Coherent control of a qubit is trap-free

Pechen, Alexander; Ильин, Николай Борисович

doi:10.1134/s0081543814040166

Cited by 26 publications

(37 citation statements)

References 28 publications

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“…This result complements the results of [6,8], where the absence of traps for this problem was proved for sufficiently large times.…”

Section: Resultssupporting

confidence: 89%

“…From the proof of Theorem 1 and Lemma 2 in [8], it follows that if L(σ x ) = L(σ y ) = L(σ z ) = 0, then there are no traps for any T . Therefore, the only possibility for the existence of traps is when…”

Section: Resultsmentioning

confidence: 98%

“…In this case f 0 = 0. In addition, in this case the Schrödinger equation (1.1) after a unitary transformation and time rescaling can be reduced to the form (see [8])…”

Section: Resultsmentioning

confidence: 99%

“…In [4,7] the existence of trapping properties was proved for n-level quantum systems with special symmetries for n > 2. A control problem for a two-level quantum system was studied in [6,8], where the absence of traps for sufficiently large times T was proved. Thus, in [6,8] we found the first class of quantum control problems that have no traps at least for sufficiently large times.…”

Section: Introductionmentioning

confidence: 99%

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Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times

Pechen

Ильин

2015

Proc. Steklov Inst. Math.

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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.

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“…This result complements the results of [6,8], where the absence of traps for this problem was proved for sufficiently large times.…”

Section: Resultssupporting

confidence: 89%

Section: Resultsmentioning

confidence: 98%

“…In this case f 0 = 0. In addition, in this case the Schrödinger equation (1.1) after a unitary transformation and time rescaling can be reduced to the form (see [8])…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times

Pechen

Ильин

2015

Proc. Steklov Inst. Math.

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“…for closed quantum systems [14,15]. In works [16,17,18] the absence of traps was proved for two-levels closed quantum systems in the case of sufficiently long T .…”

Section: The Absence Of Traps For Controlling a Qubit At Long Timesmentioning

confidence: 98%

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

Pechen¹,

Ильин²

2016

Izv. Math.

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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.

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On critical points of the objective functional for maximization of qubit observables

Pechen¹,

Ильин²

2015

Russ. Math. Surv.

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An isolated from the environment n-level controlled quantum system is described by the Schrödinger equation for unitary evolution operator U t :Here H 0 and V are n × n Hermitian matrices and f (t) ∈ L 1 ([0, T ]; R) is the control. We assume that [H 0 , V ] = 0. The goal is to find a control f (t) which maximizes the objectivewhere ρ 0 is the initial density matrix of the system, A is a Hermitian matrix and T > 0 is the final time. The functional J A describes the average value of an observable A at time T . An important problem in quantum control is the analysis of local maxima and minima (called traps) of the objective functional [1, 2, 3] because traps, if they exist, can be obstacles to find globally optimal solutions. The absence of traps in typical control problems for quantum systems was suggested in [1,2]. The absence of traps for two-level quantum systems for sufficiently large T was proved in [4,5]. The influence of constraints in the controls on the appearance of traps for two-level Landau-Zener system was investigated in [6]. For constrained control of this system with sufficiently large T only a special kind of traps may exist. These traps can be eliminated by a modification of gradient search algorithm so that in this sense the quantum control landscape practically appears as trap-free [6]. In [5] the following statement is proved.

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Coherent control of a qubit is trap-free

Cited by 26 publications

References 28 publications

Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times

Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times

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

On critical points of the objective functional for maximization of qubit observables

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