= 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 $Z_{r,j}Z_{r,j+1}$, $j=1,\ldots,m-1$, and the junction checks are $B_rB_{r+1}$ with $B_r=\prod_{j=1}^mX_{r,j}$. 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>.
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