Quantifying the barren plateau phenomenon for a model of unstructured variational ans\"{a}tze

TL;DR

This paper investigates the flatness of the objective function landscape in unstructured variational quantum circuits, providing a Monte Carlo method to estimate barren plateau severity and deriving bounds that depend on circuit parameters.

Contribution

It introduces a novel Monte Carlo estimation technique for landscape flatness and establishes new analytic bounds on barren plateaus based on circuit architecture and parameters.

Findings

Monte Carlo algorithm for landscape flatness estimation

Analytic bounds on barren plateau severity

Insights into dependence on circuit depth and architecture

Abstract

Quantifying the flatness of the objective-function landscape associated with unstructured parameterized quantum circuits is important for understanding the performance of variational algorithms utilizing a "hardware-efficient ansatz", particularly for ensuring that a prohibitively flat landscape -- a so-called "barren plateau" -- is avoided. For a model of such ans\"{a}tze, we relate the typical landscape flatness to a certain family of random walks, enabling us to derive a Monte Carlo algorithm for efficiently, classically estimating the landscape flatness for any architecture. The statistical picture additionally allows us to prove new analytic bounds on the barren plateau phenomenon, and more generally provides novel insights into the phenomenon's dependence on the ansatz depth, architecture, qudit dimension, and Hamiltonian combinatorial and spatial locality. Our analysis utilizes techniques originally developed by Dalzell et al. to study anti-concentration in random circuits.