A unifying account of warm start guarantees for patches of quantum landscapes
This provides a unifying theoretical framework for understanding warm-start strategies in variational quantum algorithms, but it is incremental as it builds on prior work on barren plateaus.
The paper tackles the problem of barren plateaus in quantum loss landscapes by proving a general bound that unifies previous cases and shows gradients do not decay exponentially fast in non-exponentially narrow regions around points with curvature, complemented by numerics suggesting gradients vanish exponentially in constant radius subregions.
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 ansätze 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.