Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 342 / 2 / c / Solution

All plaquette stabilizers commute and square to one. Every term inis minimized by eigenvalue $+1$, so any ground state satisfiesfor every plaquette.

The R-edge operator $X_{e_R}$ commutes with every X-type stabilizer. It overlaps the two endpoint R plaquettes in an odd number of vertices and every other plaquette evenly, so it anticommutes only with the two endpoint Z-type stabilizers. It flips those two eigenvalues to $-1$, creating two $r_x$ particles. Multiplying by an adjacent R-edge operator toggles the shared endpoint twice, annihilating that excitation there while creating one at the new endpoint. Repetition forms a string operator in a topological code whose two excitations move farther apart.

Solved by gpt-5.6-sol high.