Quantum gate generation by T-sampling stabilisation

摘要

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 = (u_1,..., u_m). The T -sampling stabilisation problem is introduced and solved: given any initial condition X_0 and any goal state X_(goal), find a control law u = u(X, t) such that lim_(j→∞) X(jT) = X_(goal) for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to X_(goal). This is achieved by the tracking of T -periodic reference trajectories (X_a(t), u_a(t)) of the quantum system that pass by X_(goal) using the framework of Coron's return method. The T -periodic reference trajectories X_a(t) are generated by applying controls u_a(t) that are a sum of a finite number M of harmonics of sin(2πt/T), whose amplitudes are parameterised by a vector a. The main result establishes that, for M big enough, X(jT) exponentially converges towards X_(goal) 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 = a_j 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 X_(goal) 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).