On the average-case complexity of learning states from the circular and Gaussian ensembles
Maxwell West
TL;DR
This work analyzes the average-case hardness of learning Born distributions $P_ ext{ψ}(x)=|raket{x| ext{ψ}}{ ext{ψ}}|^2$ for states drawn from the circular AI/AII and DIII fermionic Gaussian ensembles within the statistical-query model. It develops a Weingarten-free, beta/gamma-based integration approach to compute exact total-variation distances to the uniform distribution and proves explicit, exponentially strong SQ-query lower bounds for learning a fraction of the ensemble outputs. The authors show that even inverse-exponential-accurate statistical queries require doubly-exponential numbers of queries to learn a doubly-exponentially small fraction of Born distributions in these ensembles, highlighting extreme average-case hardness. The results extend prior average-case hardness findings for Haar- and matchgate-structured ensembles and demonstrate a versatile technique for exact distance calculations and concentration arguments across AI, AII, and DIII.
Abstract
Studying the complexity of states sampled from various ensembles is a central component of quantum information theory. In this work we establish the average-case hardness of learning, in the statistical query model, the Born distributions of states sampled uniformly from the circular and (fermionic) Gaussian ensembles. These ensembles of states are induced variously by the uniform measures on the compact symmetric spaces of type AI, AII, and DIII. This finding complements analogous recent results for states sampled from the classical compact groups. On the technical side, we employ a somewhat unconventional approach to integrating over the compact groups which may be of some independent interest. For example, our approach allows us to exactly evaluate the total variation distances between the output distributions of Haar random unitary and orthogonal circuits and the constant distribution, which were previously known only approximately.