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arXiv:quant-ph/0507156 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Non-Holonomic Control I

E. Brion, V. M. Akulin, D. Comparat, Ilya I. Dumer, V. Gershkovich, G. Harel, Gershon Kurizki, I. E. Mazets, P. Pillet

Published Jun 1, 2005• 1 citation
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Abstract

In this paper, we present a universal control technique, the non-holonomic control, which allows us to impose any arbitrarily prescribed unitary evolution to any quantum system through the alternate application of two well-chosen perturbations.
DOI
10.1117/12.620307

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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.
When a quantum system is driven slowly through a parametric cycle in a degenerate Hilbert space, the state would acquire a non-Abelian geometric phase, which is stable and forms the foundation for holonomic quantum computation (HQC). However, in the adiabatic limit, the environmental decoherence becomes a significant source of errors. Recently, various nonadiabatic holonomic quantum computation (NHQC) schemes have been proposed, but all at the price of increased sensitivity to control errors. Alternatively, there exist theoretical proposals for speeding up HQC by the technique of "shortcut to adiabaticity" (STA), but no experimental demonstration has been reported so far, as these proposals involve a complicated control of four energy levels simultaneously. Here, we propose and experimentally demonstrate that HQC via shortcut to adiabaticity can be constructed with only three energy levels, using a superconducting qubit in a scalable architecture. With this scheme, all holonomic single-qubit operations can be realized nonadiabatically through a single cycle of state evolution. As a result, we are able to experimentally benchmark the stability of STA+HQC against NHQC in the same platform. The flexibility and simplicity of our scheme makes it also implementable on other systems, such as nitrogen-vacancy center, quantum dots, and nuclear magnetic resonance. Finally, our scheme can be extended to construct two-qubit holonomic entangling gates, leading to a universal set of STAHQC gates.