# A stochastic Lyapunov feedback technique for propagator generation of quantum systems on U(n)

July 2013 Category : Quantum systems | All

Authors: H.B. Silveira, P.S.Pereira da Silva, P. Rouchon, Control Conference (ECC), 2013 European, pp. 2712 - 2716, 17-19 July 2013, Zurich

This work treats the problem of generating any desired goal propagator for a driftless quantum system that evolves on the unitary group U(n). The physical relevance of such control problem is the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, the paper constructs explicit stochastic control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The purpose of introducing a stochastic behaviour in the controls is to speed up convergence. The control strategy can be rigorously proved based on Lyapunov feedback and stochastic techniques. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and such that its corresponding linearised system generates the Lie algebra u(n). Their existence is ensured by the ReturnMethod of Coron, and standard Fourier series results allows them to be explicitly constructed.

BibTeX:

@Proceedings{,

author = {H.B. Silveira, P.S.Pereira da Silva, P. Rouchon},

editor = {},

title = {A stochastic Lyapunov feedback technique for propagator generation of quantum systems on U(n)},

booktitle = {Control Conference (ECC), 2013 European},

volume = {},

publisher = {},

address = {Zurich},

pages = {2712 - 2716},

year = {2013},

abstract = {This work treats the problem of generating any desired goal propagator for a driftless quantum system that evolves on the unitary group U(n). The physical relevance of such control problem is the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, the paper constructs explicit stochastic control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The purpose of introducing a stochastic behaviour in the controls is to speed up convergence. The control strategy can be rigorously proved based on Lyapunov feedback and stochastic techniques. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and such that its corresponding linearised system generates the Lie algebra u(n). Their existence is ensured by the ReturnMethod of Coron, and standard Fourier series results allows them to be explicitly constructed.},

keywords = {}}

This work treats the problem of generating any desired goal propagator for a driftless quantum system that evolves on the unitary group U(n). The physical relevance of such control problem is the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, the paper constructs explicit stochastic control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The purpose of introducing a stochastic behaviour in the controls is to speed up convergence. The control strategy can be rigorously proved based on Lyapunov feedback and stochastic techniques. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and such that its corresponding linearised system generates the Lie algebra u(n). Their existence is ensured by the ReturnMethod of Coron, and standard Fourier series results allows them to be explicitly constructed.

BibTeX:

@Proceedings{,

author = {H.B. Silveira, P.S.Pereira da Silva, P. Rouchon},

editor = {},

title = {A stochastic Lyapunov feedback technique for propagator generation of quantum systems on U(n)},

booktitle = {Control Conference (ECC), 2013 European},

volume = {},

publisher = {},

address = {Zurich},

pages = {2712 - 2716},

year = {2013},

abstract = {This work treats the problem of generating any desired goal propagator for a driftless quantum system that evolves on the unitary group U(n). The physical relevance of such control problem is the realization of arbitrary quantum gates in quantum computers. Assuming only the controllability of the system, the paper constructs explicit stochastic control laws that assure global asymptotic convergence of the propagator of the system towards the goal propagator. The purpose of introducing a stochastic behaviour in the controls is to speed up convergence. The control strategy can be rigorously proved based on Lyapunov feedback and stochastic techniques. The controls laws rely on a reference trajectory that crosses the desired goal propagator in a time-periodic fashion and such that its corresponding linearised system generates the Lie algebra u(n). Their existence is ensured by the ReturnMethod of Coron, and standard Fourier series results allows them to be explicitly constructed.},

keywords = {}}