  Double beta decays are rare nuclear processes that can occur in two modes:
two-neutrino double beta decay, observed in the Standard Model, and
neutrinoless double beta decay, a hypothetical process with profound
implications for Particle Physics. To draw reliable conclusions from their
experimental constraints, it is necessary to have accurate predictions of the
underlying hadronic interactions described by quantum chromodynamics (QCD), a
non-Abelian gauge theory with the symmetry group SU(3). QCD predictions require
non-perturbative methods for calculating observables, and lattice QCD (LQCD), a
numerical method based on QCD formulated on a finite space-time grid, is the
only reliable first-principles technique for obtaining quantitative results.
However, LQCD needs formal prescriptions to match numerical results with
observables. This thesis provides such prescriptions for double beta decays
using the finite volume effects in the LQCD framework. Matching relations that
connect two-nucleon double beta decay amplitudes to quantities accessible via
LQCD calculations, namely the nuclear matrix elements and two-nucleon energy
spectra in a finite volume are provided. The impact of uncertainties is
examined on the precision with which low-energy constants of the corresponding
effective field theories can be determined from future LQCD calculations.
  Hamiltonian simulation of QCD is another non-perturbative method of solving
QCD which can be more suitable in some cases than the conventional LQCD. The
rise of tensor network methods and quantum simulation has made Hamiltonian
simulation of lattice gauge theories (LGTs) a reality. Towards the goal of
simulating QCD, a loop-string-hadron (LSH) formulation of an SU(3) LGT with
matter in 1+1 dimensions is developed in this thesis, motivated by recent
studies that showed the LSH formulation of an SU(2) LGT to be advantageous over
other formulations.
