Quantum Gate Generation by T-Sampling Stabilization
Résumé
This paper considers right-invariant and controllable driftless quantum systems with state X(t) evolving on the unitary group U(n) and m inputs u = (u1,...,um). The T-sampling stabilization problem is introduced and solved: given any initial condition X0 and any goal state Xgoal, find a control law u = u(X, t) such that limj→∞ X(jT ) = Xgoal for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to Xgoal. This is achieved by the tracking of T-periodic reference trajectories (Xa(t),ua(t)) of the quantum system that pass by Xgoal using the framework of Coron’s Return Method. The T-periodic reference trajectories Xa(t) are generated by applying controls ua(t) that are a sum of a finite number M of harmonics of sin(2πt/T), whose amplitudes are parameterized by a vector a. The main result establishes that, for M big enough, X(jT) exponentially converges towards Xgoal for almost all fixed a, with explicit and completely constructive control laws. This paper also establishes a stochastic version of this deterministic control law. The key idea is to randomly choose a different parameter vector of control amplitudes a = aj at each t = jT, and keeping it fixed for t ∈ [jT,(j + 1)T). It is shown in the paper that X(jT) exponentially converges towards Xgoal almost surely. Simulation results have indicated that the convergence speed of X(jT ) may be significantly improved with such stochastic technique. This is illustrated in the generation of the C–NOT quantum logic gate on U(4).