Quantum gate generation by T-sampling stabilisation

Abstract

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).

Periodic Trajectory Tracking for Control-Affine Driftless Systems on Compact Lie Groups

“…. , n}, 8 An inner product on g is said to be Ad-invariant if Ad(x) is orthogonal w.r.t. it for every x ∈ G. Such an inner product always exists when G is compact, and thanks to the relationship between the adjoint maps (see footnote 5) one also has that ad(X) is skew-symmetric w.r.t.…”

A fixed point algorithm for improving fidelity of quantum gates

“…This fact justifies the need of new methods, like numerical optimal control, Lyapunov methods and so on. Lyapunov stabilization [6,7,11,12,15,[24][25][26][27] may be applicable for large n, but they generate slow solutions (in the sense that that the final time must be large) when compared to the ones that are generated by optimal control, when this last approach is applicable. Hence the study of numerical methods that can be applied for large n and also produces a fast solution is a relevant field of research, indeed.…”

“…Before describing the FPA, let us give a context of the previous contributions of the authors to this problem. Inspired by the Coron's return method [2], the authors have developed a method of quantum gate generation for driftless systems [24]. However, this method is not iterative, and hence the solutions are slow, that is, a large final time T f that depends on the rate of convergence of the Lyapunov tracking control is needed.…”

“…However, this method is not iterative, and hence the solutions are slow, that is, a large final time T f that depends on the rate of convergence of the Lyapunov tracking control is needed. In [25], the method of [24] was generalized for systems with drift. However, it also generates slow solutions when compared for instance to GRAPE [8,9].…”

“…One first obtains, by right translation of the approximate solution, a new trajectory X(t) such that its final condition X(T f ) is exactly equal to X goal . The idea is then to consider the trajectory X(t) as a reference trajectory and to apply a stabilizing feedback of the corresponding tracking problem, as done in previous works of the authors [24,25]. The novelty of this work is that we consider the construction of an adequate right-translation R ∈ U(n) that corrects the action of the feedback in order to obtain an exact generation of quantum gate.…”