Quantum tomography: Where does the magic happen?

[Jufa]

TL;DR Summary

Consider quantum state tomography of a n-qubit system. It is known that in order to perform quantum state tomography it is necessary to perform 4^n-1 measurements. Nevertheless, using a neural network substantially lowers the number of needed measurements.

My question is: How does this happen? Less measurements than 4^n-1 means that literally we don't have enough information to label the state. How can the neural network overcome this lack of information?

 

[PeterDonis]

using a neural network substantially lowers the number of needed measurements.

Please give a specific reference for this statement.

 

[Jufa]

In this paper: https://www.labxing.com/files/lab_publications/2278-1524663501-3fRuMVpV.pdf
More specifically in the paragraph in the left in page two. It says that a state that would tipically require 10^6 measurements, using a neural network (a restricted Boltzmann machine in this case) lowers the number of measurements to only 100.) To me it seems that this fact is only due to the fact that when performing tomography with the neural network they are using prior knowledge of the state which allows them to perform less measurements which sounds weird to me. They are comparing the number of measurements needed for a totally unknown state using ordinary tomography (10^6) with the number of measurements needed for partially known state using a neural network (about 100). The comparison seems unfair to me and I still don't know if for the same amount of information of an unknown (or partially unknown) state it takes less measurements for a neural network or no.