On barren plateaus and cost function locality in variational quantum algorithms

TL;DR

This paper investigates the causes of barren plateaus in variational quantum algorithms, showing that the structure of the cost function and circuit causal cone significantly influence gradient vanishing, thereby affecting optimization efficiency.

Contribution

It derives a lower bound on gradient variance based on the circuit causal cone, clarifying conditions for barren plateau emergence in variational quantum algorithms.

Findings

Gradient variance depends on the width of the circuit causal cone.

Barren plateaus are more likely with highly expressive circuits.

Cost function structure critically influences barren plateau onset.

Abstract

Variational quantum algorithms rely on gradient based optimization to iteratively minimize a cost function evaluated by measuring output(s) of a quantum processor. A barren plateau is the phenomenon of exponentially vanishing gradients in sufficiently expressive parametrized quantum circuits. It has been established that the onset of a barren plateau regime depends on the cost function, although the particular behavior has been demonstrated only for certain classes of cost functions. Here we derive a lower bound on the variance of the gradient, which depends mainly on the width of the circuit causal cone of each term in the Pauli decomposition of the cost function. Our result further clarifies the conditions under which barren plateaus can occur.