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arXiv:1904.12067 · Journal of Mathematical Physics

Algorithms for quantum control without discontinuities: Application to the simultaneous control of two qubits

Domenico D’Alessandro, Benjamin Sheller

Published Sep 1, 2019• 3 citations
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Abstract

We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries, which allows us to reduce the problem of control to a quotient space where the control system is “fully actuated.” As a result, we can prescribe a desired trajectory, which is, to some extent, arbitrary, and derive the corresponding control. We illustrate this technique with examples and focus on the application to the simultaneous control of two non-interacting spin 12 particles with different gyromagnetic ratios in zero field nuclear magnetic resonance (NMR). Our method provides a flexible toolbox for the design of control algorithms to drive the state of finite dimensional quantum systems to any desired final configuration, with smooth controls.

DOI
10.1063/1.5105351

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We propose a technique to design control algorithms for a class of finite dimensional quantum systems so that the control law does not present discontinuities. The class of models considered admits a group of symmetries, which allows us to reduce the problem of control to a quotient space where the control system is “fully actuated.” As a result, we can prescribe a desired trajectory, which is, to some extent, arbitrary, and derive the corresponding control. We illustrate this technique with examples and focus on the application to the simultaneous control of two non-interacting spin 12 particles with different gyromagnetic ratios in zero field nuclear magnetic resonance (NMR). Our method provides a flexible toolbox for the design of control algorithms to drive the state of finite dimensional quantum systems to any desired final configuration, with smooth controls.
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We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing the energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. In particular, we prove the 'normality' of the given problem and the 'regularity' of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.
We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing an energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices of dimension two, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. This includes, in particular, a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.
In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent or dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These findings do not preclude an algorithmic solvability for a particular class of quantum control problems. Moreover, any quantum control problem can be made algorithmically solvable if the set of accessible interactions (i.e., controls) is rich enough. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.