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IEEE Transactions on Automatic Control

The $K\hbox{--}P$ Problem on Tensor Products of Lie Groups and Time-Optimal Control of $n$ Quantum Bits With a Bounded Field

Francesca Albertini, Domenico D’Alessandro

Published Jul 25, 2017• 5 citations
Publisher

Abstract

We use properties of sub-Riemannian$K\hbox{--}P$problems and the optimal synthesis on$SU(2)$to solve the time-optimal control of$n$noninteracting two-level quantum systems (qubits) controlled in parallel with a control field bounded in$L_2$norm. Every qubit is controlled independently, but the total energy of the controls used is bounded. For this system, we describe the reachable set at every time. Moreover, we compute thediameter, that is, the worst-case minimum time. The mathematical problem is motivated by the control of two-level systems with different resonance frequencies, where the control fields are obtained from a single signal having bounded energy.

DOI
10.1109/tac.2017.2727225

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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.
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.
We consider the problem of controlling in minimum time a two-level quantum system which can be subject to a drift. The control is assumed to be bounded in magnitude and to affect two or three independent generators of the dynamics. We describe the time optimal trajectories in SU(2), the Lie group of possible evolutions for the system, by means of a particularly simple parametrization of the group. A key ingredient of our analysis is the introduction of the optimal front line. This tool allows us to fully characterize the time evolution of the reachable sets and to derive the worst-case operators and the corresponding times. The analysis is performed in any regime---controlled dynamics stronger than, of the same magnitude as, or weaker than the drift term---and gives a method to synthesize quantum logic operations on a two-level system in minimum time.
For the time optimal problem of an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU(2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.
Abstract In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin maximum principle, and provide a complete characterization of singular and nonsingular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.