Nuclear processes have played, and continue to play, a crucial role in unraveling the fundamental laws of nature. They are governed by the interactions between hadrons, and in order to draw reliable conclusions from their observations, it is necessary to have accurate theoretical predictions of hadronic systems. The strong interactions between hadrons are described by quantum chromodynamics (QCD), a non-Abelian gauge theory with symmetry group SU(3). QCD predictions require non-perturbative methods for calculating observables, and as of now, lattice QCD (LQCD) is the only reliable and systematically improvable first-principles technique for obtaining quantitative results. LQCD numerically evaluates QCD by formulating it on a Euclidean space-time grid with a finite volume, and requires formal prescriptions to match numerical results with physical observables.

This thesis provides such prescriptions for a class of rare nuclear processes called double beta decays, using the finite volume effects in LQCD framework. Double beta decay can occur via two different modes: two-neutrino double beta decay or neutrinoless double beta decay. The former is a rare Standard Model transition that has been observed, while the latter is a hypothetical process whose observation can profoundly impact our understating of Particle Physics. The significance and challenges associated with accurately predicting decay rates for both modes are emphasized in this thesis, and matching relations are provided to obtain the decay rate in the two-nucleon sector. These relations map the hadronic decay amplitudes to quantities that are accessible via LQCD calculations, namely the nuclear matrix elements and two-nucleon energy spectra in a finite volume. Finally, the matching relations are employed to examine the impact of uncertainties in the future LQCD calculations. In particular, the precision of LQCD results that allow constraining the low energy constants that parameterize the hadronic amplitudes of two-nucleon double beta decays is determined.

Lattice QCD, albeit being a very successful framework, has several limitations when general finite-density and real-time quantities are concerned. Hamiltonian simulation of QCD is another non-perturbative method of solving QCD that, by its nature, does not suffer from those limitations. With the advent of novel computational tools, like tensor network methods and quantum simulation, Hamiltonian simulation of lattice gauge theories (LGTs) has become a reality. However, different Hamiltonian formulations of the same LGT can lead to different computational-resource requirements with their respective system sizes. Thus, a search for efficient formulations of Hamiltonian LGT is a necessary step towards employing this method to calculate a range of QCD observables. Toward that goal, a loop-string-hadron (LSH) formulation of an SU(3) LGT coupled to dynamical matter in 1+1 dimensions is developed in this thesis. Development of this framework is motivated by recent studies of the LSH formulation of an SU(2) LGT that is shown to be advantageous over other formulations, and can be extended to higher-dimensional theories and ultimately QCD.