Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 / 2 / c

Label the physical qubits by $(i,j)$, with outer block $1\le i\le n$ and inner position $1\le j\le m$. The inner bit-flip repetition code has stabilizersand logical operators ${\overline{X}}_i={\prod }_{j=1}^m{X}_{i,j}$ and ${\overline{Z}}_i=Z_{i,1}$. Replacing each outer $X_i$ by ${\overline{X}}_i$ gives the outer stabilizers

This is the surface code on a chain of spheres. The $n+1$ pole-touching points are vertices, and the $m$ longitudes on sphere $i$ are its qubit-carrying links. At an interior touching point, the star operator is exactly $S_i^X$. Each face between adjacent longitudes has the two-edge plaquette operator $Z_{i,j}{Z}_{i,j+1}$; only $m-1$ of the $m$ face operators on each sphere are independent.

Using the conventions of part a, logical operators areThe first may be placed on any one sphere and is a dual equatorial cut crossing all $m$ longitude links. The second may use any fixed longitude and is a pole-to-pole path through all $n$ spheres. Multiplication by stabilizers deforms either representative without changing its logical action. Their minimum weights are $m$ and $n$, respectively, so the distance of a stabilizer code is