Zermelo Navigation in the Quantum Brachistochrone

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. κTr(Ĥc(t) ) = 1 for any given value of κ > 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 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. 1 Problem and Motivation 1.1 Implementation of Quantum Gates in Constrained Quantum Systems We study the speed that a quantum system can implement a desired quantum gate. Here, all systems have pure states and finite dimensional Hilbert spaces associated to them. This question has been discussed from many perspectives before, for example [1, 11, 20, 25, 30, 31, 36, 41, 46]. Notable recent works based on geometric methods are [11, 12, 45]. Our previous work [39] begins an investigation into the application of “Zermelo Navigation” to determining speed limits for implementing quantum gates in systems of the form eqn.(1), by a applying a solution in time optimal control based on Randers geometry [14]. We attempt to solve essentially the same problem as Carlini et al [12], but in a way based on intrinsic geometric structures in order to derive a first order equation for the optimal Hamiltonian driving the time evolution operator. Caneva et al [10] address the behavior of a commonly applied numerical algorithm, the Krotov method, near the quantum speed limit [10]. Nielsen [35] has highlighted a connection between Finlser geometry and quantum optimal control. Furthermore, this work indicates an interesting connection between quantum circuit complexity and Finsler geometry. A fuller bibliography on the quantum speed limit more generally can be found in the introduction to [39]. In order to implement a certain quantum information processing (QIP) task in a controlled quantum system, we consider the dynamics of the system (more precisely, the time evolution operator Ût) as given by the Schrödinger equation: dÛt dt = −iĤtÛt = −i (