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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.