Quantum gates we use like X, Y, Z, H, CNOT, etc. are all unitary. When can an arbitrary unitary operator be considered as a quantum gate?

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    $\begingroup$ Laziness, I guess. $\endgroup$ – lionelbrits Jan 20 '15 at 14:22

Quantum gates are all unitary transformations on a state of qubits. Any unitary transformation can be considered a "gate", although the ones you mention are primitive ones from which others can be constructed. More complex ones are usually referred to as circuits. The two qubit gates, $\mathrm{H}$, $\frac{\pi}{8}$ and $\mathrm{CNOT}$ are considered universal gates, because any gate set can be constructed out of those.

You may want to take a look at these lecture notes, in particular, Lemma 12. I would also suggest getting a hold of the textbook by Nielsen & Chuang.

Addendum: I said something incorrect about the Toffoli gate, which is universal for classical computation, but as Peter Shor pointed out in the comments, will not give you complex entries (both Hadamard and Toffoli are real).

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    $\begingroup$ Using just the Hadamard and Toffoli, you can never get any complex entries in the unitary matrices, so it's not true that any gate set can be constructed out of them. $\endgroup$ – Peter Shor Jan 20 '15 at 14:37
  • $\begingroup$ Thanks for clearing that up. Hopefully what I wrote is correct now. $\endgroup$ – lionelbrits Jan 20 '15 at 16:46
  • $\begingroup$ @lionelbrits was that a guess about the gate, that you were corrected? $\endgroup$ – user3483902 Jan 20 '15 at 17:52
  • $\begingroup$ Just misrecalled in haste. $\endgroup$ – lionelbrits Jan 20 '15 at 18:52
  • $\begingroup$ Note that Hadamard and Toffoli are however universal for quantum computation (see arxiv.org/abs/quant-ph/0205115 and arxiv.org/abs/quant-ph/0301040). $\endgroup$ – Norbert Schuch Jan 20 '15 at 19:21

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