Gradient projection method and stochastic search for some optimal control models with spin chains. II

This article (II) continues the research described in [Morzhin O.V. Gradient projection method and stochastic search for some optimal control models with spin chains. I (submitted)] (Article I), derives the needed finite-dimensional gradients corresponding to the infinite-dimensional gradients obtained in Article I, both for transfer and keeping problems at a certain $N$-dimensional spin chain, and correspondingly adapts a projection-type condition for optimality, gradient projection method (GPM). For the case $N=3$, the given in this article examples together with Example 3 in Article I show that: a) the adapted GPM and genetic algorithm (GA) successfully solved numerically the considered transfer and keeping problems; b) the two- and three-step GPM forms significantly surpass the one-step GPM. Moreover, GA and a special class of controls were successfully used in such the transfer problem that $N=20$ and the final time is not assigned.

💡 Research Summary

This paper, titled “Gradient projection method and stochastic search for some optimal control models with spin chains. II,” is a direct continuation of its predecessor (Article I). It focuses on developing and comparing numerical optimization strategies for quantum optimal control problems involving the transfer and maintenance of states in an N-dimensional spin chain system.

The core methodological advancement lies in bridging infinite-dimensional calculus of variations with practical finite-dimensional optimization. The authors start from the infinite-dimensional gradients for the control functions, derived in Article I via the Pontryagin maximum principle. Under the practical assumption of piecewise constant controls, Lemma 1 meticulously derives the corresponding finite-dimensional gradients. These gradients are expressed in terms of the quantum state and the adjoint state, both evaluated at the midpoints of the time discretization intervals using matrix exponentials, providing an efficient and accurate computation scheme.

Building on this, Theorem 1 states a first-order necessary optimality condition in the form of a projection-type condition for the finite-dimensional control parameter vector. This condition naturally leads to the adaptation of Gradient Projection Method (GPM) algorithms. The authors implement and compare three variants: the basic one-step form (GPM-1S), a two-step form incorporating Polyak’s momentum term (GPM-2S), and a three-step form with an additional inertial term (GPM-3S).

The numerical experiments for the case N=3 provide compelling evidence for the superiority of the multi-step approaches. In Example 1 (state transfer problem), GPM-2S and GPM-3S demonstrate a dramatic speed-up, requiring orders of magnitude fewer Cauchy problem solutions (the main computational cost) to converge compared to GPM-1S. This highlights the effectiveness of momentum-based acceleration for the topography of these quantum control landscapes.

Example 2 (state keeping problem) employs a dual strategy. First, a special class of controls, parameterized as a sum of sinusoids, is optimized using a Genetic Algorithm (GA). The GA successfully finds controls within this simple class that maintain the initial state with high fidelity. Second, starting from a sinusoidal control, the GPM variants are applied. The results again show the clear computational advantage of GPM-3S and GPM-2S over GPM-1S in reaching the stopping criteria.

Finally, the paper tackles a more challenging scenario: a state transfer problem for a larger spin chain (N=20) with a free (not pre-assigned) final time T. The problem is transformed by scaling time to a fixed interval, making T an additional optimization variable. The authors show that the combination of the parameterized control class and the GA can successfully solve this problem, demonstrating the flexibility and power of stochastic search methods for complex, high-dimensional settings where gradient information might be costly or difficult to utilize effectively.

In summary, this work makes a significant contribution to the toolbox for quantum optimal control of spin chains. It provides a rigorous link between analytical gradient theory and numerical implementation, demonstrates the superior efficiency of advanced GPM forms over the basic version, and validates the utility of genetic algorithms as a complementary, derivative-free optimization strategy capable of handling challenging problem formulations.