A single $T$-gate makes distribution learning hard

TL;DR

This paper investigates the learnability of output distributions from local quantum circuits, showing that adding a single T-gate makes the distribution hard to learn, highlighting limitations of quantum generative models.

Contribution

It demonstrates that a single T-gate can make distribution learning computationally hard, and analyzes the implications for quantum advantage and hybrid quantum-classical algorithms.

Findings

Efficient learnability of Clifford circuits can be disrupted by a single T-gate.

Learning depth d=n^{Ω(1)} quantum circuit distributions is computationally hard.

Hybrid algorithms cannot efficiently learn distributions from certain deep quantum circuits.

Abstract

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 $d=n^{\Omega(1)}$ 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 $d=n^{\Omega(1)}$ 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 $d=\omega(\log(n))$ Clifford circuits is hard. This result places limitations on the applicability of near-term hybrid quantum-classical generative modelling algorithms.