Exact synthesis of Toffoli gate from CNOT and rational single-qubit gates?

Question

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?

Are you aware of this paper: Both Toffoli and Controlled-NOT need little help to do universal quantum computation — Egretta.Thula, Aug 08 '23 at 15:52
@Egretta.Thula yes. My gateset is universal. In fact, the CNOT together with the rational pythagorean gate [[3,4],[-4,3]]/5 is already universal (https://arxiv.org/abs/2209.11793). The question is just whether it can be synthesized exactly, which Solovay-Kitaev and similar algorithms don't do. — D0r1an, Aug 08 '23 at 22:01

score 1 · Answer 1 · answered Aug 07 '23 at 14:38

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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

answered Aug 07 '23 at 14:38

Yaron Jarach

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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. – D0r1an Aug 08 '23 at 08:52

Exact synthesis of Toffoli gate from CNOT and rational single-qubit gates?

1 Answers1