  Monte Carlo calculations in the framework of lattice field theory provide
non-perturbative access to the equilibrium physics of quantum fields. When
applied to certain fermionic systems, or to the calculation of
out-of-equilibrium physics, these methods encounter the so-called sign problem,
and computational resource requirements become impractically large. These
difficulties prevent the calculation from first principles of the equation of
state of quantum chromodynamics, as well as the computation of transport
coefficients in quantum field theories, among other things.
  This thesis details two methods for mitigating or avoiding the sign problem.
First, via the complexification of the field variables and the application of
Cauchy's integral theorem, the difficulty of the sign problem can be changed.
This requires searching for a suitable contour of integration. Several methods
of finding such a contour are discussed, as well as the procedure for
integrating on it. Two notable examples are highlighted: in one case, a contour
exists which entirely removes the sign problem, and in another, there is
provably no contour available to improve the sign problem by more than a
(parametrically) small amount.
  As an alternative, physical simulations can be performed with the aid of a
quantum computer. The formal elements underlying a quantum computation - that
is, a Hilbert space, unitary operators acting on it, and Hermitian observables
to be measured - can be matched to those of a quantum field theory. In this way
an error-corrected quantum computer may be made to serve as a well controlled
laboratory. Precise algorithms for this task are presented, specifically in the
context of quantum chromodynamics.
