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CNOT gates have been realized for states living in 2-dimensional spaces (qubits).

What about higher-dimensional (qudit) states? Can CNOT gates be defined in such case? In particular, is this possible for three-dimensional states, for example, using orbital angular momentum?

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    $\begingroup$ Do you mean the CCNOT gate or a "scaled" version of the CNOT for a greater number of qubits? (Either way, yes, it exists.) $\endgroup$ – karatechop Jul 17 '18 at 22:00
  • $\begingroup$ Perhaps you could clarify what you mean by “dimension”? The term can be used in several different ways, and it’s not currently clear from cont3xt what is meant. Also, please expand the acronym OAM. $\endgroup$ – DaftWullie Jul 18 '18 at 5:04
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    $\begingroup$ @DaftWullie I think that OAM is Orbital Angular Moment. You can find an example of an OAM-CNOT (here). $\endgroup$ – D-Brc Jul 18 '18 at 6:11
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    $\begingroup$ In this paper multiple dimensions are interpreted as multiple levels qubits. It is also pointed out the study of a multi-dimensional X-gate. $\endgroup$ – D-Brc Jul 18 '18 at 6:22
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    $\begingroup$ @Goat I edited the question in an effort to make it clearer. Feel free to revert the edit if you think the previous version better reflected what you meant to ask $\endgroup$ – glS Jul 21 '18 at 16:20
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There are multiple questions implicit in this question.

How do you define an equivalent of the controlled-not for qutrits?

There are probably multiple ways that the gate can be generalised, but this paper defines it as $$ |x\rangle|y\rangle\mapsto|x\rangle|-x-y\text{ mod }3\rangle $$ I'm not sure why they use the - sign, and am instead going to take the definition $$ |x\rangle|y\rangle\mapsto|x\rangle|x+y\text{ mod }3\rangle $$ That means that we can write the unitary matrix as $$ \left(\begin{array}{ccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right). $$

Is it possible to realise this gate in 3 dimensions?

Sure, why not? This paper talks defines things slightly differently, but one could construct the gate I've specified using their formalism, and they also discuss some ideas for physical implementation. This paper may also be interesting.

Has this gate been realised?

Not to my knowledge, but I can't pretend to know everything that has every been achieved experimentally. I would point out, however, that this paper is only doing single-qudit gates, not two-qudit gates. Judging by the fact that that paper was only last year, I'd guess the two qudit generalisation hasn't been done yet in that particular physical realisation.

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  • $\begingroup$ Thank you for your kind comment. Yes, this paper is only doing single-qudit X-gate, but not cX-gate. $\endgroup$ – Goat Jul 18 '18 at 8:43
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A generalization of the $cX$ (called "controlled $X$" or " controlled $\textrm{NOT}$") gate is given as $c\tilde{X}$ in this paper. When the dimension of the Hilbert space is $d=2$, then $c\tilde{X}=cX=\textrm{CNOT}$, but $c\tilde{X}$ is also valid for $d>2$.

Then, in this paper, $d>2$ gates are interpreted in the context of OAMs.

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  • $\begingroup$ Yes,what I mean here is the cX. However, in the paper, it only gives the solution for realizing X-gate in 4dims, but not cX. $\endgroup$ – Goat Jul 18 '18 at 8:38
  • $\begingroup$ @Goat: Please see Eq. 6 of the first paper I linked. $\endgroup$ – user1271772 Jul 18 '18 at 9:18
  • $\begingroup$ Yes, Eq. 6 shows how the mode goes, here I want to ask how to realize it, to be honest, I totally do not know it can be realized or not. $\endgroup$ – Goat Jul 18 '18 at 10:46
  • $\begingroup$ @Goat: It should be possible to realize it by brute force, but I don't think anyone has done it yet. Also keep in mind there's not a lot of very interesting quantum algorithms that involve the 3-dimensional CNOT compared to the number of algorithms that involve the 2-dimensional CNOT, so it may just be that no one was so interested in it to spend the time doing the experiment. Why are you interested in it? $\endgroup$ – user1271772 Jul 18 '18 at 10:53

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