Kitaev Toric Code

The toric code places qubits on edges of an L×L lattice on a torus. Vertex stabilizers A_v = ∏σᶻ detect bit-flip errors; plaquette stabilizers B_p = ∏σˣ detect phase-flip errors. Two logical qubits are encoded in non-contractible loops — topologically protected.

L = 5
p = 8%
speed = 8
Clean qubit
X error (bit flip)
Z error (phase flip)
Both errors
Syndrome (anyonic excitation)
Logical operators: X̄₁ = ∏X along horizontal loop, Z̄₁ = ∏Z along vertical loop
Threshold: p_c ≈ 10.3% (random bond Ising model / random plaquette Z₂ gauge theory)
Anyons: e-particles (vertex violations) and m-particles (plaquette violations) braid with phase π