Experimental searches for Beyond the Standard Model physics with neutrinoless double-beta decay, nuclear recoil from dark matter, proton decay, and other hadronic processes require accurate theoretical modeling of low-energy hadronic dynamics. Lattice QCD, which relies on Monte Carlo simulation of QCD path integrals, can reliably calculate hadronic matrix elements from first principles. I’ll disucss recent applications of lattice QCD to searches for neutrinoless double-beta decay and neutron-antineutron oscillations as well as lattice QCD simulations of proton-proton fusion relevant to nuclear astrophysics. I’ll also discuss how Monte Carlo simulations of quantum field theories provide statistical models of probability distributions for quantum fields. Quantum fluctuations imprint structure into these distributions besides just the mean values of observables. My recent work suggests that properties of quantum field probability distributions can be usefully exploited in at least single-proton systems to avoid the signal-to-noise problem obstructing lattice QCD simulations of large nuclei.