Surface code

The surface code is a topological stabilizer code whose electric and magnetic excitations are bosons with mutual-semion braiding.

Join $n$ spheres pole to pole and draw $m$ longitude edges on each sphere. Qubits on edges, two-edge face $Z$ stabilizers, and vertex $X$ stabilizers realize the concatenated phase-and-bit-flip repetition code. A pole-to-pole path has weight $n$, while a dual equatorial cut has weight $m$.

For independent bit flips, summing the probabilities of all error chains in one syndrome and logical class equals, up to a class-independent factor, the partition function of a random-bond Ising model. A reference error fixes bond signs and products of star stabilizers become Ising spin configurations.

On a cylinder with equal anyon-condensing boundaries, the two topological ground states split only through a virtual string implementing a logical operator. A perturbation of strength $h$ produces a splitting of order $J$ times a path-counting factor times $(|h|/J{)}^d$, where $d$ is the shortest supported logical-string length.

The toric code has commuting star and plaquette stabilizers. Its electric and magnetic excitations are bosons with mutual-semion statistics, and their fusion is a fermion.

The surface-code anyons are $1,e,m,f=e\times m$. They obey $e^2=m^2=f^2=1$; $e$ and $m$ are bosons with mutual full-braiding phase $-1$, while their composite $f$ is a fermion.