Equivalence of quantum barren plateaus to cost concentration and narrow gorges

TL;DR

This paper proves that barren plateaus, cost concentration, and narrow gorges in quantum circuit landscapes are equivalent phenomena, enabling easier diagnosis and providing insights into quantum cost landscape limitations.

Contribution

It analytically establishes the equivalence of three landscape features in PQCs, linking them and enabling simpler diagnostics of barren plateaus.

Findings

Barren plateaus, cost concentration, and narrow gorges occur together in PQC landscapes.

Cost differences can diagnose barren plateaus more efficiently than gradients.

Quantum mechanics constrains the possible cost landscape features.

Abstract

Optimizing parameterized quantum circuits (PQCs) is the leading approach to make use of near-term quantum computers. However, very little is known about the cost function landscape for PQCs, which hinders progress towards quantum-aware optimizers. In this work, we investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minina (called narrow gorges). We analytically prove that these three phenomena occur together, i.e., when one occurs then so do the other two. A key implication of this result is that one can numerically diagnose barren plateaus via cost differences rather than via the computationally more expensive gradients. More broadly, our work shows that quantum mechanics rules out certain cost landscapes (which otherwise would be mathematically possible), and hence our results are interesting from a quantum foundations perspective.