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Zermelo Navigation in the Quantum Brachistochrone

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JournalJournal of Physics A: Mathematical and Theoretical
DateE-pub ahead of print - 23 Feb 2015
DatePublished (current) - 2015
Issue number11
Number of pages29
Early online date23/02/15
Original languageEnglish


We analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system. The family of constraints studied is that the permitted set of (time dependent) Hamiltonians is the unit ball of a norm induced by an inner product on su(n). We also consider a generalisation of this to arbitrary norms. We construct a Randers metric, by applying a theorem of Shen on Zermelo navigation, the geodesics of which are the time optimal trajectories compatible with the prescribed constraint. We determine all geodesics and the corresponding time optimal Hamiltonian for a specific constraint on the control i.e. k (Tr(Hc(t)^2) = 1 for any given value of k > 0. Some of the results of Carlini et. al. are re-derived using alternative methods. A first order system of differential equations for the optimal Hamiltonian is obtained and shown to be of the form of the Euler Poincare equations. We illustrate that this method can form a methodology for determining which physical substrates are effective at supporting the implementation of fast quantum computation.

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© 2015 IOP Publishing Ltd. This is an author produced version of a paper published in Journal of Physics A: Mathematical and Theoretical. Uploaded in accordance with the publisher's self-archiving policy.

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