A quantum approach for optimal control

TL;DR

The paper tackles nonlinear optimal control by mapping the problem to a constrained quantum system using Dirac's canonical quantization and solving for the ground state of a non-Hermitian Hamiltonian with a modified VQE. It introduces a Dirac-bracket framework to handle primary/secondary and first-/second-class constraints, and extends this to multi-dimensional problems with a structured Dirac-bracket formulation. The methodology combines spectral methods for the Schrödinger equation with VQE-based eigenvalue solutions, and demonstrates multiple 1D and 2D computational examples showing good agreement with analytic solutions or MPOPT results, while highlighting the importance of basis choice and non-Hermitian-VQE variants. Overall, the approach offers a promising quantum-aligned alternative for high-dimensional nonlinear optimal control with potential near-term applicability on NISQ-era devices.

Abstract

In this work, we propose a novel variational quantum approach for solving a class of nonlinear optimal control problems. Our approach integrates Dirac's canonical quantization of dynamical systems with the solution of the ground state of the resulting non-Hermitian Hamiltonian via a variational quantum eigensolver (VQE). We introduce a new perspective on the Dirac bracket formulation for generalized Hamiltonian dynamics in the presence of constraints, providing a clear motivation and illustrative examples. Additionally, we explore the structural properties of Dirac brackets within the context of multidimensional constrained optimization problems. Our approach for solving a class of nonlinear optimal control problems employs a VQE-based approach to determine the eigenstate and corresponding eigenvalue associated with the ground state energy of a non-Hermitian Hamiltonian. Assuming access to an ideal VQE, our formulation demonstrates excellent results, as evidenced by selected computational examples. Furthermore, our method performs well when combined with a VQE-based approach for non-Hermitian Hamiltonian systems. Our VQE-based formulation effectively addresses challenges associated with a wide range of optimal control problems, particularly in high-dimensional scenarios. Compared to standard classical approaches, our quantum-based method shows significant promise and offers a compelling alternative for tackling complex, high-dimensional optimization challenges.

Figures (10)

• Figure 1: Wave function for Example 1, with $N_c=N_s=2$ (i.e., $N=4$) and $h_w = 0.1$.
• Figure 2: Time evolution of the state and control variables for Example 1 (with $h_w=0.1$). Results from the proposed framework for $N=4$ case (dashed lines) and $N=8$ case (dashed-dotted lines) are shown, including a comparison with analytical solutions (shown in dotted lines).
• Figure 3: Wave function amplitude for Example 2, with $N_s=N_c=8$ (i.e., $N=16$) and $h_w = 2$.
• Figure 4: Time evolution of state and control variables for Example 2, with $N_s=N_c=8$ (i.e., $N=16$) and $h_w = 2$. Results from the proposed framework (shown in dotted lines) are compared with analytical solutions (shown in solid lines).
• Figure 5: Wave function amplitude for Example 3, with $N_s=N_c=16$ (i.e., $N=32$) and $h_w = 0.1$.

Theorems & Definitions (3)

• Definition 3.0.1
• Example 3.0.2
• Remark 6.3.1