International audienceWe investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e. g., in longitudinal studies of Computational Anatomy. Our approach formulates Euler-Poincaré theory in higher-order tangent spaces on Lie groups. In particular, we develop the Euler-Poincaré formalism for higher-order variational problems that are invariant under Lie group transformations. The theory is then applied to higher-order template matching and the corresponding curves on the Lie group of transformations are shown to satisfy higher-order Euler-Poincaré equations. The example of SO(3) for template matching on the sphere is presented explicitly. Various cotangent bundle momentum maps emerge naturally that help organize the formulas. We also present Hamiltonian and Hamilton-Ostrogradsky Lie-Poisson formulations of the higher-order Euler-Poincaré theory for applications on the Hamiltonian side. © 2011 Springer-Verlag

The solution to the problem of finding a time-optimal control Hamiltonian to generate a given unitary gate, in an environment in which there exists an uncontrollable ambient Hamiltonian (e.g., a background field), is obtained. In the classical context, finding the time-optimal way to steer a ship in the presence of a background wind or current is known as the Zermelo navigation problem, whose solution can be obtained by working out geodesic curves on a space equipped with a Randers metric. The solution to the quantum Zermelo problem, which is shown here to take a remarkably simple form, is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a Randers metric on the space of unitary operators. The result reveals that the optimal control in a sense 'goes along with the wind'.PACS numbers: 03.67. Ac, 42.50.Dv, 02.30.Xx The problem of finding the optimal Hamiltonian for processing a given quantum state, or implementing a quantum operation (gate), in shortest possible time subject to certain constraints, has attracted considerable attention over the past decade [1][2][3][4][5][6][7][8]. Broadly speaking, the task can be classified into two closely-related categories: (a) transforming one quantum state into another; and (b) transforming one unitary operator into another, in the shortest possible time. If the constraint is concerned merely with a limit on energy resource, then the optimal Hamiltonian is time independent, and can be found easily by noting that under a unitary motion, the shortest path coincides with the path along which the speed of evolution is also maximised [9,10]. If there are further constraints, for example, the choice of the Hamiltonian is limited, then often a time-dependent Hamiltonian that minimises an action has to be determined by variational approaches [11,12]. Finding a solution to such a variational problem is in general difficult, however, an efficient numerical regularisation scheme to obtain an approximate solution has been proposed more recently [13].For many problems related to controlling quantum systems considered in the literature, it is assumed that the experimentalist has full control over the allowable range of Hamiltonians within the constraint, whereas in a laboratory there can often be situations in which the system is immersed in an external field or potential that is beyond control (e.g., gravitational or electro-magnetic field), since a complete elimination of external fields in a laboratory can be prohibitively expensive for the given task. Evidently, this is a generic issue that needs to be addressed adequately to be able to accurately implement a rapid quantum processing.In the present paper we address this issue by finding the time-optimal control HamiltonianĤ(t) =Ĥ 0 +Ĥ 1 (t) that generates a unitary motion to transform one unitary operatorÛ I into another operatorÛ F , subject to the constraints (i) that the background HamiltonianĤ 0 cannot be controlled; (ii) that the control Hamiltonian fulfils the energy resource ...

The quantum navigation problem of finding the time-optimal control Hamiltonian that transports a given initial state to a target state through quantum wind, that is, under the influence of external fields or potentials, is analyzed. By lifting the problem from the state space to the space of unitary gates realizing the required task, we are able to deduce the form of the solution to the problem by deriving a universal quantum speed limit. The expression thus obtained indicates that further simplifications of this apparently difficult problem are possible if we switch to the interaction picture of quantum mechanics. A complete solution to the navigation problem for an arbitrary quantum system is then obtained, and the behaviour of the solution is illustrated in the case of a two-level system.

A quantum spline is a smooth curve parameterised by time in the space of unitary transformations, whose associated orbit on the space of pure states traverses a designated set of quantum states at designated times, such that the trace norm of the time rate of change of the associated Hamiltonian is minimised. The solution to the quantum spline problem is obtained, and is applied in an example that illustrates quantum control of coherent states. An efficient numerical scheme for computing quantum splines is discussed and implemented in the examples.PACS numbers: 03.67. Ac, 42.50.Dv, 02.30.Xx, 02.60.Ed Controlling the evolution of the unitary transformations that generate quantum dynamics is vital in quantum information processing. There is a substantial literature devoted to the investigation of the many aspects of quantum control [1]. The objective of quantum control is the unitary transformation of one quantum state, pure or mixed, into another one, subject to certain criteria. For example, one may wish to transform a given quantum state |ψ into another state |φ unitarily in the shortest possible time, with finite energy resource [2][3][4]. When only the initial and final states are involved, many time-independent Hamiltonians are available that achieve the unitary evolution |ψ → |φ , and we simply need to find one that is optimal. However, transforming a given quantum state |ψ along a path that traverses through a sequence of designated quantum states |ψ → |φ 1 → |φ 2 → · · · → |φ n cannot be achieved by a time-independent Hamiltonian. To realise this chain of transformations in the shortest possible time, one chooses the optimal Hamiltonian H j for each interval |φ j → |φ j+1 [3,4], and switches the Hamiltonian from H j to H j+1 when the state has reached |φ j+1 . However, instantaneous switching of the Hamiltonian is in general not experimentally feasible.In the present paper, we consider the following quantum control problem: Let a set of quantum states |φ 1 , |φ 2 , · · · , |φ m and a set of times t 1 , t 2 , · · · , t m be given. Starting from an initial state |ψ 0 at time t 0 = 0, find a time-dependent Hamiltonian H(t) such that the evolution path |ψ t passes arbitrarily close to |φ j at time t = t j for all j = 1, . . . , m, and such that the change in the Hamiltonian, in a sense defined below, is minimised. The solution to this problem will generate a continuous curve in the space of quantum states that interpolates through the designated states, just as a spline curve interpolates through a given set of data points. We thus refer to this solution as a quantum spline.There is a difference between a classical spline curve and a quantum spline. In the classical context the solution curve passes through a given set of points, whereas in the quantum context, a curve on the space of pure states in itself has no operational meaning. Thus, instead of finding a curve in the space of pure states where the designated states lie, we must find a time-dependent curve in the space of Hamiltonians that in turn wil...

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply secondorder Lagrange-Poincaré reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.