Topological Phase: Toric Code

Kitaev's toric code — anyonic excitations, error syndromes, topological degeneracy

Controls

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X errors
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Z errors
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Vertex syndromes
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Plaquette syndromes
Topological sector: 00
Toric Code: Kitaev (2003). Qubits on edges; stabilizers: vertex operators A_v = ⊗σˣ, plaquette operators B_p = ⊗σᶻ. Ground state: all A_v=B_p=+1.

Anyons: e-anyons (vertex, X-error) and m-anyons (plaquette, Z-error). They braid to give phase −1.

Click edges to flip qubits. Syndrome stars = e-anyons, syndrome squares = m-anyons. MWPM correction pairs closest anyons.

Lattice