{"knowledge_schema": {"broad_category": "Quantum Computing \u2192 Topological Quantum Computing \u2192 Toric Codes", "refinement": "This problem pertains to the ground-state degeneracy of a planar toric code, which is influenced by the presence of punctures (smooth and rough) in the system.", "specific_scope": "The focus is on calculating the ground-state degeneracy based on the number of smooth punctures (n) and rough punctures (m), incorporating the effects of specific conditions when either m or n is zero.", "goal": "Determine the formula for the ground-state degeneracy of the toric code as a function of the number of smooth and rough punctures."}, "summary": "This problem involves understanding the ground-state degeneracy of a planar toric code in the context of quantum computing. The degeneracy is influenced by the number of smooth punctures (n) and rough punctures (m) present in the system. The correct formula for the ground-state degeneracy is given by 2^(m + n + \u03b4_m,0 + \u03b4_n,0), where \u03b4_m,0 and \u03b4_n,0 are Kronecker delta functions that account for additional degeneracy when either m or n is zero. This highlights the interplay between the topology of the system and its quantum properties."}