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Journal of Physics B Atomic Molecular and Optical Physics

Nonholonomic quantum control

E. Brion, F. Carlier, G. Harel, V. M. Akulin

Published Jul 25, 2011• 18 citations

Abstract

In this paper, we review an open-loop evolution control method, called nonholonomic control, based on the alternate application of only two physical perturbations for timings which play the role of adjustable control parameters. We present the algorithm which allows one to explicitly compute the pulse sequence achieving any arbitrarily prescribed unitary evolution in a nonholonomic system, i.e. a system subject to two physical perturbations which, together with the natural Hamiltonian of the system, span the entire Lie algebra . We moreover expose two extensions of our method to open quantum systems which, respectively, aim at preserving the information stored in the system and safely processing this information. The first is based on a generalization of the quantum Zeno effect, while the second is inspired by decoupling pulse techniques. The most important feature of the methods presented here is their universality: they indeed do not rely on any specific assumption on the system, which in particular is not bound to be a collection of two-level systems, or the error model considered, as is usually the case in the literature. Numerical and physical applications of our techniques are also provided.

DOI: 10.1088/0953-4075/44/15/154001

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In this paper, we review an open-loop evolution control method, called nonholonomic control, based on the alternate application of only two physical perturbations for timings which play the role of adjustable control parameters. We present the algorithm which allows one to explicitly compute the pulse sequence achieving any arbitrarily prescribed unitary evolution in a nonholonomic system, i.e. a system subject to two physical perturbations which, together with the natural Hamiltonian of the system, span the entire Lie algebra . We moreover expose two extensions of our method to open quantum systems which, respectively, aim at preserving the information stored in the system and safely processing this information. The first is based on a generalization of the quantum Zeno effect, while the second is inspired by decoupling pulse techniques. The most important feature of the methods presented here is their universality: they indeed do not rely on any specific assumption on the system, which in particular is not bound to be a collection of two-level systems, or the error model considered, as is usually the case in the literature. Numerical and physical applications of our techniques are also provided.

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Currently, the scheme of a quantum computer based on ions in traps is accepted as one of the most realistic schemes of a scaled quantum computer. In this article, we show application of the method of nonholonomic control over the quantum evolution to this scheme. We consider this to be the most universal and simplest control method for an arbitrary scheme of a quantum computer. In this case, for arbitrary controlled evolution of the quantum system, it is sufficient to have available two lasers operating at a combined frequency equal to a frequency of internal ion transitions and selected total vibrational mode of the system of ions in a trap with different amplitudes. For each laser, it is sufficient to control only the time of their alternate operation with a sufficient accuracy.

Synthesizing an effective identity evolution in a target system subjected to unwanted unitary or nonunitary dynamics is a fundamental task for both quantum control and quantum information processing applications. Here, we investigate how single-bit, discrete-time feedback capabilities may be exploited to enact or to enhance quantum procedures for effectively suppressing unwanted dynamics in a finite-dimensional open quantum system. An explicit characterization of the joint unitary propagators correctable by a single-bit feedback strategy for arbitrary evolution time is obtained. For a two-dimensional target system, we show how by appropriately combining quantum feedback with dynamical decoupling methods, concatenated feedback-decoupling schemes may be built, which can operate under relaxed control assumptions and can outperform purely closed-loop and open-loop protocols.

We show that open-loop dynamical control techniques may be used to synthesize unitary transformations in open quantum systems in such a way that decoherence is perturbatively compensated for to a desired (in principle arbitrarily high) level of accuracy, which depends only on the strength of the relevant errors and the achievable rate of control modulation. Our constructive and fully analytical solution employs concatenated dynamically corrected gates, and is applicable independently of detailed knowledge of the system-environment interactions and environment dynamics. Explicit implications for boosting quantum gate fidelities in realistic scenarios are addressed.

We present two new algorithms that universally construct a quantum circuit approximating a controlled-U(t) up to a global phase factor of U(t), denoted by ΛθU(t), where U(t) = e−iHt is the unitary evolution operator of an unknown Hamiltonian H of the system, and is given as a black box with tunable parameter t. We show that it is possible to approximately construct ΛθU(t) with arbitrarily high accuracy. This algorithm is based on the dynamical decoupling effect [1]. If another black box consisting of a time-inverted version of U(t), namely U(−t), is also available we can implement ΛθU(t) exactly with arbitrarily high success probability. This algorithm is based on the quantum Zeno effect.