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I have a slight notation question on what the "multi" in "multi-controlled" actually stands for. These two published papers are conflicting on whether it means "mutiply" or "multiple". This is a seemingly current issue with two papers posted on arxiv in July of 2021 having different expansions: "mutiply" current paper and "multiple" current paper.

Does anyone know if there is a correct way to write this? I feel as if "multiple controlled" makes more sense, but after seeing many of the conflicting papers, I am not too sure.

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    $\begingroup$ I think both papers are referring to the same concept, a NOT operation controlled by multiple qubits. Are you simply asking which term is better English? $\endgroup$
    – jecado
    Aug 14 at 22:13
  • $\begingroup$ My pardons, I only just noticed the first two articles you linked! I think I will venture an answer, but please let me know if I've misunderstood your question. $\endgroup$
    – jecado
    Aug 14 at 23:32
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This seems more of a linguistic/English question than a quantum question, but I'll answer it all the same.

All four papers you've linked refer to the same concept, that of a unitary operator controlled by multiple qubits.

The first and third papers use the phrase "multiply-controlled", the second paper uses the term "multiple-control", and the fourth paper uses the term "multiple-controlled". The first two versions are grammatically sound; the third is not.

"Multiply-controlled" is correct because "controlled" is an adjective, correctly modified by an adverb, "multiply". "Multiple-control" is correct because "control" is a noun, correctly modified by an adjective, "multiple". "Multiple-controlled" is incorrect because "controlled" is once again an adjective, so it should not be modified by another adjective, like "multiple".

Personally I would use "multiply-controlled", since in context "controlled" serves as the adjective modifying eg. "Toffoli gate", as in "multiply-controlled Toffoli gate". But note that the term "Toffoli gate" itself consists of a noun ("Toffoli") modifying the noun "gate", so I really don't have any issue with "multiple-control Toffoli gate". Both are, I feel, perfectly clear about what they are referring to, and it does no harm to have both phrases in the literature.

But - sorry to contradict you - "multiple-controlled" does sound wrong to me! Even so, it's still perfectly clear what the phrase means, so my discomfort with it is really just a manifestation of linguistic puritanism. ;)

One More Note

Please note that the adverb "multiply", pronounced "mul-ti-plea", is entirely distinct from the verb "to multiply", pronounced "mul-ti-ply"! The adverb adds no semantic meaning beyond the adjective "multiple"; it's just a change in tense.

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    $\begingroup$ One additional linguistic comment: many style guides dictate that you omit the hyphen after an adverb ending in ly, so I would stick with "multiply controlled" and "multiple-control" $\endgroup$ Aug 15 at 1:21

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