Modeling Dynamics of Classical and Quantum Systems Using Machine Learning Techniques

Abstract

In this thesis, we examine the dynamics of many-body systems within both quantum and classical contexts. When dealing with many-body systems, the focus is typically on what are known as order parameters. These parameters indicate whether the physical system is in an ordered phase or not. Order parameters often correspond to the expectation values of local observables, or to the averages of the microscopic degrees of freedom within the system’s configuration. Specifically, we leverage machine learning methodologies to develop an interpretable and physically consistent theoretical framework for describing local and macroscopic degrees of freedom within both quantum and classical contexts. In the case of quantum system, we develop a method that is able to approximate a “dynamical generator,” that is, an operator which is able to evolve in time few body observables of the system. In particular, this generator preserves some important physical quantities of the state representing the system under investigation, such as its probabilistic interpretation. We explore the capabilities of this method in our first and second scientific paper. In the first work, we provide full access to the dynamics through the coherence vector, a representation of the quantum state. In our second work, more akin to actual experimental conditions, we supply the variational method with data obtained via state tomography through projective measurements, a routine approach in retrieving the state of quantum devices using qubits. Here, we further develop the methods used in our fist work to address the problem of learning the dynamical generator in the presence of projection noise. Given that projective measurements introduce noise into the dynamics, we naturally consider how to represent physical noise in a machine learning routine. To better understand how noise could be managed by a machine learning routine, we studied the reduced degrees of freedom of classical systems composed of many ”binary” spin degrees of freedom whose evolution is probabilistic in our third scientific paper. Average quantities of these spin variables are inherently stochastic. By taking the expectation values over numerous stochastic trajectories, order parameters emerge. Specifically, we model the dynamics of this order parameter using a stochastic differential equation of It ˆ o type. Such equations are first-order differential equations in time that include a directed force term, known as the “drift coefficient,” and a noisy force term called the “diffusion coefficient.” We con centrate on encoding the diffusion term and the drift term of this stochastic differential equation into two separate neural networks. This method proves robust in reproducing the dynamics and allows us to extract valuable information about the system, such as the critical point and the critical exponent.