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.
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.
We consider the basic quantum-control task of obtaining a target unitary operation (i.e., a quantum gate) via control fields that couple to the quantum system and are chosen to best mitigate errors resulting from time-dependent noise, which frustrate this task. We allow for two sources of noise: fluctuations in the control fields and fluctuations arising from the environment. We address the issue of control-error mitigation by means of a formulation rooted in the Martin-Siggia-Rose (MSR) approach to noisy, classical statistical-mechanical systems. To do this, we express the noisy control problem in terms of a path integral, and integrate out the noise to arrive at an effective, noise-free description. We characterize the degree of success in error mitigation via a fidelity metric, which characterizes the proximity of the sought-after evolution to ones that are achievable in the presence of noise. Error mitigation is then best accomplished by applying the optimal control fields, i.e., those that maximize the fidelity subject to any constraints obeyed by the control fields. To make connection with MSR, we reformulate the fidelity in terms of a Schwinger-Keldysh (SK) path integral, with the added twist that the ``forward'' and ``backward'' branches of the time contour are inequivalent with respect to the noise. The present approach naturally and readily allows the incorporation of constraints on the control fields---a useful feature in practice, given that constraints feature in all real experiments. In addition to addressing the noise average of the fidelity, we consider its full probability distribution. The information content present in this distribution allows one to address more complex questions regarding error mitigation, including, in principle, questions of extreme value statistics, i.e., the likelihood and impact of rare instances of the fidelity and how to harness or cope with their influence. We illustrate this MSR-SK reformulation by considering a model system consisting of a single spin-s freedom (with s arbitrary), focusing on the case of 1/f noise in the weak-noise limit. We discover that optimal error mitigation is accomplished via a universal control field protocol that is valid for all s, from the qubit (i.e., s=1/2) case to the classical (i.e., $s\ensuremath{\rightarrow}\ensuremath{\infty}$) limit. In principle, this MSR-SK approach provides a transparent framework for addressing quantum control in the presence of noise for systems of arbitrary complexity.
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.
We study the relations between classical information and the feasibility of accurate manipulation of quantum system dynamics. We show that if an efficient classical representation of the dynamics exists, optimal control problems on many-body quantum systems can be solved efficiently with finite precision. In particular, one-dimensional slightly entangled dynamics can be efficiently controlled. We provide a bound for the minimal time necessary to perform the optimal process given the bandwidth of the control pulse, which is the continuous version of the Solovay-Kitaev theorem. Finally, we quantify how noise affects the presented results.
Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.
The precise, human control of quantum systems, by its definition, must combine models of the classical and the quantum world into a calculus that supports both. Open, irreversible quantum systems must interact with closed, reversible quantum systems to predict evolutions that are partially open and closed. Inevitably, the problems of quantum measurement, the assumptions of scattering, and the role of spacetime comes under scrutiny. Such considerations have extraordinary practical value: the precise control of a quantum information is the cornerstone of scalable quantum computing. Traditionally, quantum control theory as well as a formalism of redundancy and partial measurements, known as quantum error correction, attempt to remedy systematic quantum- noise and random quantum-noise respectively, but have had mixed success. This dissertation examines how the imprecision of control in quantum and classical spin systems affects the flow of select information to a receiver and how such systems may be optimized against the imprecise scattering of control fields and spins. To this end, this dissertation intertwines the physics of state evolution with the physics of information control in classical and quantum systems. First in classical systems, a method for encoding and decoding classical spin- processing information provides an example of information flow. Then an analytic calculation of a semi-conductor spin channel's information capacity is performed. The results limit the rate of information processing and inform the design of materials for optimal spintronic information-processing in semiconductors. Next, noisy quantum interactions are described, so that the complexities of correcting small, random phase errors using traditional control theory and quantum error correction may be explained. How these noisy processes affect the relevant information flow of a quantum algorithm (derivatives of the Quantum Fourier Transform and Grover Search) is considered, several novel methods of source-coding for these quantum channels are presented and their efficacy calculated. These methods include Unitary- Fault Tolerance, Clifford operations of locally-variant basis, and an entropic controller. Together they show classical systems in the steady state can be used to control scalable, high-precision quantum-computing machines, and ultimately may eliminate all temporal control from quantum operations.
For several decades it has been appreciated that quantum computers hold incredible promise to perform calculations intractable to classical computation. However, this promise has be slow to realize. Dozens of quantum systems are currently being investigated for use in quantum information processing - none of which have yet demonstrated algorithms involving more than a handful of qubits and it remains unclear which, if any, of these systems will ultimately compose a scalable, robust quantum information processing architecture. In this thesis we employ analytical, optimal and algebraic control techniques to evaluate various quantum systems for their potential use in quantum information processing. In doing so, we have additionally identified several novel characterization procedures capable of probing both the coherent and incoherent dynamics of quantum systems. The first part of this thesis discusses work motivated by attempts to utilize donor qubits in silicon as quantum bits. We first propose a measurement of the state of a single donor electron spin using two-dimensional electron gas of a field-effect transistor and electrically detected magnetic resonance. We analyze the potential sensitivity of this measurement and show that it is a quantum nondemolition measurement of an electron-encoded state. We then present the first of two novel qubit characterization procedures. We consider the problem of rapidly characterizing a large number of similarly prepared qubits using techniques from optimal experiment design. All qubits are assumed to evolve according to the same physical processes, though the Hamiltonian parameters may vary from device to device - an inevitability in solid state qubits. We use the Cram\\'er-Rao bound on the variance of a point estimator to construct the optimal series of experiments to estimate these free parameters, and present a complete analysis of the optimal experimental configuration. Though applied to dipole- and exchange-coupled qubits, this technique is widely applicable to other systems. The second part of the thesis discusses the role that control can play in measuring and mitigating noise in qubit systems. Our first result describes a method for quickly simulating the effects of arbitrary markovian noise on qubit systems through the use of a numerically optimized, multi-state Markovian fluctuator. This ability to rapidly simulate the noisy qubit evolution allows us to compute control sequences capable of maximally decoupling the qubit from the noise source. We then introduce the second characterization procedure of the these, showing that a single measurable and controllable qubit may act as a spectrometer of dephasing noise. We show that the formalism of dynamical decoupling can be used to estimate the short-time correlation function of the noise source, while long time correlations may be estimated by a very simple series of free evolution experiments. This technique is applicable to the wide range of physical implementations which suffer from dephasing noise.The final part of this thesis demonstrates that trapped neutral atoms may be utilized for the robust simulation of complex systems exhibiting a topological phase. We present a method to simulate the toric code Hamiltonian stroboscopically, and demonstrate that our technique preserves the ground state degeneracy . Furthermore, we introduce a dissipative mechanism allowing for thermalization of the system to a finite temperature or direct cooling to the ground state manifold.