Synthesis of Linear Coherent Quantum Control Systems Using A Differential Evolution Algorithm

Abstract

We propose a new method to construct an optimal linear coherent quantum controller based on an evolutionary optimization method, namely a differential evolution algorithm. The aim is to provide a straightforward approach to deal with both nonlinear and nonconvex constraints arising in the coherent quantum controller synthesis. The solution to this control problem involves a complex algebraic Riccati equation, which corresponds to a physical realizability condition for the coherent quantum controller. The proposed method is demonstrated through an example of an entanglement control problem for a quantum network comprising two cascaded optical parametric amplifiers.Index Terms-Coherent quantum control, differential evolution, entanglement control. I. INTRODUCTIONDeveloping reliable computational methods has been an important aspect of solving quantum control problems. In this regard, we particularly refer to [1]- [3] where it is shown that a dynamic coherent quantum controller must satisfy a physical realizability condition in order to exhibit a meaningful quantum dynamic. This requirement leads to a quantum controller synthesis problem with a nonconvex nonlinear constraint, which is often difficult to solve numerically. This issue has indeed been addressed in [2] where a rank constrained LMI approach (see [4]) is used to synthesize a physically realizable coherent quantum LQG controller. The approach of [2], however, is applied only to solve a relaxed feasibility version of the original quantum LQG control problem, which is a nonconvex nonlinear optimization problem and remains to be solved. Also, there is a concern about the success of the rank constrained LMI approach, which is dependent on a suitable initial point to begin the numerical iteration. Moreover, when dealing with higher order quantum control systems, the approach of [2] will lead to complicated rank constrained LMI problems.These concerns provide our impetus to present in this technical note a tractable algorithm to solve a coherent quantum feedback control problem for a class of non-commutative linear quantum stochastic systems. Inspired by [2] and [5], we thus propose to use an evolutionary optimization approach, that is the differential evolution (DE) algorithm (see [6]), to solve the quantum feedback control problem. The aims are to obtain a straightforward algorithm and to avoid a critical dependence on a suitable initial point to start the numerical iteration. Consequently, we reformulate the given quantum control problem as a constrained nonlinear optimization problem. In this case, the Manuscript

Composite Differential Evolution for Constrained Evolutionary Optimization

“…Currently, DE has been successfully applied to solve optimization problems in a considerable number of fields, such as electrical and power systems [24], [25], manufacturing science and operational research [26], [27], automotive design [28], and controller design [13], [14], [15].…”

“…In terms of both the modified DE/rand-to-best/1/bin and DE/current-to-best/1/bin, the information of the "best" individual in the population is utilized to guide the search, thus accelerating the convergence. As shown in Equation (14), the modified DE/rand-to-best/1/bin is derived by replacing the second x t r1 in Equation (5) with a randomly selected individual x t r2 from the population:…”

“…Note that no other single algorithm can accomplish this [12]. More importantly, its search ability has been demonstrated in many real-world applications [13], [14], [15].…”

A numerical approach to optimal coherent quantum LQG controller design using gradient descent

“…A numerical procedure for finding suboptimal controllers for this problem was proposed in [21], and algebraic equations for the optimal CQLQG controller were obtained in [33]. We also mention that coherent quantum LQG control settings were considered in [15] for a class of quantum systems (with annihilation operators only), in the context of evolutionary optimization for entanglement control [7], and also for different scenarios of plant-controller coupling in [35]. Despite the previous results, the CQLQG control problem does not lend itself to an "elegant" solution (for example, in the form of decoupled Riccati equations as in the classical case [12]) and remains a subject of research.…”

Implicit Lyapunov‐based control strategy for closed quantum systems with dipole and polarizability coupling

“…In the past decades, there have been intensive applications in quantum control theory, such as quantum computation [1], nuclear magnetic resonance (NMR) [2], quantum chemistry [3], and quantum optics [4,5]. Increasing research activities have appeared in the field of quantum control, for example, controllability of quantum systems [6,7], coherent control [8,9], optimal control [10,11], Lyapunov control [12][13][14][15][16][17][18][19][20][21][22], sliding mode control [23,24], robust control [25], switching control [26], learning control [27], and closed-loop feedback control [28][29][30][31].…”