For Pauli errors, “suitably local” means that every pairwise product has support too small to carry a logical string:
$$
\boxed{
\operatorname{wt}(E_a^\dagger E_b)<d
\quad\text{for every }a,b.}
$$
A sufficient geometric statement is $|\operatorname{supp}E_a\cup\operatorname{supp}E_b|<d$; in particular, each error having weight below $d/2$ suffices. Such a product is either a stabilizer or creates an excitation detected by at least one stabilizer. It is never a nontrivial logical operator, so the ground-space projector of the <surface code> obeys the <Knill--Laflamme condition> for the entire set.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-342/2/d/solution