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.
Open quantum systems