# How to translate between continuous variable model and discrete model?

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?

• 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
Commented Aug 20, 2022 at 10:08

Here is a partial answer. In general, the connection between continuous and discrete variables gets pretty complicated... However, if you restrict yourself to Gaussian unitaries on the CV side, and Clifford unitaries on the odd prime dimensional discrete side, both are projective representations of the affine symplectic group. In the CV case, Gaussian unitaries are represented by real affine symplectomorphisms. In the odd-prime dimensional qudit case, the Clifford group is represented by $$\mathbb{F}_p$$ affine symplectomorphisms.
This connection does not give you a direct way to translate between the CV and discrete gate-sets, because there is no homorphism between fields of different characteristics. However, both groups are generated by the same gates (parameterized by real numbers or integers modulo $$p$$) and have the same abstract structure. Your examples fall into this category, even in the qubit case... although there are annoying complications with even characteristic. Moreover, the continuous variable versions of states such as the GHZ-state, must be convoluted with Gaussian noise to be interpreted as bounded linear operators... otherwise, they are just formal "infinitely-squeezed" states.