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Russian Microelectronics

Quantum Control Landscapes and Traps

Boris Olegovich Volkov, Alexander Nikolaevich Pechen

Published Dec 1, 2023• 1 citation
Publisher

Abstract

Abstract Quantum control is a necessary tool for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems for various tasks. Traps are points of local but not global optimum of the objective functional for a given quantum control problem. In a more general sense, traps are critical points of the objective functional which are hard to escape by local search algorithms. Here a review of some results of the analysis of possibility of having traps in landscapes of coherently controlled closed quantum systems is given. In one-qubit case, there are no traps. For special multilevel quantum systems, higher-order traps may appear.

DOI
10.1134/s1063739723600796

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Abstract Quantum control is a necessary tool for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems for various tasks. Traps are points of local but not global optimum of the objective functional for a given quantum control problem. In a more general sense, traps are critical points of the objective functional which are hard to escape by local search algorithms. Here a review of some results of the analysis of possibility of having traps in landscapes of coherently controlled closed quantum systems is given. In one-qubit case, there are no traps. For special multilevel quantum systems, higher-order traps may appear.
Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.
Abstract An important problem in quantum control is the analysis of quantum control landscapes and its trapping features. The notion of a trap in quantum control landscape was introduced in [H. Rabitz, M. Hsieh, C. Rosenthal, Quantum optimally controlled transition landscapes, Science 303, 1998 (2004)]. Later, the notion of higher-order traps was introduced [A. Pechen and D.J. Tannor, Are there traps in quantum control landscapes?, Phys. Rev. Lett. 106, 120402 (2011)] and such higher-order traps were shown to exist for a variety of important applications for physical systems such as for example Λ-atoms widely used in quantum optics and superconducting transmon qubit systems used as one of the potential platform for quantum computation. In this note, we provide several results on the existence of higher-order traps in quantum control landscapes.
Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.
Abstract In this work we study extrema of objective functionals for ultrafast manipulation by a qubit. Traps are extrema of the objective functionals which are optimal for manipulation by quantum systems only locally, not globally. Prior work has devoted a large amount of effort to the analysis of traps for quantum systems controlled by laser pulses which are long enough, and, for example, manipulation by a qubit with long control pulses was shown to be trap-free. Ultrafast femtosecond and attosecond control has now become widely applicable, which makes the analysis of traps on the ultrafast time scale a necessity. We complete such analysis for a qubit and show that ultrafast state transfer in a qubit remains trap-free for a wide range of the initial and final states of the qubit. We prove that for this range the probability of transition between the initial and the final states has a saddle but no traps.
Abstract A proof is presented that almost all closed, finite dimensional quantum systems have trap free (i.e. free from local optima) landscapes for a large and physically general class of circumstances, which includes qubit evolutions in quantum computing. This result offers an explanation for why gradient-based methods succeed so frequently in quantum control. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator, and thus in the case of observables as well. Singular controls have been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause control optimization progress to halt, and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity (full rank) assumption of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This mild condition is shown to be sufficient for a quantum system’s landscape to be trap free. The control landscape is shown to be trap free for all but a null set of Hamiltonians using a geometric technique based on the parametric transversality theorem. Numerical evidence confirming this analysis is also presented. This new result is the analogue of the work of Altifini, wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation. These collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps.
The problems of optimizing the value of an arbitrary observable of a two-level system at both a fixed time and the shortest possible time is theoretically explored. Complete identification and classification along with comprehensive analysis of globally optimal control policies and traps (i.e., policies which are locally but not globally optimal) are presented. The central question addressed is whether the control landscape remains trap-free if control constraints of the inequality type are imposed. The answer is astonishingly controversial: Although the traps are proven always to exist in this case, in practice they become trivially escapable once the control time is fixed and chosen long enough.