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IEEE International Conference on Quantum Control, Computing and Learning (qCCL)

Second-Order Adjoint Method for Quantum Optimal Control

Harish S. Bhat

Published Jun 25, 2025• 0 citations
Publisher

Abstract

We derive and implement a second-order adjoint method to compute exact gradients and Hessians for a prototypical quantum optimal control problem, that of solving for the minimal energy applied electric field that drives a molecule from a given initial state to a desired target state. For small to moderately sized systems, we demonstrate a vectorized GPU implementation of a second-order adjoint method that computes both Hessians and gradients with wall times only marginally more than those required to compute gradients via commonly used first-order adjoint methods. Pairing our secondorder adjoint method with a trust region optimizer (a type of Newton method), we show that it outperforms a first-order method, requiring significantly fewer iterations and wall time to find optimal controls for four molecular systems.

DOI
10.1109/qccl65142.2025.11158804

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We derive and implement a second-order adjoint method to compute exact gradients and Hessians for a prototypical quantum optimal control problem, that of solving for the minimal energy applied electric field that drives a molecule from a given initial state to a desired target state. For small to moderately sized systems, we demonstrate a vectorized GPU implementation of a second-order adjoint method that computes both Hessians and gradients with wall times only marginally more than those required to compute gradients via commonly used first-order adjoint methods. Pairing our secondorder adjoint method with a trust region optimizer (a type of Newton method), we show that it outperforms a first-order method, requiring significantly fewer iterations and wall time to find optimal controls for four molecular systems.
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time-dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently. The accurate numerical computation of optimal controls of infinite dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2 versus H1 based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. A matrix-free cascadic BFGS algorithm is introduced in the L 2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented. A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov-Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the non-convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high-energy state transitions. Control of bound-to-bound and bound-to-continuum transitions in open quantum systems and vector field control of two-dimensional systems is presented. An efficient space-time spectral discretization of the time-dependent Schrodinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined.
One of the most popular methods for solving numerical optimal control problems is the Krotov method, adapted for quantum control by Tannor and coworkers. The Krotov method has the following three appealing properties: (1) monotonic increase of the objective with iteration number, (2) no requirement for a line search, leading to a significant savings over gradient (first-order) methods, and (3) macrosteps at each iteration, resulting in significantly faster growth of the objective at early iterations than in gradient methods where small steps are required. The principal drawback of the Krotov method is slow convergence at later iterations, which is particularly problematic when high fidelity is desired. We show here that, near convergence, the Krotov method degenerates to a first-order gradient method. We then present a variation on the Krotov method that has all the advantages of the original Krotov method but with significantly enhanced convergence (second-order or quasi-Newton) as the optimal solution is approached. We illustrate the method by controlling the three-dimensional dynamics of the valence electron in the Na atom.
A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nanoelectronic devices and quantum computing. This method is based on a reduced Hessian Krylov–Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified, proving the existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the nonconvex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this paper is able to provide fast and accurate controls for high-energy state transitions.
Accurate derivatives are important for locally traversing and converging efficiently in quantum optimization landscapes. By deriving analytically exact control derivatives (gradient and Hessian) for unitary control tasks, we show here that the computational feasibility of meeting this accuracy requirement depends on the choice of propagation scheme and problem representation. Even when exact propagation is sufficiently cheap it is, perhaps surprisingly, much more efficient to optimize the (appropriately) approximate propagators: Approximations in the dynamics are traded off for significant complexity reductions in the exact derivative calculations. Importantly, past the initial analytical considerations, only standard numerical techniques are explicitly required with straightforward application to realistic systems. These results are numerically verified for two concrete problems of increasing Hilbert space dimensionality. The best schemes obtain unit fidelity to machine precision, whereas the results for other schemes are separated consistently by orders of magnitude in computation time and in worst-case ten orders of magnitude in achievable fidelity. Since these gaps continually increase with system size and complexity, this methodology allows numerically efficient optimization of very high-dimensional dynamics, e.g., in many-body contexts, operating in the high-fidelity regime.
The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.
Efficient optimization of quantum systems is a necessity for reaching fault-tolerant thresholds. A standard tool for optimizing simulated quantum dynamics is the gradient-based grape algorithm, which has been successfully applied in a wide range of different branches of quantum physics. In this work, we derive and implement exact second-order analytical derivatives of the coherent dynamics and find improvements compared to the standard of optimizing with the approximate second-order Broyden-Fletcher-Goldfarb-Shanno algorithm. We demonstrate performance improvements for both the best and the average errors of constrained unitary gate synthesis on a circuit-QED system over a broad range of different gate durations.