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Zermelo navigation in the quantum brachistochrone. (English) Zbl 1348.37053

Summary: 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 \(\mathfrak{su}(n)\). We also consider a generalization 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. \(\kappa \mathrm{Tr}(\hat{H}_{c}(t)^{2})=1\) for any given value of \(\kappa > 0\). Some of the results of A. Carlini et al. [“Time-optimal unitary operations”, Phys. Rev. A (3) 75, No. 4, Article ID 042308, 4 p. (2007; doi:10.1103/PhysRevA.75.042308)] 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-Poincaré 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.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
81P68 Quantum computation
81Q80 Special quantum systems, such as solvable systems
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
53C22 Geodesics in global differential geometry
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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