# Definition of distance d of the QECC $T(S)$

Definition of distance $$d$$:

$$d$$ of $$T(S)$$ is the weight of the smallest Pauli operator $$N$$ in $$N(S)$$ \ $$S$$. $$S$$ is the stabilizer, $$T(S)$$ is the corresponding QECC, and $$N(S)$$ are all errors commute with elements in S.

I'd really appreciate it if someone could help clarify it!

The weight of an error in QEC is the number of qubits, on which the error acts non-trivially. For example, the error $$X\otimes Y\otimes Id$$ has weight two. The weight is important, because usually we know that a code can correct errors up to some weight.

• That is a really concise and great answer! Thanks!! Sep 9 '21 at 7:13