Unitary Transformations and Quantum Dynamics

Nam-Anh Nguyen, T. T. Nguyen‐Dang

Tak‐San Ho, Jason Dominy, HERSCHEL A. RABITZ

We consider the control problem of generating unitary transformations, which is especially relevant to current research in quantum information processing and computing, in contrast to the usual state-to-state or the more general observable expectation value control problems. A previous analysis of optimal control landscapes for unitary transformations from a kinematic perspective in the finite-dimensional unitary matrices is extended to a dynamical one in the infinite-dimensional function space of the time-dependent external field. The underlying dynamical landscape is defined as the Frobenius square norm of the difference between the control unitary matrix and the target matrix. A nonsingular adaptation matrix is introduced to provide additional freedom for exploring and manipulating key features, specifically the slope and curvature, of the control landscapes. The dynamical analysis reveals many essential geometric features of optimal control landscapes for unitary transformations, including bounds on the local landscape slope and curvature. Close examination of the curvatures at the critical points shows that the unitary transformation control landscapes are free of local traps and proper choices of the adaptation matrix may facilitate the search for optimal control fields producing desired unitary transformations, in particular, in the neighborhood of the global extrema.

Lev Deych

Michael Hsieh, Re-Bing Wu, HERSCHEL A. RABITZ +1 more

The reliable and precise generation of quantum unitary transformations is essential for the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface-optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of nondissipative and controllable dynamics, the landscape topology is trap free, which implies that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis, which incorporates specific constraints in the Hamiltonian that correspond to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to a solution by optimal control.

Hsiang Shun Chou

Jakob Stoustrup, Oliver Schedletzky, Steffen J. Glaser +3 more

We report a general bound for the efficiency of polarization or coherence transfer between quantized states under unitary transformations. This bound is more general than the so-called universal bound on spin dynamics [O. W. S\o{}rensen, J. Magn. Reson. 86, 435 (1990)] that applies only to transformations between Hermitian states. The new bound on transfer efficiency under unitary transformations between general non-Hermitian operators depends exclusively on the singular values of the matrix representations of these operators.