# CNOT, Hadamard and Φ quantum gates

How can somebody use these gates to create other gates like pauli's or other gates? Also how can somebody go from pauli's to CNOT, Hadamard and Φ quantum gates?

• By combining them in quantum circuits? I'm not really sure what you're asking, here. – David Richerby Feb 10 '16 at 1:44
• some gates are "functionally complete" & are enough to build full digital logic, suspect thats the point of the question... – vzn Feb 10 '16 at 16:24
• see this related answer cs.stackexchange.com/a/345/157 – Ran G. Feb 25 '16 at 8:45

• Factor each single-qubit operation $U$ into $A$, $B$, $C$, $\theta$ such that $ABC = I$ but $AXBXC e^{i \theta} = U$. Use that to replace controlled-$U$ gates with uncontrolled single-qubit operations separated by $CNOT$ gates.
• Approximate each uncontrolled single-qubit operation with a sequence of $H$ and $T$ gates (which is possible because they correspond to rotations that can approximate any other rotation).
What you're left with is a circuit containing only $CNOT$, $H$, and $T$ gates that approximates the original operation. You need more gates to get a better approximation, but you can get as close as desired. Also the overhead isn't too terrible.