Learning properties of quantum states without the IID assumption

Area of Science:

• Quantum Information Science
• Machine Learning Theory

Background:

• Current quantum state learning often assumes independent and identically distributed (i.i.d.) data.
• This assumption limits applicability to real-world quantum systems where states may be correlated.

Purpose of the Study:

• To develop a general framework for learning properties of quantum states beyond the i.i.d. assumption.
• To adapt existing learning algorithms for non-i.i.d. quantum data and improve their sample complexity.

Main Methods:

• Theoretical framework development for general quantum state learning.
• Analysis of sample complexity scaling with non-i.i.d. data.
• Leveraging permutation invariance and randomized single-copy measurements.
• Derivation of a new quantum de Finetti theorem.

Main Results:

• Algorithms for i.i.d. quantum states can be generalized to non-i.i.d. states with polynomial increase in sample complexity.
• Sample complexity can be improved to polylogarithmic for algorithms using non-adaptive, single-copy measurements.
• Generalization of the classical shadow framework to the non-i.i.d. setting with improved sample efficiency.
• A new quantum de Finetti theorem is derived that scales favorably with Hilbert space dimension.

Conclusions:

• The proposed framework enables robust quantum state learning from arbitrary data distributions.
• The findings significantly enhance the practical applicability of quantum machine learning algorithms.
• The new de Finetti theorem offers a powerful tool for analyzing measurement statistics in complex quantum systems.