TY - JOUR T1 - A single T-gate makes distribution learning hard Y1 - 2022 A1 - Marcel Hinsche A1 - Marios Ioannou A1 - Alexander Nietner A1 - Jonas Haferkamp A1 - Yihui Quek A1 - Dominik Hangleiter A1 - Jean-Pierre Seifert A1 - Jens Eisert A1 - Ryan Sweke AB -

The task of learning a probability distribution from samples is ubiquitous across the natural sciences. The output distributions of local quantum circuits form a particularly interesting class of distributions, of key importance both to quantum advantage proposals and a variety of quantum machine learning algorithms. In this work, we provide an extensive characterization of the learnability of the output distributions of local quantum circuits. Our first result yields insight into the relationship between the efficient learnability and the efficient simulatability of these distributions. Specifically, we prove that the density modelling problem associated with Clifford circuits can be efficiently solved, while for depth d=nΩ(1) circuits the injection of a single T-gate into the circuit renders this problem hard. This result shows that efficient simulatability does not imply efficient learnability. Our second set of results provides insight into the potential and limitations of quantum generative modelling algorithms. We first show that the generative modelling problem associated with depth d=nΩ(1) local quantum circuits is hard for any learning algorithm, classical or quantum. As a consequence, one cannot use a quantum algorithm to gain a practical advantage for this task. We then show that, for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth d=ω(log(n)) Clifford circuits is hard. This result places limitations on the applicability of near-term hybrid quantum-classical generative modelling algorithms.

UR - https://arxiv.org/abs/2207.03140 ER -