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Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

Time-optimal control of a 3-level quantum system and its generalization to an n-level system

Dong Eui Chang, Rodolphe J. Sepulchre

Published Jul 1, 2007• 2 citations
Publisher

Abstract

We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimal control problem for a general n-level quantum system. For the three- level system we find all optimal controls by finding two types of symmetry in the problem: Z2x S3discrete symmetry and S1continuous symmetry, and exploiting them to solve the problem through discrete reduction and symplectic reduction. We then study the geometry, in the same framework, which occurs in the time-optimal control of a general n-level quantum system.

DOI
10.1109/acc.2007.4282658

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We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimal control problem for a general n-level quantum system. For the three- level system we find all optimal controls by finding two types of symmetry in the problem: Z2x S3discrete symmetry and S1continuous symmetry, and exploiting them to solve the problem through discrete reduction and symplectic reduction. We then study the geometry, in the same framework, which occurs in the time-optimal control of a general n-level quantum system.
Abstract. We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimal control problem for a general n-level quantum system. For the three-level system we find all optimal controls by finding two types of symmetry in the problems: Z2×S3 discrete sym-metry and S1 continuous symmetry, and exploiting them to solve the problem through discrete reduction and symplectic reduction. We then study the geometry, in the same framework, which occurs in the time-optimal control of a general n-level quantum system.
We study the time optimal control problem for the evolution operator of an n-level quantum system from the identity to any desired final condition. For the considered class of quantum systems the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. From a mathematical perspective, such a problem is a sub-Riemannian K-P problem, whose underlying symmetric space is SU(n)/S(U(n-1) x U(1)). Following the method of symmetry reduction, we consider the action of S(U(n-1) xU(1)) on SU(n) as a conjugation X ---> AXA^{-1}. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (Lambda-systems). We solve this latter problem by reducing it to an integer quadratic optimization problem with linear constraints.
In this paper, we solve the problem of simultaneously driving in minimum time to arbitrary final conditions, N two level quantum systems subject to independent controls. The solution of this problem is obtained via an explicit description of the reachable set of the associated control system on SU(2). The treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on Lie groups.
We consider the problem of time-optimal control for a three-level quantum system where one level is coupled by the control field to the lowest two, which are not coupled to each other. A bound is assumed on the norm of the control at every time. Such a problem belongs to the class of KP sub-Riemannian problems for which we can perform a symmetry reduction and reduce to a Riemannian problem on the associated quotient space. We prove several properties of such a quotient space in our case, including the fact that it is an example of an almost-Kähler manifold that is not Kähler. We provide the explicit form of the optimal controls for any unitary transformation on the lowest two levels and discuss the geometric and practical implications of this result.
Sub-Riemannian Geodesics in SU(n)/S(U(n−1)×U(1))SU(n)/S(U(n-1) \times U(1))SU(n)/S(U(n−1)×U(1)) and Optimal Control of Three Level Quantum Systems
We study the time optimal control problem for the evolution operator of an n-level quantum system. For the considered models, the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. The resulting problem is a sub-Riemannian K-P problem, (as introduced in articles by U. Boscain etal. and by V. Jurdjevic), whose underlying symmetric space is SU(n)/S(U(n - 1) × U(1)). Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of S(U(n - 1) × U(1)) on SU(n) as a conjugation X → KXK-1. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (A-systems). We reduce the latter problem to an integer quadratic optimization problem with linear constraints, which we solve completely for a specific set of final data.
In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.