UC Santa Barbara

Given a quantum system that we seek to understand, finding the ground state is often the first and most informative task we can assume, from which we can then examine dynamics, excitations, entanglement geometry, and so on. But there are several different precise senses in which we could ask to "find the ground state", and depending on the exact system and the question, the resulting task could be quite easy or difficult. This thesis examines four distinct problems, drawing from a toolkit of Bayesian statistics, the Density Matrix Renormalization Group, convex optimization, Fourier analysis, and computational complexity theory.

In one setting, we are given a set of local interactions and asked only, is this Hamiltonian frustrated? This question can be very easy or difficult depending on the types of interactions permitted by the symmetry of the system. We show that in fact there are in fact "natural" (in a precise, mathematical sense) interaction types of many different difficulties, including complexity classes BQP and QCMA. In the second setting, we have repeatedly measured an unknown quantum state, and we are tasked with determining the most probable state given the measurements. We show that this task is in fact exponentially difficult (NP-hard) in the dimension of the Hilbert space. In the third setting, we examine one-dimensional fermionic systems, and show how Gaussian Fermionic Matrix Product States and DMRG can be combined with Hartree-Fock iteration to find approximate groud states very quickly. In the final setting, we use a quantum computer to perform binary measurements of Green's functions and wish to reconstruct the whole function. We show that although classical statistical techniques give an acceptable reconstruction, imposing physicality constraints greatly enhances the sample efficiency and reconstruction quality.