Theoretical Developments in Lattice Gauge Theory for Applications in Double-beta Decay Processes and Quantum Simulation

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.