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Is it possible to implement a Toffoli gate exactly using just CNOT gates and single qubit complex rational gates (i.e. with entries in $\mathbb{Q}(i)$), possibly with ancillas?

I know this works with a CS (controlled square root of Z) gate. So I have been trying to implement the CS gate, but some simple constructions like $ABC=I$ and $AXBXC=S$ did not work out (as in my SAGE script reports a dimension of $-1$ for the corresponding ideal).

Is there maybe a number theoretic reason why this is impossible?

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Is this helpful - How do you implement the Toffoli gate using only single-qubit and CNOT gates? ?

These might also be helpful: https://arxiv.org/abs/0803.2316, https://arxiv.org/pdf/quant-ph/9503016.pdf

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  • $\begingroup$ Thank you for the answer, but I was already familiar with these references. I am looking specifically for a construction of the Toffoli gate using just CNOT and single qubit gates with complex rational entries, which excludes the T gate as $e^{i\pi/4} \notin \mathbb{Q}(i)$. There just might not exist an exact solution, but I don't know how to prove it. $\endgroup$
    – D0r1an
    Aug 8, 2023 at 8:52

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