Quantum Splines

Abstract

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...

Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework

“…In this section, we give an overview of a quantum mechanical application presented in [6], where more details can be found. We consider an n + 1-level quantum system with Hilbert space H = C n+1 .…”

“…However, in this paper, we will restrict ourselves to the case of finite-dimensional Lie groups and object manifolds. A natural finite-dimensional instance for illustrating these ideas arises in quantum control [6], where quantum state vectors evolve under the action of the unitary group. The generator curve ξ (t) in this case corresponds to the Hamiltonian operator, which is controlled in experiments.…”

“…In general, one expects therefore that exact matching may not be possible and works instead with a soft constraint. Two of the authors have recently studied a problem of similar nature in quantum control, see [6]. There, one considers the group of unitary matrices acting on quantum state space, with the goal of finding the optimal experimental manipulation of the system such that the evolution of an initial state passes near a sequence of given target states.…”

“…In some applications, for example in computational anatomy [3] and quantum control [6], the optimal trajectory is generated via a group action. That is, a curve g(t) in a Lie group G acts on a point Q 0 in an object manifold and generates a curve q(t) = g(t)Q 0 , which passes through a sequence of target points at prescribed times.…”

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“…By recasting the effective Hamiltonian as a sum of a Hermitian operator (H H ≡ (H + H † )/2) and an antiHermitian operator (H A ≡ (H − H † )/2), i.e. H = H H + H A , the dynamic equation of the system can be expressed as [21] ∂ρ ∂t…”