Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 2 / c

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.