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SIAM Journal on Control and Optimization

Optimal Control of Quantum Systems in Fermion Fields: The Pontryagin-Type Maximum Principle (I)

Penghui Wang, Shan Wang

Published May 14, 2026• 1 citation
Publisher

Abstract

Abstract. In this paper, we derive a Pontryagin-type maximum principle for optimal control of quantum systems in fermion fields. These systems have gained significant prominence in numerous quantum applications ranging from physical chemistry to multidimensional nuclear magnetic resonance experiments. Furthermore, we establish the existence and uniqueness of solutions to backward quantum stochastic differential equations driven by fermion Brownian motion. The application of noncommutative martingale inequalities and the martingale representation theorem enables this achievement.

DOI
10.1137/24m1644237

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Abstract. In this paper, we derive a Pontryagin-type maximum principle for optimal control of quantum systems in fermion fields. These systems have gained significant prominence in numerous quantum applications ranging from physical chemistry to multidimensional nuclear magnetic resonance experiments. Furthermore, we establish the existence and uniqueness of solutions to backward quantum stochastic differential equations driven by fermion Brownian motion. The application of noncommutative martingale inequalities and the martingale representation theorem enables this achievement.
In this paper, the Pontryagin-type maximum principle for optimal control of quantum stochastic systems in fermion fields is obtained. These systems have gained significant prominence in numerous quantum applications ranging from physical chemistry to multi-dimensional nuclear magnetic resonance experiments. Furthermore, we establish the existence and uniqueness of solutions to backward quantum stochastic differential equations driven by fermion Brownian motion. The application of noncommutative martingale inequalities and the martingale representation theorem enables this achievement.
ABSTRACT This manuscript focuses on the study of optimal quantum control problem for the quantum stochastic model in noncommutative space. The existence and uniqueness of the quantum stochastic differential equation driven by a fermion Brownian motion with bounded delay in finite time interval are established by employing the noncommutative Burkholder–Gundy inequality, generalised Minkowski inequality and generalised Banach fixed‐point theorem. The proposed system is formulated to control the quantum stochastic differential equation and to maximise the probability that the quantum state will reach the target state. By using the Pontryagin maximum principle, optimal quantum control theory is established for the proposed system. Furthermore, an application of the algorithm is presented through numerical simulations which validate the theoretical results.
In the present paper, by using the relaxed transposition method[29], we solve the second-order adjoint equations, corresponding to the optimal control of quantum stochastic systems in fermion fields, which plays the fundamental roles in the study of the Pointryagin-type maximum principle in quantum stochastic optimal control. The second-order adjoint equation is a backard operator valued quantum stochastic differential equation, which has no definition in the algebra of bounded operators, and the solution derived from the relaxed transposition method makes sense in W*-topology.
This paper aims to establish second order necessary conditions for optimal control in quantum stochastic systems. We employ a variational approach, analogous to methods in classical stochastic control, to analyze systems governed by quantum stochastic differential equations driven by fermionic Brownian motion, where the control enters both the drift and diffusion terms. This result provides a theoretical foundation for further exploration of optimization problems and their practical applications in the field of quantum stochastic control.
This paper aims to establish second order necessary conditions for optimal control in quantum stochastic systems. We employ a variational approach, analogous to methods in classical stochastic control, to analyze systems governed by quantum stochastic differential equations driven by fermionic Brownian motion, where the control enters both the drift and diffusion terms. This result provides a theoretical foundation for further exploration of optimization problems and their practical applications in the field of quantum stochastic control.
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This tutorial aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This tutorial describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin Maximum Principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.