Construction of Explicit classifier in Feature Hilbert space

Hi @Maria_Schuld, I am a beginner in in the quantum machine learning and was going through the paper QML in feature hilbert space. At one point I am little confused, it will be great if either of you help in clarifying.

In the paper at page #7 the statement “Since this probability depends on the displacement and squeezing intensity, it is better to define two probabilities, say p(n1 = 2; n2 = 0) and p(n1 = 0; n2 = 2), as
a one-hot encoded output vector (o0; o1).”
Here my doubt is why we are considering 2. Is it representing probability or photon count. I am bit confused. if you kindly give a brief explanation to it.

Thanks and Regards

Thanks for the question @Satanik_Mitra!

I think I can answer that question (hopefully a suitable proxy for @Maria_Schuld, as I’m the other author on that paper :wink:).

The (ideal) squeezing operation always creates photons in even numbers. If you squeeze the vacuum state, you end up with a superposition state like
c_0\vert n=0\rangle + c_2 \vert n=2 \rangle + c_4\vert n=4 \rangle + \dots
The odd photon numbers have zero probability no matter what the squeezing, so we don’t want to use their probabilities as a feature!

Now, if you introduce displacement, you can start having odd photon-numbers having nonzero probability in addition to the even numbers.

So because of the behaviour of odd photon numbers in the case of no displacement, we’ve chosen to exclude them from consideration when building the model.

Thanks @nathan for the help. Can you please describe the intuition behind choosing the ansatz of quadratic and cubic phases. can we go with gaussian gates only?

Hi @Satanik_Mitra,

The basic intuition here is that if we went with purely Gaussian gates, everything would be efficiently classically simulable (so less interesting :smile:). You’d expect little difference in performance to classical kernels. With cubic phase gates (or quartic phase), you start making things much harder to simulate classically, and hence can explore classically inaccessible kernels.