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Cayley Commutator-free Methods for Krotov-Type Algorithms in Quantum Optimal Control

Boris Wembe, Usman Ali, Torsten Meier, Sina Ober-Blöbaum

Abstract

This paper presents a class of structure-preserving numerical methods for quantum optimal control problems, based on commutator-free Cayley integrators. Starting from the Krotov framework, we reformulate the forward and backward propagation steps using Cayley-type schemes that preserve unitarity and symmetry at the discrete level. This approach eliminates the need for matrix exponentials and commutators, leading to significant computational savings while maintaining higher-order accuracy. We first recall the standard linear setting and then extend the formulation to nonlinear Schrödinger and Gross-Pitaevskii equations using a Cayley-polynomial interpolation strategy. Numerical experiments on state-transfer problems illustrate that the CF-Cayley method achieves the same accuracy as high-order exponential or Cayley-Magnus schemes at substantially lower cost, especially for longtime or highly oscillatory dynamics. In the nonlinear regime, the structure-preserving properties of the method ensure stability and norm conservation, making it a robust tool for large-scale quantum control simulations. The proposed framework thus bridges geometric integration and optimal control, offering an efficient and reliable alternative to existing exponential-based propagators.

Cayley Commutator-free Methods for Krotov-Type Algorithms in Quantum Optimal Control

Abstract

This paper presents a class of structure-preserving numerical methods for quantum optimal control problems, based on commutator-free Cayley integrators. Starting from the Krotov framework, we reformulate the forward and backward propagation steps using Cayley-type schemes that preserve unitarity and symmetry at the discrete level. This approach eliminates the need for matrix exponentials and commutators, leading to significant computational savings while maintaining higher-order accuracy. We first recall the standard linear setting and then extend the formulation to nonlinear Schrödinger and Gross-Pitaevskii equations using a Cayley-polynomial interpolation strategy. Numerical experiments on state-transfer problems illustrate that the CF-Cayley method achieves the same accuracy as high-order exponential or Cayley-Magnus schemes at substantially lower cost, especially for longtime or highly oscillatory dynamics. In the nonlinear regime, the structure-preserving properties of the method ensure stability and norm conservation, making it a robust tool for large-scale quantum control simulations. The proposed framework thus bridges geometric integration and optimal control, offering an efficient and reliable alternative to existing exponential-based propagators.
Paper Structure (19 sections, 3 theorems, 33 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 19 sections, 3 theorems, 33 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation and Necessary Conditions
  3. Quantum Optimal Control Problem
  4. Pontryagin Framework and Necessary Conditions
  5. Krotov-Type Iterative Method
  6. Cayley commutator-free method (CFCT)
  7. Commutator-Free Cayley Schemes in the Linear Case
  8. Commutator-Free Cayley Schemes in the Nonlinear Case
  9. Algorithmic idea
  10. One-step update
  11. Cayley--Polynomial Algorithm.
  12. CF-Cayley Methods in Quantum Optimal Control
  13. Incorporation into Krotov’s Algorithm
  14. Example 1: Linear Schrödinger Equation for Non-interacting Trapped Cold Atoms in a Driven Optical Lattice
  15. Example 2: Nonlinear Schrödinger (Gross--Pitaevskii) Equation for Interacting Bose-Einstein Condensates
  16. ...and 4 more sections

Key Result

Lemma V.1

Let $\psi(t)$ be the solution of system eq:system_dynamics, with $-1 \notin \sigma(\psi(t)\psi_0^{-1})$where $\sigma(A)$ defined the spectrum of the matrix A. for any $t \in [0,T]$. Let $\psi(t) = \,{\rm Cay}(\Omega(t))\psi_0$, then $\Omega$ is the solution of the system: with the Lie bracket (commutator) of two matrices $A$ and $B$ defined by $[A,B] = A\cdot B - B\cdot A$.

Figures (2)

  • Figure 1: Quantum state transfer with the linear Schrödinger equation for non-interacting cold atoms in a driven parabolic lattice. The initial Gaussian wave packet at $x_0 = 0.0$, the target state, and the corresponding solutions of the Schrödinger equation are shown in the top-left panel, while the evolution of the control function is displayed in the top-right panel. The bottom-left panel presents a different target state obtained from an initial Gaussian wave packet centered at $x_0 = -25.0$, and the associated control function is shown in the bottom-right panel.
  • Figure 2: Quantum state transfer governed by the Gross-Pitaevskii equation under the fixed driving control $u(t)= u_c \sin(t)$. Starting from a Gaussian initial state localized at $x=0$, the system evolves toward the target state. The real and imaginary components of the target state are shown in the left-panel and right-panel, respectively, and are reproduced with high fidelity by both the Runge-Kutta-Munthe-Kaas (RKMK4) and Non-Linear CF-Cayley (CaylPol) methods.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma V.1
  • Lemma V.2
  • Lemma V.3