Intro to CVQC

This paper talks a bit more in depth about literally anything CVQC related. Some important highlights:

Kets now become functions: $| 0, 1 ⟩ \to | x ⟩$
Now operators/gates becomes functionals, that take functions $| x ⟩$ and map them to new functions $| x^{'} ⟩$
Now instead of having finite dimensions, we have infinite dimensions, so now our operators cannot take the form of closed matrices (or even infinitely sized matrices)
Now we measure via the eigenfunctions, namely the functions such that $\hat{f} ψ_{f} = f ψ_{f}$ . We used to measure only certain values of 0, 1 based on the eigenvalues of these, but now we can measure a continuum of values.
Inner products remain similar:

⟨ x_{1} | x_{2} ⟩ = \int_{- \infty}^{\infty} ψ_{x_{1}}^{*} (q) ψ_{x_{2}} (q) d q

HOWEVER now norms are not so obvious. Notice that $| p | p ⟩ \to \infty$ using this definition so then we have to use the Dirac Delta function
Describes the Fourier Gate which is:

F | x ⟩ = \frac{1}{\sqrt{2 π ℏ}} \int \exp (\frac{i}{ℏ} x y) | y ⟩ d y

where the standard definition is:

$X (ω) = F (x (t)) = \int_{- \infty}^{\infty} x (t) e^{- i ω t} d t$

this is the analog of the Hadamard Gate.