= Toric code
{title2=$H=-J_e\sum_sA_s-J_m\sum_pB_p$}
{wiki}
The <toric code> is a <stabilizer code> with a <qubit> on every edge of a two-dimensional lattice on a <torus>. A vertex check multiplies <Pauli operators> $X$ on its incident edges, while a face check multiplies <Pauli operators> $Z$ around its boundary:
$$
A_s=\prod_{j\in\operatorname{star}(s)}X_j,\qquad B_p=\prod_{j\in\partial p}Z_j.
$$
The checks commute because a vertex and face share either zero or two edges. The code is their common $+1$ <eigenspace>, equivalently the space of <ground states> of
$$
H=-J_e\sum_sA_s-J_m\sum_pB_p,\qquad J_e,J_m>0.
$$
On the <torus>, $\prod_sA_s=\prod_pB_p=I$ gives two redundant checks. The remaining independent checks leave a four-dimensional code space, encoding two <logical qubits>. Products of <Pauli operators> around noncontractible loops act on that information. <Fault-tolerant quantum computation by anyons by A. Yu. Kitaev>[Section 1, equations 1–4 and protected-subspace dimension, printed pp. 2–5]{ref}
A violated vertex or face check is an excitation. <Open strings> create pairs of excitations at their endpoints; closed noncontractible strings can change a logical state without creating local excitations. The electric and magnetic excitations are <bosons> with mutual full-braiding phase $-1$, while their composite is a <fermion>. <Fault-tolerant quantum computation by anyons by A. Yu. Kitaev>[Section 2, string operators and anyon statistics]{ref}
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