Free Fermion Distributions Are Hard to Learn
Alexander Nietner
TL;DR
The paper tackles the problem of learning the Born distributions of free-fermion states measured in the occupation-number basis, focusing on the particle-number non-preserving regime. It shows an information-theoretic exponential lower bound for learning from expectation values and proves LPN-based computational hardness for learning from samples, highlighting a separation between learning via local observables and full distribution learning. The results leverage the free-fermion/match-gate correspondence and Wick’s theorem to contrast efficient tomography with distribution learning while tying hardness to circuit depth $d$ and to parity-based reductions. Under the LPN assumption, learning from classical samples remains intractable, with best-known algorithms resembling $2^{O(n/ ext{log} )}$-time attacks, underscoring fundamental limits depending on data-access models. The work thus delineates the landscape of learnability for quantum-distribution tasks and motivates further study of fixed-particle-number cases and average-case hardness in quantum settings.
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.