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I have been studying quantum universality and came across the following statement:

The Clifford set alone is not a universal quantum gate set, as it can be efficiently simulated classically.

A set of gates is universal when it can approximate with arbitrary precision any unitary operation (please correct me if this is wrong). I don't understand how this concept of universality is related to the ability of efficient classic simulation.

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We assume that there are tasks that a universal quantum computer can do efficiently, and a classical computer cannot. Therefore, if one can simulate efficiently the action of the Clifford group with a classical computer, it cannot be a quantum universal gate set.

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  • $\begingroup$ Thanks for your answer, Yaron. So the question of universality is not only concerning whether a set of gates can do any unitary operation, but whether the set can do it efficiently. Is that correct? This might be too much for a comment, but are there easy counter examples of unitary operations that the Clifford set cannot reproduce? $\endgroup$
    – grav.field
    Jun 27, 2023 at 17:34
  • $\begingroup$ There many examples- T gate, Toffoli gate, Controlled swap. In general, any gate that takes a stabilizer into a non-stabilizer state. However, you might be interested in this paper: arxiv.org/pdf/1207.0046.pdf, where they give some non-Clifford noise channels that might be approximated with Clifford operations and Pauli measurements. $\endgroup$ Jun 28, 2023 at 5:28
  • $\begingroup$ Thanks for the suggestion, I will take a look at that paper! $\endgroup$
    – grav.field
    Jun 30, 2023 at 4:17

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