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Optimal quantum engineering

Zoller, Jonathan

Published Jan 1, 2018• 1 citation
Publisher

Abstract

Optimal control of complex quantum systems in theory and experiment is the focus of this thesis. The theoretical aspect of our work was concerned with the development of algorithms that are capable of precisely tailoring the shapes of control pulses to meet various control goals while respecting all physical constraints, both fundamental and problem specific. Having these effective tools at hand, we computed control inputs for a range of physical systems using simulation-based time propagation. We extended the dressed chopped random basis (dCRAB) optimization method, previously developed in our group, such that boundary conditions and other constraints that arise from experimental setups of quantum systems can be regarded. Therefore, we wrote a software suite that enables experimentalists to improve their control pulses by directly interfacing the lab with our control algorithm.

DOI
10.18725/oparu-10199

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Optimal control of complex quantum systems in theory and experiment is the focus of this thesis. The theoretical aspect of our work was concerned with the development of algorithms that are capable of precisely tailoring the shapes of control pulses to meet various control goals while respecting all physical constraints, both fundamental and problem specific. Having these effective tools at hand, we computed control inputs for a range of physical systems using simulation-based time propagation. We extended the dressed chopped random basis (dCRAB) optimization method, previously developed in our group, such that boundary conditions and other constraints that arise from experimental setups of quantum systems can be regarded. Therefore, we wrote a software suite that enables experimentalists to improve their control pulses by directly interfacing the lab with our control algorithm.
This thesis details work performed on the improvement of experimental control of quantum systems through the use of optimal control theory. We pursue this objective through both conceptual and practical routes. We first devise improved methods of optimal control for quantum systems. We then apply them, as well as more established methods, to experimental situations where robustness to inaccuracy and error is sought. We develop a method for improving the computational efficiency of optimal control theory in the context of quantum control. This approach makes use of the iterative nature of optimisation algorithms, by using perturbation theory to minimise the effort needed to calculate the time propagator associated with a controlled Hamiltonian. We also create a new way of defining a figure of merit when optimising for robustness against error in an experiment. This method specifically applies to errors that have a relatively small effect on the controlled system's Hamiltonian. We compare this with an ensemble method based around randomly sampling several points on the control landscape, and find these methods to be largely equivalent for small errors, with the ensemble method being superior for larger error. On the applied side, we design radio frequency magnetic pulses that create robustness in spin ensembles to inhomogeneous broadening and variations of coupling strength to the control field for two applications. In the first, we create reliable gates in a situation where limitations are imposed on the control that can be applied. In the second application, we design controls for radio frequency magnetic pulses to create a superradiant state in a spin ensemble. Finally, we collaborate with the experiment to measure the electrons electric dipole moment that is being performed at Imperial College London. For this collaboration, we shape the envelope of the radio frequency control pulses used in the experiment to create superpositions of Zeeman and Stark-shifted excited states in YbF molecules. This shaping allows us to impart pre-determined phases to these superpositions, in a manner that is robust to errors in the control pulse envelope. We obtain sets of robust and sensitive pulses that have been sent to the experiment for testing purposes.
The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse-shaping experiments allow us to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analysed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets.
Accurate manipulations of an open quantum system require a deep knowledge of its controllability properties and the information content of the implemented control fields. By using tools of information and quantum optimal control theory, we provide analytical bounds (information-time bounds) to characterize our capability to control the system when subject to arbitrary sources of noise. Moreover, since the presence of an external noise field induces open quantum system dynamics, we also show that the results provided by the information-time bounds are in very good agreement with the Kofman-Kurizki universal formula describing decoherence processes. Finally, we numerically test the scaling of the control accuracy as a function of the noise parameters, by means of the dressed chopped random basis (dCRAB) algorithm for quantum optimal control.
This thesis is concerned with the control of quantum systems.Given a Hamiltonian model of a quantum system, we are interested in finding controls-typically shaped electromagnetic pulses-that steer the evolution of the system toward a desired target operation.For this we employ a numerical optimisation method known as the GRAPE algorithm.For particular experimental systems, we design control schemes that respect constraints of robustness and addressability, and are within the reach of the experimental hardware.A general procedure is given for specifying a Hamiltonian model of a driven N -level system and converting it to an appropriate rotating frame.This is then applied together with the numerical algorithm to design improved schemes for two different systems, where laser fields manipulate orbital and hyperfine states of Pr 3+ and Rb.The generation of cluster states in Ising-coupled systems is also studied.We find that, in the ideal case, the solution of evolving only under the coupling Hamiltonian is not time-optimal.This surprising result is in contrast to the known cases for unitary gates.For a symmetrised three-qubit example, we provide a geometrical interpretation of this.Numerically optimised control schemes are then developed for a nonideal coupling topology, modelling an experimental configuration of trapped ions.Controls for the implementation of the two-qubit Deutsch and Grover algorithms are designed for a pair of 13 C nuclear spins at a nitrogen vacancy center in diamond.These implementations are robust to experimental errors, and found to be reproduced with high accuracy on a VFG-150 pulse generator.We also consider two-qubit gate synthesis in a system of superconducting qubits coupled by mii xii CONTENTS C.4 Examples of time-optimal gates . . . . . . . .
Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.
Applying sophisticated optimization methods to control problems in various systems offers a structured approach for achieving system efficiency and reliability, while ensuring the feasibility of physics-based constraints. This dissertation primarily focuses on developing optimization algorithms for control problems in complex systems with mixed-integer variables and parameter uncertainties. In Chapter 1, we provide a detailed introduction about the problem motivation and summarize our problems and contributions. Specifically, we develop new optimization models and algorithms for network-based traffic control and binary control in quantum systems, detailed as follows. Traffic signal control is an effective way of mitigating traffic congestion, where its complexity escalates in large cities due to numerous intersections and varying traffic conditions. Advancements in communication technologies within transportation infrastructures have made distributed traffic signal control at the network level possible and worth investigating. In Chapter 2, we build a stochastic optimization model for network-level traffic signal control under traffic demand uncertainty and solve it using decentralized algorithms. We compare the results with state-of-the-art traffic control methods via testing instances of real-world traffic networks and data. Quantum computing and quantum systems provide a novel way of significantly accelerating computation and their operations largely rely on quantum pulse control optimization to attain desired states or status. The optimization of binary quantum pulse control can potentially enhance the performance of classical quantum variational techniques and improve solution quality. In Chapter 3, we study a discrete-valued binary quantum control problem and propose an algorithmic framework to solve the problem with penalty on switches. In Chapter 4, we explore the binary quantum control problem in a continuous time horizon by optimizing both control functions and control time intervals. Moreover, the time-varying noise in quantum systems and the wide use of inhomogeneous quantum ensembles highlight the need for quantum controls considering uncertainties. In Chapter 5, we develop a stochastic optimization model for the binary quantum control problem with uncertain Hamiltonian controllers and solve it using gradient-based methods with rounding algorithms. We test the performance of our approaches and benchmark with state-of-the-art control methods in all three chapters. Lastly, in Chapter 6, we conclude the dissertation and state future research directions.