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

Oleg V. Morzhin

TL;DR

This work addresses quantum information transfer along an $N$-site spin chain driven by a controlled external parabolic field, formulating both transfer and keeping objectives under pointwise control constraints. It develops infinite-dimensional gradients and a projection-type linearized PMP to drive gradient-projection updates for piecewise-continuous controls, including GPM-1S/2S/3S variants, with exact state propagation for piecewise-constant controls via matrix exponentials. The paper demonstrates a stochastic search using a genetic algorithm on a special control class for the $N=3$ case, achieving high-fidelity transfer ($I_1\approx 8\times 10^{-4}$) and highlighting the potential of these methods beyond standard GRAPE/Krotov approaches. Overall, it extends constrained gradient-based optimization in quantum control to spin-chain transfer problems and lays groundwork for scaling to larger chains and more complex keeping constraints.

Abstract

This article (I) considers the known optimal control model of a quantum information transfer along a spin chain with controlled external parabolic magnetic field, with an arbitrary length. The article adds certain lower and upper pointwise constraints on controls, adds the problem of keeping the signal at the last spin, considers various classes of controls. For these problems under piecewise continuous controls, the projection-type linearized Pontryagin maximum principle, gradient projection method's constructions in its one- and two- and three-step forms were adapted by analogy with [Morzhin O.V., Pechen A.N. J. Phys. A: Math. Theor., 2025]. Moreover, an example with a genetic algorithm's successful use under a special class of controls is given.

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

TL;DR

This work addresses quantum information transfer along an -site spin chain driven by a controlled external parabolic field, formulating both transfer and keeping objectives under pointwise control constraints. It develops infinite-dimensional gradients and a projection-type linearized PMP to drive gradient-projection updates for piecewise-continuous controls, including GPM-1S/2S/3S variants, with exact state propagation for piecewise-constant controls via matrix exponentials. The paper demonstrates a stochastic search using a genetic algorithm on a special control class for the case, achieving high-fidelity transfer () and highlighting the potential of these methods beyond standard GRAPE/Krotov approaches. Overall, it extends constrained gradient-based optimization in quantum control to spin-chain transfer problems and lays groundwork for scaling to larger chains and more complex keeping constraints.

Abstract

This article (I) considers the known optimal control model of a quantum information transfer along a spin chain with controlled external parabolic magnetic field, with an arbitrary length. The article adds certain lower and upper pointwise constraints on controls, adds the problem of keeping the signal at the last spin, considers various classes of controls. For these problems under piecewise continuous controls, the projection-type linearized Pontryagin maximum principle, gradient projection method's constructions in its one- and two- and three-step forms were adapted by analogy with [Morzhin O.V., Pechen A.N. J. Phys. A: Math. Theor., 2025]. Moreover, an example with a genetic algorithm's successful use under a special class of controls is given.

Paper Structure

This paper contains 5 sections, 17 equations, 2 figures.

Table of Contents

  1. Introduction-I
  2. Optimal Control Models with a Spin Chain
  3. Infinite-Dimensional Gradients, Projection-Type Linearized PMP, Zero Control, and GPM Forms
  4. System's States Represented via Matrix Exponentials. Stochastic Search for the Transfer Problem with $N=3$
  5. Conclusion-I

Figures (2)

  • Figure 1: $F(\psi^{u=0}(t); \psi_{\rm g})$ vs $t$: (a) in the transfer problem (Example 1, $\psi_0 \neq \psi_{\rm g}$); (b) in the keeping problem (Example 2, $\psi_0 = \psi_{\rm g}$).
  • Figure 2: For the transfer problems with the PConst. approximation of the special simple class (\ref{['sect2_f6']}), the GA results (Example 3) in the terms of the resulting $u_1, u_2$ (subfig. (b,c)), and $\psi^u_j$, $j=1,2,3$ (subfig. (a), (d)--(i)).