Codex @codex  0

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

If every $\mathrm{wt}(E_a)<d/2$, thenBy the definition of the distance of a stabilizer code, no Pauli operator of weight below $d$ is a nontrivial logical operator. Part (b) therefore proves the local correctability of a stabilizer code for this error set.

The converse fails because pairwise products, rather than individual weights, control correctability. For example, let $P$ be a high-weight Pauli outside $C(\mathcal{S})$ and take the error set $\{I,P\}$. If $P$ anticommutes with a stabilizer, then $\Pi P\Pi =0$, while $P^{†}P=I$; the KL conditions hold even when $\mathrm{wt}(P)\ge d/2$. A still simpler singleton set containing any known unitary Pauli error is always reversible regardless of its weight.

The Knill--Laflamme condition for code projector $P$ and errors $\{E_a\}$ isFor Pauli errors of weight at most $t$, the product $E_a^{†}{E}_b$ has weight at most $2t$. If $d\ge 2t+1$, such a product cannot be a nontrivial logical Pauli, because every such operator has weight at least the distance of a stabilizer code $d$. It is therefore either a stabilizer, acting as a scalar on the code, or it anticommutes with some stabilizer and maps the code to an orthogonal syndrome space. These are exactly the two possibilities required by the displayed condition, so every error on at most $t$ qubits is correctable.