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Section: New Results

Exponential stochastic stabilization of a two-level quantum system via strict Lyapunov control

Participants: Gerardo Cardona, Alain Sarlette, Pierre Rouchon

In this result, we address the fundamental task of stabilizing the state of a quantum system towards a target eigenstate of a continuous-time quantum nondemolition measurement. The starting point is that a static output feedback does not allow us to stabilize this system, while more complicated procedures were not able to provide a convergence rate. Our main idea is to introduce a dynamic feedback controller of moderate complexity, where (i) feedback gains depend on estimated state and progressively go to zero as one approaches the target; and (ii) the feedback involves noise (in this paper from the measurement back-action but in further extensions possibly just independent noise). With this controller we show, providing a Lyapunov function close to the Bures distance measure, that the system converges exponentially towards the target eigenstate. This result, restricted to a proof-of-principle on the qubit, was published in [19].

This has laid the basis for further work, presented on posters and to be published next year, where we have shown that:

- the optimal convergence rate, equal to information gain, can be achieved with this feedback;

- the procedure extends to N-level systems, with noise just independent instead of coming from the measurement backaction;

- the procedure can be exploited towards continuous-time measurement-based quantum error correction