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arXiv:2211.09167 · Physical Review A

Time-optimal control of two-level quantum systems by piecewise constant pulses

E Dionis, Dominique Sugny

Published Mar 17, 2023• 18 citations
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Abstract

We apply an extension of the Pontryagin maximum principle to derive time-optimal controls of two-level quantum systems by means of piecewise constant pulses. Global optimal solutions are obtained for state-to-state transfer in the cases with one and two controls. Exact quantum speed limits are established as a function of the sampling period. We observe numerically an exponential convergence towards the minimum time in the continuous limit when this period goes to zero. We show that this convergence is only polynomial for a linearized quantum system. We discuss the experimental impact of this result.

DOI
10.1103/physreva.107.032613

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We apply an extension of the Pontryagin maximum principle to derive time-optimal controls of two-level quantum systems by means of piecewise constant pulses. Global optimal solutions are obtained for state-to-state transfer in the cases with one and two controls. Exact quantum speed limits are established as a function of the sampling period. We observe numerically an exponential convergence towards the minimum time in the continuous limit when this period goes to zero. We show that this convergence is only polynomial for a linearized quantum system. We discuss the experimental impact of this result.
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We study the time-optimal robust control of a two-level quantum system subjected to field inhomogeneities. We apply the Pontryagin maximum principle and we introduce a reduced space onto which the optimal dynamics is projected down. This reduction leads to a complete analytical derivation of the optimal solution in terms of elliptic functions and elliptic integrals. Necessary optimality conditions are then obtained for the original system. These conditions are verified numerically and lead to the optimal control protocol. Various examples, ranging from state-to-state transfer to the generation of a NOT gate, illustrate this study. The connection with other geometric optimization approaches that have been used to solve this problem is also discussed. Finally, we show how this method can be combined with numerical optimization algorithms to design robust control for higher-dimensional quantum systems, as illustrated by a Bose-Einstein condensate in an optical lattice.
Motivated by problems in quantum control, we introduce a novel method for studying time-optimal control problems of affine systems with unbounded controls. Intuitively, we construct a relaxation of our problem as follows: for each control of finite amplitude, we appropriately rescale the time variable so that, in the limit, an instantaneous jump in the state of the system becomes a trajectory containing both the state before and after the jump. The main advantage of this method is the direct applicability of the classical Pontryagin maximum principle. Several examples, including the generation of noiseless and noisy qubit gates, as well as population transfer in open two-level systems, are discussed.
Using optimal control, we establish and link the ultimate bounds in time (referred to as the quantum speed limit) and energy of two- and three-level quantum nonlinear systems which feature 1:2 resonance. Despite the unreachable complete inversion, by using the Pontryagin maximum principle, we determine the optimal time, pulse area, or energy for a given arbitrary accuracy. We show that the third-order Kerr terms can be absorbed in the detuning in order to lock the dynamics to the resonance. In the two-level problem, we determine the nonlinear counterpart of the optimal $\ensuremath{\pi}$-pulse inversion for a given accuracy. In the three-level problem, we obtain an intuitive pulse sequence similar to the linear counterpart but with different shapes. We prove the (slow) logarithmic increasing of the optimal time as a function of the accuracy.
Abstract We consider a two-level open quantum system whose dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad equation with Hamiltonian and dissipation superoperator depending, respectively, on coherent and incoherent controls. Results about reachability, controllability, and minimum-time control are obtained in terms of the Bloch parametrization. First, we consider the case when the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls. Second, for zero coherent control and for incoherent control lying in the class of constant functions, the reachability and controllability sets of the system are exactly described and some analytical results on the minimum-time control are found. Third, we consider a series of increasing values of the final time and the corresponding classes of controls with zero incoherent control and with coherent control equal to zero until a switching time instant and to a cosine function after it. The corresponding reachable points in the Bloch ball are numerically obtained and visualized. Fourth, a known method for estimating reachable sets is adapted and used to analyze the situation where the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls while, as shown numerically, are not optimal.
In this paper, we discuss a quantum control algorithm, devised by the authors, inspired in Rabitz and Krotov algorithms and applied to Nuclear Magnetic Resonance (NMR) That algorithm is based on a numerical method for iterative optimization.Specifically, we address the determination of external optimal pulses (controls) to minimal cost, over a two-level quantum system.We use the numerical approximation to find the optimal controls in the case of two external electromagnetic fields, integrating the associated equations of the Pontryagin Maximum Principle.That algorithm unifies and generalizes the Rabitz and Krotov algorithms.We compare the efficiency of these algorithms with the solutions found by analytical methods.