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.
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.
Bayesian inference is a widely used technique for real-time characterization of quantum systems. It excels in experimental characterization in the low data regime, and when the measurements have degrees of freedom. A decisive factor for its performance is the numerical representation of the Bayesian probability distributions. In this work, we explore advanced statistical methods for this purpose, and numerically analyze their performance against the state-of-the-art in quantum parameter learning. In particular, we consider sequential importance resampling, tempered likelihood estimation, Markov Chain Monte Carlo, random walk Metropolis (RWM), Hamiltonian Monte Carlo (HMC) and variants (stochastic gradients with and without friction, energy conserving subsampling), block pseudo-marginal Metropolis-Hastings with subsampling, hybrid HMC-RWM approaches, and Gaussian rejection filtering. We demonstrate advantages of these approaches over existing ones, namely robustness under multi-modality and high dimensionality. We apply these algorithms to the calibration of superconducting qubits from IBMQ, surpassing the standard quantum limit and achieving better results than Qiskit's default tools. In Hahn echo and Ramsey experiments, we reduce the uncertainty by factors of 10 and 3 respectively, without increasing the number of measurements; conversely, we match the performance of Qiskit's methods while using up to to 99.5% less experimental data. We additionally investigate the roles of adaptivity, dataset ordering and heuristics in quantum characterization. Our findings have applications in challenging quantum characterization tasks, namely learning the dynamics of open quantum systems.
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.
In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer–Rao lower bound, certifying its own performance.
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.
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a k-body Hamiltonian H whose correlation matrix has a spectral gap Δ can be estimated to precision ε with only O~(n3k/(εΔ)3/2) measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of H starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of m Pauli operators with weight ≤k to precision ε using only O(ε−23klogm) measurements, which quadratically improves a recent result by Cotler and Wilczek.