Limits of quantum generative models with classical sampling hardness

TL;DR

This paper investigates the limitations of quantum generative models, showing that models with anticoncentration are not trainable on average, while those with sparse outputs can be trained, impacting the pursuit of quantum advantage.

Contribution

It reveals a fundamental trade-off between anticoncentration and trainability in quantum generative models, challenging assumptions about their classical hardness and quantum advantage.

Findings

Anticoncentrating models are not trainable on average.

Sparse distribution models can be trained effectively.

Trade-offs relate to quantum process verification.

Abstract

Sampling tasks have been successful in establishing quantum advantages both in theory and experiments. This has fueled the use of quantum computers for generative modeling to create samples following the probability distribution underlying a given dataset. In particular, the potential to build generative models on classically hard distributions would immediately preclude classical simulability, due to theoretical separations. In this work, we study quantum generative models from the perspective of output distributions, showing that models that anticoncentrate are not trainable on average, including those exhibiting quantum advantage. In contrast, models outputting data from sparse distributions can be trained. We consider special cases to enhance trainability, and observe that this opens the path for classical algorithms for surrogate sampling. This observed trade-off is linked to verification of quantum processes. We conclude that quantum advantage can still be found in generative models, although its source must be distinct from anticoncentration.