Surface code on a chain of spheres

Join $n$ spheres pole to pole, with $m$ longitude edges on each sphere and one qubit per edge. For this surface code, omit the Pauli X gates star checks at the two end poles. Include the Pauli Z gates checks on each two-edge face and the Pauli X gates star checks at the $n-1$ interior junctions. The boundary choice is essential for encoding one logical qubit.

Label an edge by $(r,j)$, with sphere $r$ and longitude $j$. Independent face checks are $Z_{r,j}{Z}_{r,j+1}$, $j=1,\dots ,m-1$, and the junction checks are $B_r{B}_{r+1}$ with $B_r={\prod }_{j=1}^m{X}_{r,j}$. There are $n(m-1)+(n-1)=nm-1$ independent stabilizer generators on $nm$ qubits, so the code space has dimension two. These are exactly the checks of a concatenated phase-and-bit-flip repetition code.

A pole-to-pole $Z$ path has weight $n$, and an equatorial dual $X$ cut has weight $m$. They cross once and have zero anticommutator, giving logical operations and quantum code distance $\min (m,n)$. If every pole star is instead included, the star rank is $n$ and the code dimension falls to one; the closed chain then carries no logical qubit.