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2024 / Topological quantum states of matter are a central focus in modern condensed matter physics and quantum information science. The study of topological order in quantum spin systems is closely linked to the concept of gauge symmetries. In this dissertation, we conduct detailed studies of two quantum spin systems with local gauge symmetries. Specifically, we numerically investigate a recently proposed theoretical model that realizes local Z2 gauge symmetry with only two-body interactions.We confirm the existence of topological order and explore other physical properties within the model in depth. In the study of the U(1) toric code, we enhance the conventional Kitaev’s toric code by introducing global U(1) symmetry. The study emphasizes the interplay between local Z2 gauge symmetry and constraints arising from global U(1) symmetry. This examination uncovers the presence of topological degeneracy and reveals an unconventional UV/IR mixing phenomenon within the system. We explore possible implementations of the U (1) toric code in superconducting wires and discuss potential future research directions based on the insights gained from our investigations of these two models.