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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?

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

  • Decompose a huge multi-qubit operation into many single-qubit gates, but with lots of controls allowed on each single-qubit gate. Think of it as doing Gaussian elimination: you're factoring the matrix into a bunch of simple row operations.
  • Gradually reduce the number of controls to at-most-one-per-gate, by repeatedly using a construction based on the fact that every operation has a square root and an inverse.
  • 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.

For details of the constructions, and proofs that the errors don't compound in a way that destroys the efficiency of the approximation, you'll probably have to get a textbook or read papers. There's a lot of details.

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