Why is phase gate a member of universal gate set?

Question

According to Solovay-Kitaev theorem it is possible to approximate any unitary quantum gate by sequence of gates from small set of another gates. The approximation can be done with an arbitrary accuracy $\epsilon$.

One of such set of gates is composed of Hadamard gate, phase gate ($S$), $\pi/8$ gate ($T$) and CNOT gate. However, it is also true that $S=T^2$ because $T$ gate is a rotation around $z$ axis by $\pi/4$ and $S$ gate a rotation by $\pi/2$ around the same axis.

Since $S$ gate can be composed of two $T$ gates, why do we add $S$ gate to the set? It seems that a set containing only $H$, $T$ and CNOT is equivalent. What am I missing?

Answer 1

You might want a small set of gates, but it doesn't necessarily mean that you want the smallest set possible. When you talk about a fault-tolerant quantum computer, what you really want to do is minimise the number of $T$ gates (typically the thing that is hard to implement). Other gates from, for example, the Clifford group, are (relatively) easy to implement, so you would much rather implement $S$ rather than $T^2$.