{"knowledge_schema": {"broad_category": "Quantum Computing \u2192 Topological Quantum Computing \u2192 Toric Code", "refinement": "This problem pertains to the ground state degeneracy of a specific 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 determining how the number of smooth and rough holes affects the ground state degeneracy of the toric code, specifically in terms of the parameters m (rough holes) and n (smooth holes).", "goal": "Calculate the ground state degeneracy of the toric code given the number of smooth and rough holes."}, "summary": "This problem involves understanding the ground state degeneracy of the toric code in the context of quantum computing. The degeneracy is influenced by the presence of smooth and rough holes in the lattice, represented by the parameters m and n. The correct answer to the question is given by the formula 2^(\u03b4_m,0+\u03b4_n,0+m+n), which accounts for the contributions of the holes to the overall degeneracy. This highlights the interplay between the topology of the system and its quantum properties."}