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We know that a universal gate set for CV quantum computation is that of squeezing, beam splitters, and some nonlinear gate (e.g. this answer). We also know from Knill, Laflamme, and Milburn that linear optics plus some postselection is enough to universally do CV quantum computation (e.g. this answer). Is there a direct way to implement an arbitrary controlled operation in CV quantum computation? More specifically, can this be done using only Gaussian states and, perhaps, postselection (which, I guess, makes states non-Gaussian)?

The controlled gates I am envisioning are controlled Gaussian operations such as displacements that depend on whether or not a control mode is vacuum or nonvaccum. An operator of the form $$cD\sim|0\rangle\langle 0|\otimes \hat{D}(\beta_1)+ |\alpha\rangle\langle \alpha|\otimes \hat{D}(\beta_2)$$ would be ideal, where I realize there will be problems with the overlap $\langle \alpha|0\rangle$.

Is this type of thing directly possible, or do we have to convert between a whole bunch of things to make progress (eg create TMSV, post select on a single photon in one mode, use the single photon to control some phase shift, then use the phase shift to do some Gaussian operation)?

Even a gate like $$cD\sim|0\rangle\langle 0|\otimes \hat{D}(\beta_1)+ |1\rangle\langle 1|\otimes \hat{D}(\beta_2)$$ would be helpful - I could see this being easier.

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