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If I understand correctly, the discrete and continuous variable (CV) version of quantum computation are equivalent. However, the continuous aspect of the CV model makes me wonder to what extent can both models be compared. Given a discrete-gate quantum algorithm, is it well understood how to translate it into a CV circuit?

I have seen the some algorithms (Shor, teleportation, dense coding, error correction) applied to both but I still do not get if there is 1-to-1 correspondence between a discrete gate-based circuit and a linear optics circuit. Is there a procedure to translate an CV algorithm into a discrete one and viceversa? Or is it an inherently hard problem?

Examples:

What is the equivalent of entangling two qubits using a Hadamard and a CNOT gate (Bell state)? Is it two-mode squeezing? What about a GHZ?

Mauricio
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    just a comment: boson sampling, at least in its original formulation, doesn't really use "continuous variables". You only deal with discrete events: occupation numbers in different modes. Though the bosonic statistics is crucial, hence why it's convenient to describe it using second quantisation. But you can also describe everything in standard "kets and unitaries" notation if you so wish – glS Aug 20 '22 at 10:08

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