Relations

product_of → hgp

HGP(C_L, C_L^T), where C_L is the square cyclic neighbor-parity map, reproduces the ordered Toric check matrices exactly. These periodic square factors are not the open rep_L factors used by the planar Surface construction recipe.

References: tillich2009quantum, kovalev2012improved

same_bulk_phase_as → surface

Both use the square-lattice star and plaquette bulk. Cutting the torus creates rough and smooth boundaries and changes absolute cycles into relative strings; closing those boundaries restores two periodic homology directions. Bulk equivalence is not code-space identity.

References: kitaev2003fault, bravyi1998quantum, dennis2002topological

same_bulk_phase_as → honeycomb

At the appropriate frames, contracting the just-measured Honeycomb matching and applying a depth-one local Clifford basis change gives Toric stabilizers on a coarser lattice. The full Honeycomb object remains a time-periodic measurement protocol with spacetime detectors and an anyon automorphism.

References: hastings2021dynamically, aasen2022automorphism