Connecting ansatz expressibility to gradient magnitudes and barren plateaus

TL;DR

This paper establishes a fundamental link between the expressibility of parameterized quantum circuits and their trainability, showing that highly expressive ans"{a}tze tend to have flatter gradients and are harder to optimize.

Contribution

It extends the barren plateau analysis to arbitrary ans"{a}tze, deriving bounds that connect expressibility with gradient variance and trainability.

Findings

Highly expressive ans"{a}tze exhibit flatter cost landscapes.

Expressibility correlates with gradient magnitude and barren plateau onset.

Numerical results support the theoretical bounds and implications.

Abstract

Parameterized quantum circuits serve as ans\"{a}tze for solving variational problems and provide a flexible paradigm for programming near-term quantum computers. Ideally, such ans\"{a}tze 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 ans\"{a}tze that form exact 2-designs, to arbitrary ans\"{a}tze. 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 2-design. Our resulting bounds indicate that highly expressive ans\"{a}tze 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.