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On page 11 of this study they write that "In order to calculate macroscopic values according to Eq.(11), point-wise addition of the distribution function ... is required. To perform this operation quantumly, a quantum circuit composed of controlled H, SWAP, and X gate is applied ... while an addition procedure is conducted by applying the Hadamard operation." They then go on to give an explicit circuit to do this operation (Fig. 7).

My question is, what do they mean by "point-wise addition?" Is this the same as using an adder function as discussed here? If not, what are they doing differently? It seems to me like they may be adding probability amplitudes, but that doesn't seem possible since I don't think it is a unitary operation.

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    $\begingroup$ I usually have seen pointwise in the context of manifolds, basically addition is performed at each point, like how you add two vector fields. Since it seems to be some kind of lattice maybe the lattice points are the points. Not sure how this translates (are the operators defined for each individual lattice point or for the whole manifold?) $\endgroup$
    – Emil
    May 11, 2022 at 6:22
  • $\begingroup$ Do you understand the method they used in this paper to perform the point-wise addition? $\endgroup$ May 12, 2022 at 14:59
  • $\begingroup$ Not really, but something being pointwise is not a big deal, for example $sin(x)+cos(x)$ is "pointwise addition" of two functions, x being the point (for all x in the domain being implicit.). So you could just as well write $(sin + cos)(x)$ $\endgroup$
    – Emil
    May 12, 2022 at 16:12

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