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In an old paper from 2003, D. Aharanov simplified Shi's proof that CCNOT (or Toffoli) gates + Hadamard gates together form a computationally universal gate set for quantum computing. The CCNOT gate is well-known to be computationally universal for reversable classical computing; another similarly well-known such gate is the CSWAP (or Fredkin) gate.

Can we plug in the Fredkin gate into Aharanov's paper to prove the universality of CSWAP + Hadamard for quantum computing?

In her proof of the universality of Toffoli + Hadamard, Aharanov considers $\Lambda(U)$ as a controlled version of $U$ - she writes the Toffoli gate as $\Lambda^2[X]$. Does something similar work for CSWAP?

Toffoli + Hadamard is an interesting computationally universal quantum gate set, at least because there's never any imaginary amplitudes (unless provided by ancilla qubits). The Fredkin gate is an interesting computationally universal classical gate, because the one's count (Hamming weight) is conserved between input and output.

Mark Spinelli
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    I believe you can apply $CSWAP$ to the same Real-Imaginary encoding Aharanov used. This should make a $\Lambda [iI]$ gate which is equivalent to $S\otimes I$ Then, since you already have $H$; you have the Clifford group. Finally, use the result that Clifford plus anything ($CSWAP$ in this case) is universal. – Jonas Anderson Mar 04 '24 at 05:16
  • @JonasAnderson I'm starting to think that this is on the right track. Can you expand it to a formal answer? – Mark Spinelli Mar 05 '24 at 14:43
  • I tried sketching the idea out and now I'm not sure if it works. The $CSWAP$ doesn't implement the encoded gate I thought it did. – Jonas Anderson Mar 06 '24 at 13:15

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