Quantum tomography: Where does the magic happen?

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SUMMARY

The discussion centers on the efficiency of quantum tomography when utilizing neural networks, specifically a restricted Boltzmann machine, to reduce the number of required measurements. Traditional methods necessitate approximately 10^6 measurements for a completely unknown quantum state, while the neural network approach reduces this to around 100 measurements. This significant reduction is attributed to the incorporation of prior knowledge about the state, raising questions about the fairness of comparing measurement requirements for known versus unknown states. The paper referenced provides detailed insights into this methodology and its implications.

PREREQUISITES
  • Understanding of quantum tomography principles
  • Familiarity with neural networks, specifically restricted Boltzmann machines
  • Knowledge of measurement theory in quantum mechanics
  • Basic comprehension of prior knowledge application in machine learning
NEXT STEPS
  • Read the referenced paper on quantum tomography and neural networks
  • Explore the mechanics of restricted Boltzmann machines in depth
  • Investigate the implications of prior knowledge in quantum state measurements
  • Learn about alternative quantum tomography techniques and their measurement requirements
USEFUL FOR

Researchers in quantum computing, machine learning practitioners, and physicists interested in optimizing quantum state measurements through advanced methodologies.

Jufa
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TL;DR
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?
 
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Jufa said:
using a neural network substantially lowers the number of needed measurements.

Please give a specific reference for this statement.
 
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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.
 

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