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arXiv:0801.0746

Comparative Analysis of Control Strategies

Sonia G. Schirmer, Peter J. Pemberton-Ross, Xiaoting Wang

Published Jan 4, 2008• 2 citations
arXivHTMLPDF

Abstract

Different ways of modelling quantum control systems, formulating control problems and solving the resulting problems are considered and compared. In particular, we compare the performance of geometric and optimal control, as well as iterative techniques for optimal control design versus local gradient optimization using a Lyapunov-type potential function for two problems of general interest: global control of qubits and entanglement generation in the form of Bell state preparation.

DOI
10.48550/arxiv.0801.0746

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Different ways of modelling quantum control systems, formulating control problems and solving the resulting problems are considered and compared. In particular, we compare the performance of geometric and optimal control, as well as iterative techniques for optimal control design versus local gradient optimization using a Lyapunov-type potential function for two problems of general interest: global control of qubits and entanglement generation in the form of Bell state preparation.
Some of the limitations of popular optimal control algorithms such as Lyapunov control, the Krotov method and other iterative algorithms based on functional gradients are considered. Lyapunov control, while elegant and generally effective for ideal systems, struggles for realistic systems. Iterative techniques such as the Krotov method tend to be more effective, but the inclusion of an energy penalty term implies that the resulting solutions are not critical points of the objective functional alone, and do not maximize it. These problems can be avoided by using functional gradients without penalty terms, but even these techniques fail for some problems.
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.
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.
This dissertation is a study of several aspects of discrete time quantum control. After reviewing the necessary background from quantum mechanics and control theory we construct a simple model and solve a control problem on it as a proof of concept for the discrete control formulation developed herein. We apply optimal control to solve a state preparation problem on a physically motivated time discretized single qubit system with simplified dynamics to enable the Bellman equation to be solved both numerically and approximately analytically. However for more than one qubit classical theories of control become problematic, a notable difficulty being the exponential increase in the size of the state space with the number of particles. To study workarounds for this problem we quantify how much information a quantum measurement can extract from the system and then use the results to show that a naive classical technique for controlling a multiparticle system performs poorly when the system has "quantum" properties. We then consider the problem of protecting the entanglement of a bipartite quantum system using discrete open loop control with a technique that seems to have no classical analog. The states of the bipartite system, called X-states, have a special structure which is generalized to states of systems with more than two qubits. We develop the properties of these generalized X-states which are a class of mixed states with interesting multipartite entanglement properties
Globally controlled superconducting qubit systems are regaining popularity due to their potential to solve one of the problems that most threatens the scalability of these platforms: the wiring problem. In fact, current quantum computers based on this technology rely on a local approach to qubit control, which, as the number of qubits increases, results in a cable density that interferes with the processor’s proper functioning. Adopting a global control approach makes it possible to reduce the number of cables and thus improve scalability. Since globally controlled architectures capable of universal quantum computation have already been proposed, the natural next step in the study of these platforms is the application of optimal control techniques. Indeed, given the global nature of the controls, optimization takes place across the entire system simultaneously, offering possibilities that would otherwise be unattainable with local control. The main objective of this work is to investigate these possible applications, focusing primarily on three aspects: digital quantum computation, quantum simulation, and the scalability of optimized pulses.
This dissertation is a study of several aspects of discrete time quantum control. After reviewing the necessary background from quantum mechanics and control theory we construct a simple model and solve a control problem on it as a proof of concept for the discrete control formulation developed herein. We apply optimal control to solve a state preparation problem on a physically motivated time discretized single qubit system with simplified dynamics to enable the Bellman equation to be solved both numerically and approximately analytically. However for more than one qubit classical theories of control become problematic, a notable difficulty being the exponential increase in the size of the state space with the number of particles. To study workarounds for this problem we quantify how much information a quantum measurement can extract from the system and then use the results to show that a naive classical technique for controlling a multiparticle system performs poorly when the system has quantum properties. We then consider the problem of protecting the entanglement of a bipartite quantum system using discrete open loop control with a technique that seems to have no classical analog. The states of the bipartite system, called X-states, have a special structure which is generalized to states of systems with more than two qubits. We develop the properties of these generalized X-states which are a class of mixed states with interesting multipartite entanglement properties