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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.
Pauli gates are but a particularly convenient choice of an orthogonal basis of Hermitian traceless 2x2 matrices. You can make infinitely many other choices which won't significantly affect the math. You can take similar sets of orthogonal Hermitian traceless operators in arbitrary dimensions, which I suppose you might consider as generalisation of Pauli matrices in higher dimensions. See e.g. the relevant Wikipedia page.
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?".
