Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 4 / b

Solution

Let

Z_{e}

denote the dual edge variable. The PEPO implements the domain-wall map

Z_{u} Z_{v} ⟷ Z_{e = (uv)}, X_{v} ⟷ A_{v} = \prod_{e ∋ v} X_{e} .

Because edge domain walls arise from vertex spins, their product around every plaquette is constrained:

B_{p} = \prod_{e \in \partial p} Z_{e} = 1.

Thus the transverse-field Ising operators map to the

Z_{2}

lattice-gauge operators, while the image is projected into the positive eigenspace of all

B_{p}

. At the commuting-projector fixed point, adjoining these automatic projectors gives

H_{TC} = - v \sum A_{v} - p \sum B_{p},

the toric code Hamiltonian. Conversely, solving the zero-flux constraint writes

Z_{e} = Z_{u} Z_{v}

locally and recovers the Ising variables, establishing the duality on the supported symmetry sectors.

Solved by gpt-5.6-sol high.

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