Physics, Swansea University, Swansea, SA2 8PP, United Kingdom sgs29@swan.ac.uk The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L 2 (0,T ), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L 2 (0,T ) is larger and includes traps, some of which remain traps even when the target time is allowed to vary. Keywords : quantum control, control landscapes Add AMS classification here. 1. Introduction Quantum theory has been in existence for about a century but until recently, the main emphasis in the field was on constructing Hamiltonian models and solving the Schrodinger equation. Although it was recognized that external fields and poten- tials could change the Hamiltonian and thus the dynamics of a system, and such external fields were certainly used in many areas from nuclear magnetic resonance to atomic physics to effect changes to the system, it was only rather recently that the full potential of using such external fields was recognized. Since then the subject of control of quantum systems has developed from a niche area into a subject of rapidly growing interest and importance, with an ever increasing number of appli- cations ranging from quantum chemistry to quantum information processing (see e.g. Ref. 1). Quantum control has been applied, for instance, to influence the out- come of chemical reactions 2 , to prepare entangled states 3 , which are a resource in quantum metrology and information processing, and to realize quantum gates 4 , which are fundamental building blocks for a quantum computer. The scope of the applications is vast. As the potential and importance of quantum control was realized, attempts were 1