Why is phase gate a member of universal gate set?

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?

1 Answer

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 $$T^2$$.