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It seems to me that the "disjunction gate" (aside: is that its proper name?) can be thought of as the combination of three gates, G1, G2, and G3, where G2 is the CCNOT gate, and $G1 = G3 = ¬_a \otimes ¬_b \otimes I_c$. For context, see the bottom two rows of the following figure that I created in draw.io: Quantum computing summary

Is there a name for the gate $¬_a \otimes ¬_b \otimes I_c$ (the one I'm calling NaNbIc)?

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    $\begingroup$ I like how "the gate that does not not nothing" sounds, I suggest you start using that. :) $\endgroup$ – JiK Dec 28 '20 at 16:42
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The gate in which you're interested:
enter image description here
would more often be called $X_1X_2I_3$ rather than $¬_a ¬_b I_c$, because we use $X$ to denote the NOT gate more often than we use $¬$.

However it is very unlikely that $X_1X_2I_3$ is given a new name, because there's nothing going on other than just two $X$ gates. CNOT has a name because the "C" part in CNOT denotes something that isn't quite so easily captured by something like $X,Y$ or $Z$. It seems that you have some familiarity with classical gates, is there any name for the 3-bit gate that does NOT to the first two bits and nothing to the third?

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  • $\begingroup$ Thank you for the very thorough explanation! $\endgroup$ – Ben Hocking Dec 28 '20 at 0:29
  • $\begingroup$ No problem, and with pleasure! $\endgroup$ – user1271772 Dec 28 '20 at 0:29
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Welcome to QCSE. You seem to have gotten some of the gist of quantum gates but don't be surprised if every such gate is promoted to having a specific, universally recognized name.

A reason some gates such as $\mathsf{CCNOT}$ (Toffoli) or $\mathsf{CCSWAP}$ (Fredkin) have such a name is because they have been found to be useful and are a shorthand for the specific functions they perform. Indeed, the properties of both the $\mathsf{CCNOT}$ gate and the $\mathsf{CCSWAP}$ gates were investigated in the context of reversible (classical) computing as studied in the 70's/early 80's, prior to/parallel to the beginnings of quantum computing.

Furthermore other lessons from classical computer engineering may form a context for quantum computing. For example, you properly identified a quantum $\mathsf{SWAP}$ gate, but classically promoting such an operation to a formal "gate" appears to be overkill. A classical computer engineer wishing to implement such a classical $\mathsf{SWAP}$ gate wouldn't likely consider any specific layout of CMOS transistors implementing a swap; rather, she would instead just consider jumping one input around another with a set of vias/metal runners. However, a quantum computer engineer (if there is such a term) would indeed be worried about how she implements the gate by means of the appropriate microwave/laser pulses, for example. A quantum $\mathsf{SWAP}$ is something entirely different from a classical swap; thus it's appropriate for us to give it a formal name.

Looking more at your specific gate, each input is a single-qubit gate, and there's no way for any entanglement to be created with such a gate. For example, if each of $\vert a\rangle$, $\vert b\rangle$, and $\vert c\rangle$ were unentangled prior to the operation of the $\mathsf{HOCKING}$ gate, they would be afterwards.

Thus don't be surprised if your gate doesn't have a specific, universally recognized name.

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    $\begingroup$ +1 for the excellent explanation. Alas, I can only accept one answer, and I think $X_1X_2I_3$ sounds a little less egotistical for me than the HOCKING gate. ;) $\endgroup$ – Ben Hocking Dec 28 '20 at 11:09

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