Why is the Toffoli gate not sufficient for universal quantum computation?

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I know that there are papers (cf. arXiv:quant-ph/0205115) out there which prove that the Toffoli gate by itself is not enough for universal quantum computation, but I haven't had the time to go through the whole proof. Could someone give me the crux of the proof or the intuition for this?

Sanchayan Dutta

Posted 2019-11-24T14:40:37.187

Reputation: 14 463

Answers

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The Toffoli gate is just a permutation. If you start in a known basis state, application of a Toffoli just changes it into another basis state, one that you can easily calculate classically (after all, it’s a decision based on looking at 3 bit values). Repeating that doesn’t change anything.

To make it universal, you need to add something like Hadamard which introduces superposition.

DaftWullie

Posted 2019-11-24T14:40:37.187

Reputation: 35 722