Square planar surface code

Fix an integer $d\ge 2$. Take the square grid with vertices $(i,j)$ for $0\le i,j\le d$, unit horizontal and vertical edges, and $d^2$ square faces. Let $R$ consist of the left and right vertical boundary edges together with their vertices. The physical qubits are the edges not in $R$; there are $2d^2$ such edges. Order vertices and faces lexicographically by their coordinates, and order physical edges lexicographically by their lower endpoint, with horizontal edges before vertical edges when endpoints tie.

Let $Q_0$ be the binary vector space on vertices outside $R$, $Q_1$ the binary vector space on physical edges, and $Q_2$ the binary vector space on faces. Define ${\mathrm{\partial }}_1:Q_1\to Q_0$ by sending each physical edge to the sum of its endpoints that are outside $R$. Define ${\mathrm{\partial }}_2:Q_2\to Q_1$ by sending each face to its boundary with any edge in $R$ deleted.

The X-check matrix is $H_X={\mathrm{\partial }}_1$, with rows indexed by vertices outside $R$ and columns by physical edges. The Z-check matrix is $H_Z={\mathrm{\partial }}_2^{\mathsf{T}}$, with rows indexed by faces and columns by physical edges. Thus X checks are the rows of $H_X$ and Z checks are the rows of $H_Z$. The maps satisfy ${\mathrm{\partial }}_1{\mathrm{\partial }}_2=0$. The rough boundaries are the left and right sides; the top and bottom are the complementary dual boundaries.

The relative cellulation fixes every boundary convention. In particular, rough edges are removed from the physical register, and only vertices outside the rough boundary index X checks.