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IEEE Conference on Decision and Control (CDC)

Pulse control of affine systems with applications to quantum control

I. Beschastnyi, Lamberto Dell’Elce, J.-B. Pomet, Ludovic Sacchelli, D. Tinoco

Published Dec 9, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1109/cdc57313.2025.11312971

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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.
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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.
As technological advances allow us to peer into and beyond microscopic phenomena, new theoretical developments are necessary to facilitate this exploration. In particular, the potential afforded by utilizing quantum resources promises to dramatically affect scientific research, communications, computation, and material science. \n \nControl theory is the field dedicated to the manipulation of systems, and quantum control theory pertains to the manoeuvring of quantum systems. Due to the inherent sensitivity of quantum ensembles to their environment, time-optimal solutions should be found in order to minimize exposure to external sources. Furthermore, the complexity of the Schr\\"odinger equation in describing the evolution of a unitary operator makes the analytical discovery of time-optimal solutions rare, motivating the development of numerical algorithms. \n \nThe seminal result of classical control is the Pontryagin Maximum Principle, which implies that a restriction to bounded control amplitudes admits two classifications of trajectories: bang-bang and singular. Extensions of this result to generalized control problems yield the same classification and hence arise in the study of quantum dynamics. While singular trajectories are often avoided in both classical and quantum literature, a full optimal synthesis requires that we analyze the possibility of their existence. \n \nWith this in mind, this treatise will examine the issue of time-optimal quantum control. In particular, we examine the results of existing numerical algorithms, including Gradient Ascent Pulse Engineering and the Kaya-Huneault method. We elaborate on the ideas of the Kaya-Huneault algorithm and present an alternative algorithm that we title the Real-Embedding algorithm. These methods are then compared when used to simulate unitary evolution. \n \nThis is followed by a brief examination on the conditions for the existence of singular controls, as well possible ideas and developments in creating geometry based numerical algorithms.
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.
Optimal control provides powerful tools and concepts that can be applied to control quantum systems. It has been used extensively to improve the performance of quantum processes in a variety of active areas in quantum technologies. This paper reviews the necessary conditions of optimality in the form of a Pontryagin Maximum Principle (PMP), and introduces its formulations for quantum systems. After formulating an optimal control problem for a two-level quantum system, we then study the application of PMP to obtain an optimal control strategy.