{"knowledge_schema": {"broad_category": "Quantum Computing \u2192 Topological Quantum Codes \u2192 Toric Code", "refinement": "This problem pertains to the ground state degeneracy of the toric code, which is a type of topological quantum error-correcting code characterized by its use of smooth and rough holes.", "specific_scope": "The focus is on determining the ground space degeneracy based on the number of smooth holes (n) and rough holes (m) in the toric code lattice.", "goal": "Identify the correct expression for the ground space degeneracy of the toric code given the parameters n and m."}, "summary": "The problem involves understanding the ground space degeneracy of the toric code, which is influenced by the presence of smooth and rough holes in the lattice. The goal is to determine the correct formula for this degeneracy based on the number of smooth holes (n) and rough holes (m). The candidates provided offer different expressions, and the correct answer reflects the relationship between these parameters in the context of the toric code's topological properties."}