Characterization of the Critical Sets of Quantum Unitary Control Landscapes

Details Characterization of the Critical Sets of Quantum Unitary Control Landscapes Characterization of the Critical Sets of Quantum Unitary Control Landscapes

2011-02-17

arxiv.org/abs/1102.3502v1

Physics

Quantum Physics

Scientific paper

This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled, Schrodinger equation. We examine the critical topology of the kinematic landscapes JF (U) = ||(U-W)A||^2 and JP (U) = ||A||^4 - |Tr(AA'W'U)|^2 defined on the unitary group U(N). The parameter matrix A is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of these functions includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U(N). These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes. Finally, landscapes based on the intrinsic (geodesic) distance on U(N) and the projective unitary group PU(N) are considered.

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