Scientific Machine Learning for the Automated Discovery of Quantum Control Schemes and Phase Diagrams

Scientific machine learning denotes the integration of machine learning into traditional scientific methods and has become a powerful tool in recent years. This thesis establishes innovative machine-learning methods that combine knowledge from physics, numerical analysis, and computer science for the automated discovery of quantum control schemes and phase diagrams.
Conceptually, it is straightforward to determine the time evolution of a quantum system for a fixed initial state given its (time-dependent) Hamiltonian or Lindbladian. Depending on the physical context, we will describe the dynamics by an ordinary or stochastic differential equation. Controlling the (stochastic) dynamics of a quantum system requires solving the inverse problem and is indispensable in fields such as quantum metrology and information processing. However, solving the control problem by deriving a performant control scheme from scratch is generally hard. In particular, it is desirable to develop feedback controllers that can react to fluctuations in the system, making them extremely powerful control systems. Up to now, there has been no general ready-made approach for designing efficient control strategies because existing approaches based on black-box reinforcement learning are difficult to optimize. In the first part of this thesis, we propose an automated control-scheme design based on the differentiable programming paradigm, which allows us to exploit prior knowledge about the structure of the physical system. Specifically, we employ a controller in the form of a neural network that selects the control drive to be applied in each timestep based on the current quantum state or the observed measurement record. The neural network parameters are optimized in a series of epochs based on gradient information computed by (adjoint) sensitivity methods. We demonstrate our method in various scenarios, such as state preparation and stabilization of a qubit subjected to homodyne detection. The homodyne-detection signal contains only minimal information on the actual state of the system, masked by unavoidable photon-number fluctuations.
In the second part, we develop two data-driven methods to automatically identify phase boundaries in physical systems. The first method is based on training a predictive model such as a neural network to infer the parameters of a physical system from its state. The deviation of the inferred parameters from the correct underlying parameters will be most susceptible and point in opposite directions in the vicinity of phase boundaries. Therefore, peaks in the vector-field divergence of the model's predictions reveal phase transitions. This prediction-based method is applicable to phase diagrams of arbitrary parameter dimensions without prior information about the phases. We apply the method to the two-dimensional Ising model, Wegner's Ising gauge theory, a generalized toric code, the Falicov-Kimball model, and the dissipative Kuramoto-Hopf model.
As a second method, we introduce a physically motivated, computationally favorable, and interpretable approach based on an (appropriate) choice of input features. The method relies on the difference between mean input features as an indicator for phase transitions and does not utilize predictive models. Crucially, this mean-based method provides direct physical insights into the revealed phase diagram without prior labeling or knowledge of its phases. As an example, we consider the physically rich ground-state phase diagram of the Falicov-Kimball model. Note that the large number of phases in this model renders the analysis of this phase diagram by standard methods a complex and tedious task. In particular, supervised machine-learning methods are bound to fail because phase labels are not known in advance.