We explore methodologies for characterizing and controlling small quantum systems. We are interested in starting with a description of a quantum system, designing estimators for parameters of the system, developing robust and high-fidelity gates for the system using knowledge of these parameters, and experimentally verifying the performance of these gates. A strong emphasis is placed on using rigorous statistical methods, especially Bayesian ones, to analyze quantum system data. Throughout this thesis, the Nitrogen Vacancy system is used as an experimental testbed. Characterization of system parameters is done using quantum Hamiltonian learning, where we explore the use of adaptive experiment design to speed up learning rates. Gates for the full three-level system are designed with numerical optimal control methods that take into account imperfections of the control hardware. Gate quality is assessed using randomized benchmarking protocols, including standard randomized benchmarking, unitarity benchmarking, and leakage/loss benchmarking.
We explore methodologies for characterizing and controlling small quantum systems. We are interested in starting with a description of a quantum system, designing estimators for parameters of the system, developing robust and high-fidelity gates for the system using knowledge of these parameters, and experimentally verifying the performance of these gates. A strong emphasis is placed on using rigorous statistical methods, especially Bayesian ones, to analyze quantum system data. Throughout this thesis, the Nitrogen Vacancy system is used as an experimental testbed. Characterization of system parameters is done using quantum Hamiltonian learning, where we explore the use of adaptive experiment design to speed up learning rates. Gates for the full three-level system are designed with numerical optimal control methods that take into account imperfections of the control hardware. Gate quality is assessed using randomized benchmarking protocols, including standard randomized benchmarking, unitarity benchmarking, and leakage/loss benchmarking.
This thesis explores adaptive inference as a tool to characterize quantum systems using experimental data, with applications in sensing, calibration, control, and metrology. I propose and test algorithms for learning Hamiltonian and Kraus operators using Bayesian experimental design and advanced Monte Carlo techniques, including Sequential and Hamiltonian Monte Carlo. Application to the characterization of quantum devices from IBMQ shows a robust performance, surpassing the built-in characterization functions of Qiskit for the same number of measurements. Introductions to Bayesian statistics, experimental design, and numerical integration are provided, as well as an overview of existing literature.
The characterization of noisy quantum circuits is an important step in the development of large-scale quantum computers. As experimental quantum architectures approach the threshold for fault-tolerant quantum computing, obtaining a benchmark of process fidelities is needed to verify the integrity of the system and ensure functionality. Additionally, the ability to successfully implement a quantum code of choice may depend on the type of noise that is present in the system. It is therefore necessary to characterize not only the fidelity of the process, but also the structure of the noise. \n \nRandomized benchmarking is a characterization protocol which extracts partial information from a noisy quantum process. This thesis gives a detailed review of the randomized benchmarking protocol, and presents the results of three collaborative projects which use techniques similar to randomized benchmarking to obtain previously hidden information. The first protocol is designed to obtain a benchmark on loss errors. This protocol also provides information on the detector efficiency and an additional constraint on estimated parameters from randomized benchmarking. The second protocol is an extension of the first, under the special case of coherent leakage errors, for which we obtain an estimate of the leakage rate. The third protocol provides a method of testing for spatial correlations in noisy processes. This test may be used to determine the correlation structure of errors, and the error correcting codes needed for successful experiments in a particular architecture. All protocols presented in this thesis are scalable, platform-independent and insensitive to state preparation and measurement errors. We also provide results of numerical tests of these protocols for simulated noise, which show their accuracy and robustness.
Quantum information processing offers potential improvements to a wide range \nof computing endevaors, including cryptography, chemistry simulations and \nmachine learning. The development of practical quantum information processing \ndevices is impeded, however, by challenges arising from the apparent exponential \ndimension of the space one must consider in characterizing quantum \nsystems, verifying their correct operation, and in designing useful control \nsequences. In this work, we address each in turn by providing useful \nalgorithms that can be readily applied in experimental practice. \n \nIn order to characterize the dynamics of quantum systems, we apply statistical \nmethods based on Bayes' rule, thus enabling the use of strong prior \ninformation and parameter reduction. We first discuss an \nanalytically-tractable special case, and then employ a numerical algorithm, \nsequential Monte Carlo, that uses simulation as a resource for characterization. We \ndiscuss several examples of SMC and show its application in nitrogen vacancy \ncenters and neutron interferometry. \nWe then discuss how characterization techniques such as SMC can be used to \nverify quantum systems by using credible region estimation, model selection, \nstate-space modeling and hyperparameterization. Together, these techniques \nallow us to reason about the validity of assumptions used in analyzing quantum \ndevices, and to bound the credible range of quantum dynamics. \n \nNext, we discuss the use of optimal control theory to design robust control \nfor quantum systems. We show extensions to existing OCT algorithms that allow \nfor including models of classical electronics as well as quantum dynamics, \nenabling higher-fidelity control to be designed for cutting-edge experimental \ndevices. Moreover, we show how control can be implemented in parallel across \nnode-based architectures, providing a valuable tool for implementing \nproposed fault-tolerant protocols. \n \nWe close by showing how these algorithms can be augmented using quantum \nsimulation resources to enable addressing characterization and control design \nchallenges in even large quantum devices. In particular, we will introduce a \nnovel genetic algorithm for quantum control design, MOQCA, that utilizes \nquantum coprocessors to design robust control sequences. Importantly, MOQCA is \nalso memetic, in that improvement is performed between genetic steps. We then \nextend sequential Monte Carlo with quantum simulation resources to enable \ncharacterizing and verifying the dynamics of large quantum devices. By using \nnovel insights in epistemic information locality, we are able to learn \ndynamics using strictly smaller simulators, leading to an algorithm we call \nquantum bootstrapping. We demonstrate by using a numerical example of learning \nthe dynamics of a 50-qubit device using an 8-qubit simulator.
Robust control of a quantum system is essential to utilize the current noisy quantum hardware to its full potential, such as quantum algorithms. To achieve such a goal, a systematic search for an optimal control for any given experiment is essential. The design of optimal control pulses requires accurate numerical models and, therefore, accurate characterization of the system parameters. We present an online Bayesian approach for quantum characterization of qutrit systems, which automatically and systematically identifies optimal experiments that provide maximum information on the system parameters, thereby greatly reducing the number of experiments that need to be performed on the quantum testbed. Unlike most characterization protocols that provide point-estimates of the parameters, the proposed approach is able to estimate their probability distribution. The applicability of the Bayesian experimental design technique was demonstrated on test problems, where each experiment was defined by a parameterized control pulse. In addition to this, we also present an approach for iterative pulse extension, which is robust under uncertainties in transition frequencies and coherence times, and shot noise, despite being initialized with wide uninformative priors. Furthermore, we provide a mathematical proof of the theoretical identifiability of the model parameters and present conditions on the quantum state under which the parameters are identifiable. The proof and conditions for identifiability are presented for both closed and open quantum systems using the Schrödinger equation and the Lindblad master equation, respectively.
Quantum computers promise to be a revolutionary new technology. However, in order to realise this promise many hurdles must first be overcome. In this thesis we investigate two such hurdles: the presence of noise in quantum computers and limitations on the connectivity and control in large scale quantum computing architectures.In order to combat noise in quantum devices we must first characterize this noise. To do this several diagnostic tools have been developed over the last two decades. The current industry standard for such a diagnostic tool is called randomized benchmarking. Randomized benchmarking doesn't give a full characterization of the noise afflicting the quantum device but rather attempts to give some indication of of the device's average behavior, captured in a quantity called the average fidelity. Because it does not endeavor to characterize every small detail of the noise it can be efficiently applied even to very large quantum devices. However, with this power also comes increased complexity. Randomized benchmarking has a lot of moving parts, and some fairly strong assumptions must be made in order to guarantee its correctness. In this thesis we attempt to justify these assumptions and if possible remove or weaken them, making randomized benchmarking a more robust and general tool. In chapter 6 of this thesis we investigate the finite statistics of randomized benchmarking. We prove strong bounds on the number of samples needed to perform rigorous randomized benchmarking. To do this we make use tools from representation theory. In particular we use a characterization of certain representations of the Clifford group, which we develop in chapter 5. In chapter 7 we re-use these tools to also bound the number of samples needed to perform rigorous unitarity randomized benchmarking, a newer variant of randomized benchmarking quickly gaining in popularity. These results retroactively justify the use of randomized benchmarking in an experimental setting and also provide guidance on optimal statistical practices in the context of randomized benchmarking. In chapter 8 we expand upon the standard randomized benchmarking protocol and formulate a new class of protocols which we call character randomized benchmarking. This new class of protocols removes a critical assumption made in standard randomized benchmarking, making character randomized benchmarking vastly more generally applicable. To show the advantages of character randomized benchmarking we implement it in an experiment characterizing the noise in a Si/SiGe quantum dot device. This experiment is detailed in chapter 9. Finally we deal with the second main topic of this thesis in chapter 10. Large scale quantum computer will, like classical computers, face limitations in the connectivity between different parts of the computer. This is due to a fundamental law in computer design called Rent's rule, which states that the number of wires connecting a (quantum) computer chip to the outside world is much smaller than the number of components in that chip. This means the individual components of the chip can not be controlled individually in parallel. Given that parallelism is absolutely critical for the functioning of quantum computers this is a serious problem for the development of large scale quantum computers. Luckily it is possible to organize quantum computing devices in such a way that they can be controlled using a relatively small amount of input wires. One example of such an organization is called a crossbar architecture. Recently a proposal was made for a crossbar architecture quantum computer in quantum dots, and in chapter 10 of this thesis we investigate in detail the advantages and disadvantages of such an architecture. We focus in particular on its effect on standard quantum error correction procedures, a key part of a functioning quantum computer, and one where parallel control of all parts of the quantum device is essential.
We are currently approaching the point where quantum systems with 15 or more qubits will be controllable with high levels of coherence over long timescales. One of the fundamental problems that has been identified is that, as the number of qubits increases to these levels, there is currently no clear way to use efficiently the information that can be obtained from such a system to make diagnostic inferences and to enable improvements in the underlying quantum gates. Even with systems of only a few bits the exponential scaling in resources required by techniques such as quantum tomography or gate-set tomography will render these techniques impractical. Randomized benchmarking (RB) is a technique that will scale in a practical way with these increased system sizes. Although RB provides only a partial characterization of the quantum system, recent advances in the protocol and the interpretation of the results of such experiments confirm the information obtained as helpful in improving the control and verification of such processes. This thesis examines and extends the techniques of RB including practical analysis of systems affected by low frequency noise, extending techniques to allow the anisotropy of noise to be isolated, and showing how additional gates required for universal computation can be added to the protocol and thus benchmarked. Finally, it begins to explore the use of machine learning to aid in the ability to characterize, verify and validate noise in such systems, demonstrating by way of example how machine learning can be used to explore the edge between quantum non-locality and realism.