Time optimal simultaneous control of two level quantum systems

M. A. Jafarizadeh, Fahimeh Naghdi, M. R. Bazrafkan

Shuang, Cong Cong, Jie Jie +3 more

Dominique Sugny

Francesca Albertini, Domenico D’Alessandro

In this paper, we solve the problem of simultaneously driving in minimum time to arbitrary final conditions, N two level quantum systems subject to independent controls. The solution of this problem is obtained via an explicit description of the reachable set of the associated control system on SU(2). The treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on Lie groups.

Francesca Albertini, Domenico D’Alessandro

We consider a two-level quantum system (quantum bit), S, which can only be controlled by modulating the difference between the two energy levels. We propose a control scheme that utilizes an additional quantum bit, A, as an auxiliary system, and obtains manipulation of S through the interaction with A. For this scheme, we derive the minimum time control for any unitary transformation on the target system S, and the form of the optimal control law. We find that the optimal time protocol involves a preliminary preparation of the state of A, the control of S using interaction with A, and no simultaneous control of S and A. The optimal time is the same as the one for a controllable two level system.

Raffaele Romano, Domenico D’Alessandro

Abstract In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin maximum principle, and provide a complete characterization of singular and nonsingular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.