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

For Pauli errors, “suitably local” means that every pairwise product has support too small to carry a logical string:A sufficient geometric statement is $|\mathrm{supp}{E}_a\cup \mathrm{supp}{E}_b|<d$; 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.