On the average-case complexity of learning states from the circular and Gaussian ensembles

TL;DR

This paper proves that learning the distributions of quantum states from circular and Gaussian ensembles is computationally hard on average, using a novel integration method over compact groups.

Contribution

It establishes average-case hardness results for learning quantum states from specific ensembles and introduces a new technique for integrating over compact groups.

Findings

Proves average-case hardness of learning states from circular and Gaussian ensembles.

Develops a new method for exact integration over compact groups.

Evaluates total variation distances exactly for Haar random circuits.

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.