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arXiv:1909.12958 · Izvestiya Mathematics

On extrema of the objective functional for short-time generation of single-qubit quantum gates

Alexander Nikolaevich Pechen, N B Il'in

Published Aug 22, 2016• 11 citations
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Abstract

In the present work the extrema of the objective functional for the problem of generation of quantum gates (logical elements for quantum computations) for two--level systems are investigated for short duration of the control. The problem of existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operators and Hamiltonians we obtain a new estimate for the lower boundary of the control duration which guarantees the absence of traps.

DOI
10.1070/im8567

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In the present work the extrema of the objective functional for the problem of generation of quantum gates (logical elements for quantum computations) for two--level systems are investigated for short duration of the control. The problem of existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operators and Hamiltonians we obtain a new estimate for the lower boundary of the control duration which guarantees the absence of traps.
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.
We consider the Mayer maximization problem for an objective functional that describes the average value at some fixed time of a quantum-mechanical observable for a two-level quantum system (qubit). In the previous studies we proved that for sufficiently large times the objective functional has no local maxima that are not global maxima. Such local maxima that are not global are called traps. In this paper we prove that for sufficiently short times under certain conditions traps for this problem do exist.
In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in case of a saddle, the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian (Theorem 3 in [22]).
There is a strong interest in optimal manipulating of quantum systems by external controls. Traps are controls which are optimal only locally but not globally. If they exist, they can be serious obstacles to the search of globally optimal controls in numerical and laboratory experiments, and for this reason the analysis of traps attracts considerable attention. In this paper we prove that for a wide range of control problems for two-level quantum systems all locally optimal controls are also globally optimal. Hence we conclude that two-level systems in general are trap-free. In particular, manipulating qubits—two-level quantum systems forming a basic building block for quantum computation—is free of traps for fundamental problems such as the state preparation and gate generation.
The analysis of traps, that is, locally but not globally optimal controls, for quantum control systems has attracted great interest in recent years. The central problem that has remained open is to demonstrate for a given system either the existence or the absence of traps. We prove the absence of traps and hence completely solve this problem for the important tasks of unconstrained manipulation of the transition probability and unitary gate generation in the Landau-Zener system, a system with a wide range of applications across physics, chemistry, and biochemistry. This finding provides an example of a controlled quantum system which is completely free of traps. We also discuss the impact of laboratory constraints due to decoherence, noise in the control pulse, and restrictions on the available controls, which, when being sufficiently severe, can produce traps.