Data encoding in CV setting


I would like to better understand how data encoding is performed in the CV setting. In I read that with continuous variable quantum computing models we usually use Squeezing and Displacement embeddings, so I would like to focus on them.

For Squeezing encoding, I know from that we start from the decomposition of a squeezed state in the Fock basis
|z>=\frac{1}{\sqrt{\cosh r}}\sum_{n=0}^{\infty}\frac{\sqrt{(2n)!}}{2^nn!}(-e^{i\varphi}\tanh r)^n|2n>,\qquad z=re^{i\varphi},
and using the squeezing feature map with phase encoding \phi:|x>\to|(c,\boldsymbol x)>, we get the kernel
\kappa(\boldsymbol{x},\boldsymbol{x'};c)=\prod_{i=1}^{N}<(c,x_i)|(c,x_i')>, with <(c,x_i)|(c,x_i')>=\sqrt{\text{sech}(c) \text{ sech}(c)}\sum_{n=0}^{\infty}\frac{(2n)!}{2^{2n}n!^2}(e^{i(x_i'-x_i)}\tanh c\tanh c)^n=
\sqrt{\frac{\text{sech} (c)\text{ sech} (c)}{1-\exp(i(x'_i-x_i)\tanh c\tanh c)}}, where c is the squeezing strength.

My question is about Displacement encoding instead: we know that the decomposition of a coherent state in Fock basis is
|\alpha>=e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n>, \alpha\in\mathbb{C}. What kind of feature map should we use in this case? What is the resulting kernel?

Thanks in advance!

Hi Valeria — For the displacement encoding you end up with a Gaussian kernel, since |\langle \alpha| \beta \rangle| \sim \exp(\frac12 |\alpha - \beta|^2).


Thanks a lot @Nicolas_Quesada !

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