Free Fermion Distributions Are Hard to Learn

TL;DR

This paper demonstrates the computational and information-theoretic hardness of learning free fermion distributions, especially in the particle number non-preserving case, highlighting fundamental complexity challenges in quantum system learning.

Contribution

It establishes the first hardness results for learning free fermion distributions, both from expectation values and samples, under specific assumptions.

Findings

Hardness results for learning free fermion distributions from expectation values.

Computational hardness based on the LPN assumption for sampling-based learning.

Highlights fundamental complexity in quantum system distribution learning.

Abstract

Free fermions are some of the best studied quantum systems. However, little is known about the complexity of learning free-fermion distributions. In this work we establish the hardness of this task in the particle number non-preserving case. In particular, we give an information theoretical hardness result for the general task of learning from expectation values and, in the more general case when the algorithm is given access to samples, we give a computational hardness result based on the LPN assumption for learning the probability density function.