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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 = (u1, …, um). The T-sampling stabilisation problem is introduced and solved: given any initial condition X0 and any goal state , find a control law u = u(X, t) such that
for the closed-loop system. The purpose is to generate arbitrary quantum gates corresponding to
. This is achieved by the tracking of T-periodic reference trajectories
of the quantum system that pass by
using the framework of Coron’s return method. The T-periodic reference trajectories
are generated by applying controls
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
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
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).
Notes
1. A brief summary of this result without proofs was presented in Silveira, Pereira da Silva, and Rouchon Citation(2013).
2. The qubit (quantum bit) is the quantum analogue of the usual bit in classical computation theory (Nielsen & Chuang, Citation2000).
3. Recall that an n-square complex matrix X belongs to U(n) if and only if X†X = I, and S is in if and only if S† = −S (skew-Hermitian), where S† is the conjugate transpose of S.
4. Given with B invertible and AB− 1 = B− 1A (i.e. A and B− 1 commute), one defines A/B≜AB− 1 = B− 1A. It is easy to see that A and B− 1 commute whenever A and B commute.
5. It follows from Equation (EquationA1(A1) ) in Appendix A that one may always write
, where exp (ιθj), j = 1, …, n, are the eigenvalues of
.
6. Another advantage of this strategy in two steps was verified in simulations. In some cases, it may generate smaller inputs when compared with the one-step procedure, even if it is combined with a global phase change.
7. One could restate Theorem B.1 for smooth manifolds without any problem. However, the notation becomes awful, and a notion of distance must be chosen, for instance, the one that U(n) inherits from .
8. At this moment, one is considering that K is compact set with respect to the topology of the Euclidean space . As
is an embedded manifold, the topology of U(n) is equivalent to the subspace topology induced by
, and so this distinction is of minor importance.
9. Simple computations show that lima → ±πα(a) = ±∞ and , for a ∈ −(π, π).
10. Note that in this argument Z* ∈ G is a fixed matrix.
11. Note that is a polynomial function in the entries Xij of X = (Xij), and as it is not identically zero, the set
of its roots is closed and it has Lebesque measure zero (Caron & Traynor, Citation2005).
12. Their existence is the heart of Coron’s return method.
13. The derivative of an even function is an odd function and vice-versa. In particular, the derivative at zero of an even function is zero.
14. For an elementary proof of this fact, see e.g. Caron and Traynor Citation(2005). This result holds in fact for analytic functions in a much more general situation (see Federer Citation(1969)).
Additional information
Funding
and FUNPESQUISA/UFSC. The second author was partially supported by 
, 
and USP-COFECUB. The third author was partially supported by ‘Agence Nationale de la Recherche’ (ANR), Projet Blanc EMAQS number ANR-2011-BS01-017-01 and USP-COFECUB.