# Why do we have only 3 Pauli gates X, Y and Z

This question is out of curiosity thus might not be of much importance.

We have Pauli X, Y, Z gate which rotate the phase by π along X, Y and Z basis. Just wondering why not do we have these 3 gates why not more since there are vectors with more than 3 computational basis.

The set of $$n\times n$$ Hermitian matrices is a vector space of dimension $$n^2$$, while the set of traceless $$n\times n$$ Hermitian matrices has dimension $$n^2-1$$. So for $$n=2$$ you have dimension $$2^2-1=3$$, which I suppose you might take as the answer to the question: "why only 3 Pauli gates?".