The Planar Surface Code

A torus is convenient theory but impossible hardware — you cannot wire a chip into a doughnut. The planar surface code keeps all the local structure of the toric code but lays it on a flat patch with boundaries. Opening up the surface costs one logical qubit (we go from $k=2$ to $k=1$ ), but the surviving logical qubit is exactly what a 2D chip with nearest-neighbour couplings can host. This is the code being built in real superconducting and ion-trap devices.

From torus to patch

Cut the torus open into a rectangular sheet. The interior keeps the familiar checks: star operators $A_v = \prod_{e \ni v} X_e$ at vertices and plaquette operators $B_p = \prod_{e \in \partial p} Z_e$ at faces. The novelty is what happens at the edges of the patch, where stabilizers are truncated to act on fewer qubits.

Two kinds of boundary

A planar surface code has two distinct boundary types, and they must alternate around the patch:

Rough boundary (sometimes " $X$ -boundary"). Edges dangle so that truncated plaquette ( $Z$ -type) checks sit at the boundary; a $Z$ -string can terminate here without leaving an anyon. An $m$ -type flux can be absorbed.
Smooth boundary (sometimes " $Z$ -boundary"). Truncated star ( $X$ -type) checks sit at the boundary; an $X$ -string (dual) can terminate here. An $e$ -type charge can be absorbed.

A standard planar patch has two opposite rough boundaries and two opposite smooth boundaries.

The single logical qubit

On the planar patch the logical operators are boundary-to-boundary strings, not closed loops:

\bar Z = \prod_{e \in \gamma_Z} Z_e , \qquad \bar X = \prod_{e \in \gamma_X} X_e ,

where $\gamma_Z$ is a $Z$ -string connecting the two rough boundaries and $\gamma_X$ is an $X$ -string (on the dual lattice) connecting the two smooth boundaries. These two strings must cross an odd number of times — at least once — so

\{\bar X, \bar Z\} = 0,

giving one pair of anticommuting logical Paulis: exactly one logical qubit, $k=1$ . The torus had two independent cycle directions and hence $k=2$ ; the disk-like patch has only one nontrivial way to connect opposite boundaries of each type, so $k=1$ .

Distance on the patch

As on the torus, the logical operators are the shortest such strings. For a patch with $d$ rows and $d$ columns of data qubits along the relevant direction, the shortest boundary-to-boundary string has length $d$ , so the code distance is $d$ and the patch realizes parameters of order

[[\,n, k, d\,]] = [[\,O(d^2),\; 1,\; d\,]].

The rotated surface code packs this most efficiently, using $n = d^2$ data qubits for distance $d$ . The next lesson makes the distance-to-lattice-size relationship precise; for now the message is that the planar code trades one logical qubit for physical realizability while keeping every check local and the distance tunable by patch size.