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arXiv:1410.4906 · Physical Review A

Geometric analysis of minimum-time trajectories for a two-level quantum system

Raffaele Romano

Published Dec 1, 2014• 19 citations
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Abstract

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.

DOI
10.1103/physreva.90.062302

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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.
We study how to generate in minimum time special unitary transformations for a two-level quantum system under the assumptions that (i) the system is subject to a constant drift, (ii) its dynamics can be affected by three independent, bounded controls, and (iii) the bounds on the controls are asymmetric, that is, the constraint on the control in the direction of the drift is independent of that on the controls in the orthogonal plane. Using techniques recently developed for the analysis of SU(2)\text{SU}(2)SU(2) transformations, we fully characterize the reachable sets of the system and the optimal control strategies for any possible target transformation.
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.
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.
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.
A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated with the tools available, which is one of a number of possible quantum speed limits. We examine this problem from the perspective of quantum control, where the system of interest is described by a drift Hamiltonian and set of control Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups, and differential geometry and formulates the problem in terms of geodesics on a differentiable manifold. We provide explicit lower bounds on the quantum speed limit for the case of an arbitrary drift, requiring only that the control Hamiltonians generate a topologically closed subgroup of the full unitary group, and formulate criteria as to when our expression for the speed limit is exact and not merely a lower bound. These analytic results are then tested and confirmed using a numerical optimization scheme. Finally, we extend the analysis to find a lower bound on the quantum speed limit in the common case where the system is described by a drift Hamiltonian and a single control Hamiltonian.
We consider a quantum control problem involving a spin-1/21/21/2 particle in a magnetic field. The magnitude of the field is held constant, and the direction of the field, which is constrained to lie in the x−−yx--yx−−y plane, serves as a control parameter that can be varied to govern the evolution of the system. We analytically solve for the time dependence of the control parameter that will synthesize a given target SU(2) transformation in the least possible amount of time, and we show that the time-optimal solutions have a simple geometric interpretation in terms of the fiber bundle structure of SU(2). We also generalize our time-optimal solutions to a control problem that includes a constant bias field along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{z}}$ axis and to the case of inhomogeneous control, in which a single control parameter governs the evolution of an ensemble of spin-1/21/21/2 systems.