Connecting ansatz expressibility to gradient magnitudes and barren plateaus
Abstract
Parameterized quantum circuits serve as ansätze for solving variational problems and provide a flexible paradigm for programming nearterm quantum computers. Ideally, such ansätze should be highly expressive so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well established barren plateau phenomenon, which holds for ansätze that form exact 2designs, to arbitrary ansätze. Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2design. Our resulting bounds indicate that highly expressive ansätze exhibit flatter cost landscapes and therefore will be harder to train. Furthermore, we provide numerics illustrating the effect of expressiblity on gradient scalings, and we discuss the implications for designing strategies to avoid barren plateaus.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.02138
 Bibcode:
 2021arXiv210102138H
 Keywords:

 Quantum Physics;
 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 Main text: 10 pages, 4 figures. Appendices: 10 pages, 2 figures