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Planar and periodic surface-code distances differ
Statement
At linear size three, the square planar surface code has asymmetric Z and X distances and , whereas the periodic toric code has distance for both logical species and encodes two qubits. Thus the planar surface-code calculation cannot be used as the periodic torus calculation.
Facts & Assumptions
Given: The relative square-patch convention and the periodic toric-code convention, both with linear size three.
Square planar surface code: the planar patch has left and right rough boundaries; its physical qubits and checks use relative boundary maps, not periodic incidence.
Square planar surface-code distance and path count: for a planar patch, the code encodes one qubit, has Z distance , and has X distance .
Toric code distance is L: the periodic toric code has distance for both primal and dual nontrivial cycles.
The periodic toric code encodes two qubits: the periodic toric code encodes two qubits.
Noncontractible toric logical loops: the periodic code has nontrivial primal Z and dual X loop representatives, each supported on edges.
Counterexample
Set . By [F2], the planar code has Z distance and X distance , and encodes one qubit. For the toric code, [F3] gives the lower bound for every nontrivial logical Pauli, while [F5] supplies a nontrivial three-edge representative of each pure species. Thus both species have distance ; [F4] gives code dimension four. The distance profiles and code dimensions differ at the same linear size.
The reason is already present in the hypotheses: [F1] uses relative boundary maps and rough edges, while [F3] uses periodic incidence on a closed torus. Therefore planar boundary paths and their asymmetric distance theorem do not establish torus winding classes, check relations, or toric distance. The distinction is a finite combinatorial one and does not require a physical interpretation.
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Sources
- Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)