Toric code

Example

Let $L \ge 2$ and divide the torus into an $L \times L$ square grid with periodic boundary conditions. The grid has $L^2$ vertices, $2L^2$ edges and $L^2$ faces, called plaquettes. Put one qubit on each edge, so $n = 2L^2$. For a vertex $v$, let $A_v$ be the product of the Pauli matrices $X_e$ over the four edges $e$ that meet $v$. For a plaquette $p$, let $B_p$ be the product of $Z_e$ over the four edges $e$ on the boundary of $p$. The toric code is the stabilizer code whose stabilizer $S$ is generated by all $A_v$ and $B_p$.

Check of the hypotheses: a vertex and a plaquette share either $0$ or $2$ edges, so $A_v$ and $B_p$ commute. Two $X$-type operators commute, and so do two $Z$-type operators. So $S$ is abelian. Every element of $S$ is a product of an $X$-type and a $Z$-type operator with sign $+1$, so $-I \notin S$.

Parameters: the only relations among the generators are $\prod_v A_v = I$ and $\prod_p B_p = I$. So $S$ has $2L^2 - 2$ independent generators and the code has $k = 2L^2 - (2L^2 - 2) = 2$ logical qubits. The logical operators are products of $Z$ along a loop that winds around the torus. They also include products of $X$ along such a loop of the dual grid. The shortest such loops have length $L$, so the distance is $L$ and the toric code is a $[\![2L^2, 2, L]\!]$ code.