This thesis make significant contributions to both the numerical and analytical aspects of particle physics, reducing the noise associated with matrix calculations in quantum chromodynamics (QCD) and modeling multi-quark mesonic matters that could be used to investigate particles previously unseen in nature. Several methods are developed that can reduce the statistical uncertainty in the extraction of hard-to-detect lattice QCD signals from disconnected diagrams. The most promising technique beats competing methods by 1700 percent, leading to a potential decrease in the computation time of quark loop quantities by an order of magnitude. This not only increases efficiency but also works for QCD matrices with almost-zero eigenvalues, a region where most QCD algorithms break down. This thesis also develops analytical solutions used to investigate exotic particles, specifically the Thomas-Fermi quark model, giving insight into possible new states formed from mesonic matter. The main benefit of this model is that it can work for a large number of quarks which is currently almost impossible with lattice QCD. Patterns of single-quark energies are observed which give the first a priori indication that stable octa-quark and hexadeca-quark versions of the charmed and bottom Z-meson exist.

This thesis make significant contributions to both the numerical and analytical aspects of particle physics, reducing the noise associated with matrix calculations in quantum chromodynamics (QCD) and modeling multi-quark mesonic matters that could be used to investigate particles previously unseen in nature. Several methods are developed that can reduce the statistical uncertainty in the extraction of hard-to-detect lattice QCD signals from disconnected diagrams. The most promising technique beats competing methods by 1700 percent, leading to a potential decrease in the computation time of quark loop quantities by an order of magnitude. This not only increases efficiency but also works for QCD matrices with almost-zero eigenvalues, a region where most QCD algorithms break down. This thesis also develops analytical solutions used to investigate exotic particles, specifically the Thomas-Fermi quark model, giving insight into possible new states formed from mesonic matter. The main benefit of this model is that it can work for a large number of quarks which is currently almost impossible with lattice QCD. Patterns of single-quark energies are observed which give the first a priori indication that stable octa-quark and hexadeca-quark versions of the charmed and bottom Z-meson exist.