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I was trying to find a reference for the 7 T-gate decomposition of the Toffoli gate given by Cirq. The decomposition originates from the the one used for CCZPowGate as given in the doc string here

https://github.com/quantumlib/Cirq/blob/master/cirq-core/cirq/ops/three_qubit_gates.py#L100

This is mentioned as an adjacency respecting decomposition by which I understand that the CNOTs are nearest neighbour

Most of the literature I have seen on Toffoli gate decompositions seem to use long range CNOTs for example in Neilson and Chuang or here

https://arxiv.org/abs/2101.04764 .

Does anyone have an idea about the source of the decomposition used?

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I'm the one who added this code to Cirq. I found that decomposition as part self-answering a question from 2018. I found it by messing around in Quirk, trying different strategies.

Norbert Schuch has a comment on the answer saying the decomposition appears in his thesis in 2002. I wasn't able to get his thesis to check as the link he gave is dead now, but I don't doubt he also found it or that the original discovery predates mine by more than a decade.

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  • $\begingroup$ Thanks Craig, that is very helpful! $\endgroup$
    – Madhav Vijayan
    Jul 29, 2022 at 2:07

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