A unifying account of warm start guarantees for patches of quantum landscapes

TL;DR

This paper provides a unifying theoretical framework for understanding the existence of regions with substantial gradients in quantum loss landscapes, which is crucial for effective warm-start strategies in variational quantum algorithms.

Contribution

It introduces a general lower-bound on loss variance that unifies previous results and applies to physically-motivated ans"atze, advancing understanding of barren plateaus.

Findings

Loss variance cannot decay exponentially fast near certain points.

Barren plateau regions have exponentially vanishing gradients in constant radius subregions.

Warm-start strategies must approach regions of attraction increasingly closely as problem size grows.

Abstract

Barren plateaus are fundamentally a statement about quantum loss landscapes on average but there can, and generally will, exist patches of barren plateau landscapes with substantial gradients. Previous work has studied certain classes of parameterized quantum circuits and found example regions where gradients vanish at worst polynomially in system size. Here we present a general bound that unifies all these previous cases and that can tackle physically-motivated ans\"atze that could not be analyzed previously. Concretely, we analytically prove a lower-bound on the variance of the loss that can be used to show that in a non-exponentially narrow region around a point with curvature the loss variance cannot decay exponentially fast. This result is complemented by numerics and an upper-bound that suggest that any loss function with a barren plateau will have exponentially vanishing gradients in any constant radius subregion. Our work thus suggests that while there are hopes to be able to warm-start variational quantum algorithms, any initialization strategy that cannot get increasingly close to the region of attraction with increasing problem size is likely inadequate.