  In lattice quantum field theory studies, parameters defining the lattice
theory must be tuned toward criticality to access continuum physics. Commonly
used Markov chain Monte Carlo (MCMC) methods suffer from critical slowing down
in this limit, restricting the precision of continuum extrapolations. Further
difficulties arise when measuring correlation functions of operators widely
separated in spacetime: for most correlation functions, an exponentially severe
signal-to-noise problem is encountered as the operators are taken to be widely
separated. This dissertation details two new techniques to address these
issues. First, we define a novel MCMC algorithm based on generative flow-based
models. Such models utilize machine learning methods to describe efficient
approximate samplers for distributions of interest. Independently drawn
flow-based samples are then used as proposals in an asymptotically exact
Metropolis-Hastings Markov chain. We address incorporating symmetries of
interest, including translational and gauge symmetries. We secondly introduce
an approach to "deform" Monte Carlo estimators based on contour deformations
applied to the domain of the path integral. The deformed estimators associated
with an observable give equivalent unbiased measurements of that observable,
but generically have different variances. We define families of deformed
manifolds for lattice gauge theories and introduce methods to efficiently
optimize the choice of manifold (the "observifold"), minimizing the deformed
observable variance. Finally, we demonstrate that flow-based MCMC can mitigate
critical slowing down and observifolds can exponentially reduce variance in
proof-of-principle applications to scalar $\phi^4$ theory and $\mathrm{U}(1)$
and $\mathrm{SU}(N)$ lattice gauge theories.
