Yeh, Hsiu-Chung2022-09-132022-09-132022-06https://hdl.handle.net/11299/241631University of Minnesota Ph.D. dissertation. June 2022. Major: Physics. Advisor: Alex Kamenev. 1 computer file (PDF); vii, 94 pages.We investigate quantum fluctuation driven rare events by studying emptiness formation probability (EFP) and quantum optimization in this dissertation. EFP is the probability that no particle resides in a macroscopic interval of a given size $2R$. We employ instanton technique to numerically study weakly interacting Bose liquid, where EFP exhibits a non-trivial crossover from the Poisson to Gaussian behavior in wide range $\rho_0^{-1} < R < \xi$ (here $\rho_0$ is the average density and $\xi$ is the correlation length). We also consider interacting quantum liquids with polytropic equation of state $P(\rho) \sim \rho^\gamma$, where $\rho$ is density and $P$ is pressure. Analytic instanton solutions are found for a certain infinite sequence of rational polytopic indexes $\gamma$ and EFP can be analytically continued to any value of $\gamma \geq 1$. All our findings agree with previous results and moreover explore unstudied cases. In the study of quantum optimization, we suggest an iterative quantum protocol, allowing to approximately solve optimization problems with a glassy energy landscape. It is based on a periodic cycling around the tricritical point of the many-body localization (MBL) transition. This ensures that each iteration leads to a non-exponentially small probability to find a lower local energy minimum. The other key ingredient is to tailor the cycle parameters to a currently achieved optimal state (the ``reference'' state) and to reset them once a deeper minimum is found. We show that the required time scales algebraically with the system size and the required optimization precision. The corresponding exponents are related to critical indices of the MBL transition. It is likely that the latter provide bounds on the performance of any approximate optimization algorithms.enThesis or Dissertation