Quantum Computing and Machine Learning Approaches to Quantum Many-Body Physics

Abstract

Lattice field theory provides a framework for which to explore properties of quantum field theories non-perturbatively. However for certain lattice calculations, for example when considering real-time dynamics or fermionic systems at finite density, sign problems occur which render those calculations intractable. One approach to solving the sign problem is to avoid it altogether by instead considering a simulation of the field theory on a quantum computer. For bosonic field theories, a procedure of qubitizing the bosonic fields is a necessary first step. The infinite-dimensional Hilbert space of the bosonic fields must be properly truncated as to encode those fields on a finite-dimensional Hilbert space spanned by the qubits on the quantum computer. This thesis first discusses various strategies of making such a truncation. Ideally, the truncation yields a discrete spin system that contains a critical point in the same universality class as the untruncated field theory. That way, the physics of the original field theory is reproduced in the continuum limit of the truncated theory without needing to take a second limit of removing the truncation. Simulations of different models arising from various truncation strategies of the (1+1)-dimensional O(3) nonlinear sigma model are performed and different qubitizations for SU(2) gauge fields are considered and proposed. Due to a lack of an efficient method for solving many-body systems in more than one dimension, numerical simulations of these SU(2) qubitizations are unavailable. The second half of the thesis explores the use of machine learning techniques in providing effective ways to solve quantum many-body problems. Neural network structures, such as feed-forward networks and restricted Boltzmann machines are universal approximators for continuous and discrete functions respectively. Therefore, they can be used as flexible wave function ansatze. Gradient descent algorithms can be applied to variationally search the general functional space spanned by neural-network-based ansatze for ground states of interacting, many-body systems. An ansatz is constructed explicitly for a system of indistinguishable bosons in one dimension and tested by comparing numerical results with analytic solutions of several exactly-solvable models. An extension of these neural-network ansatze to systems of identical bosons and fermions and discrete spin systems in higher dimensions would allow for concrete simulations of systems ranging from nuclei and qubitization models.