برنامه ریزی پویا و راه حل های ویسکوزیته برای کنترل بهینه سیستم های اسپین کوانتومی

The purpose of this paper is to describe the application of the notion of viscosity solutions to solve the Hamilton–Jacobi–Bellman (HJB) equation associated with an important class of optimal control problems for quantum spin systems. The HJB equation that arises in the control problems of interest is a first-order nonlinear partial differential equation defined on a Lie group. Hence we employ recent extensions of the theory of viscosity solutions to Riemannian manifolds in order to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems. The convergence of these approximations is proven using viscosity solution methods. In order to illustrate the techniques developed, these methods are applied to an example problem.

Recently, there has been considerable attention directed at the problem of obtaining time optimal trajectories for open loop control of quantum spin systems [1], [2], [3] and [4]. These problems arise from applications which include NMR spectroscopy (to produce a time optimal trajectory), and the optimal construction of quantum circuits [5] and [6] (to minimize the number of logic gates required to construct a desired unitary transformation). These spin systems have the mathematical structure of a bilinear right invariant system on the special unitary group. Owing to the importance of the applications, there have been various approaches to solving these problems which utilize Lie theoretic arguments [2] and [7], calculus of variations [4] and [8], and dynamic programming [9] and [10]. In the dynamic programming approach, under appropriate regularity assumptions, the optimal cost function (value function) is the solution to a Hamilton–Jacobi–Bellmann (HJB) equation [11], [12] and [13]. For many problems of interest this value function can be demonstrated to be non-differentiable. Hence there is the need for a more general notion of a solution to such partial differential equations (PDEs). A popular and successful concept of such a weak solution of nonlinear PDEs is the well-studied theory of viscosity solutions [14] and [15] on Euclidean spaces. Because the quantum spin problem leads to an HJB equation defined on a Lie group, we use extensions of viscosity solution theory to Riemannian manifolds [16], [17] and [18] in order to interpret the solutions of this equation. For a detailed introduction to this topic, we refer the reader to [14] and [19] and the references contained therein. In this article, we build up the components required for a rigorous application of viscosity solution theory on manifolds for quantum systems. This commences with an explanation of a discretization method based on the triangulation of manifolds [20] to solve the HJB equation for the optimal spin control problem. We then use viscosity solution concepts to prove the convergence of the solution obtained by this triangulation-based discretization scheme to the solution of the original HJB equation. The structure of this article is as follows. We begin by describing the quantum spin control problem in Section 2. This is followed in Section 3 by a study of the regularity properties of the value function which play an important role in the solution of the associated HJB equation. After motivating the need for more generalized solutions of the HJB equation using an example system with a non-differentiable value function, we explain the use of the notion of viscosity solutions on Lie groups in Section 4. Results pertinent to the existence and uniqueness of such solutions are recalled from relevant literature and are modified to the framework of the problems introduced. In order to solve these optimal control problems numerically, we make use of the notion of triangulation of the group on which the system evolves. This concept and the proofs of convergence of the approximations to the actual solution using viscosity solution notions are introduced and developed in Section 5. In Section 6, the ideas developed are then applied to solve an example control problem on SU(2)SU(2) for which sample optimal trajectories and the value functions from the simulations are obtained. We conclude with comments and possible extensions in Section 7.

In this paper, we have introduced a rigorous framework for the numerical techniques involved in using the dynamic programming technique from optimal control theory for the control of quantum spin systems evolving on compact Lie groups. Numerical simulations were performed by triangulation of the group which, due to the well-studied numerical procedures available for tesselation of surfaces, enable better numerical speed and efficiency of implementation. In addition, the solution can be made more accurate at points of the group where such accuracy is desired (e.g., around the origin in Fig. 3 where the solution is non-differentiable). The dynamic programming methods provide a framework that can, in principle, be used for systems with an arbitrary number of qubits, unlike limitations on the Lie theoretic methods. In addition, alternative numerical techniques that use the calculus of variations are subjected to issues in the entrapment at local minima — a drawback absent in the current approach. The value function iteration methods, when used on any grid, suffer from the curse of dimensionality, and hence become intractable for higher-dimensional systems. For instance, the number of spatial dimensions in a quantum spin-1/2 system with nn qubits grows as 4n−14n−1. Possible directions of future work may involve a study of methods such as fast marching [24] or meshless techniques to improve the speed of computations. Inspired by the dynamic programming framework in this article, and the curse of dimensionality-free approaches in [25], new methods are currently being developed for reduced-dimensionality approximation techniques to quantum control [26] and [27]. There exist classes of control problems such as those involving quantum systems with bounded controls and drift for which the value function is discontinuous. Viscosity solution techniques for such discontinuous cost functions may be used to provide the technical framework for the use of impulsive controls in the dynamic programming approach to quantum control.