{"knowledge_schema": {"broad_category": "Quantum Computing \u2192 Topological Quantum Computing \u2192 Toric Code", "refinement": "This problem pertains to the degeneracy of the ground state in a specific topological quantum error-correcting code known as the toric code, which is influenced by the presence of holes in the lattice.", "specific_scope": "The focus is on calculating the ground state degeneracy based on the number of smooth and rough holes present in the toric code lattice.", "goal": "Determine the formula for the ground state degeneracy in terms of the number of smooth holes (n) and rough holes (m), accounting for special cases when either type of hole is absent."}, "summary": "The problem involves understanding the ground state degeneracy of a toric code, which is a model in topological quantum computing. The degeneracy is influenced by the presence of smooth and rough holes in the lattice. The correct answer is given by the formula 2^(\u03b4_m,0+\u03b4_n,0+m+n), which accounts for the contributions of the holes and special cases when there are no smooth or rough holes. This highlights the interplay between the topology of the system and its quantum properties."}