False traps on quantum-classical optimization landscapes

TL;DR

This paper investigates the causes of false traps in quantum-classical optimization landscapes, revealing that parameter sufficiency alone does not prevent local optima and linking trap emergence to loss of quantum distinguishability.

Contribution

It develops a comprehensive framework for analyzing critical points in quantum optimization landscapes and demonstrates that false traps can occur even with sufficient parameters, challenging previous assumptions.

Findings

False traps can occur despite sufficient parameters.

A connection between landscape topology and quantum distinguishability is established.

The work provides practical guidance for quantum optimization problems.

Abstract

Optimization is ubiquitous in quantum information science and technology, however, the corresponding optimization landscape can encounter false traps, i.e., local but not global optima, likely to prevent used optimizers from finding optimal solutions. Such traps are believed to arise from parameter insufficiency and are expected to disappear when tunable parameters are sufficiently abundant. In this work, we investigate optimization landscapes of quantum optimization problems, and especially obtain that the parameter sufficiency is not enough to ensure the absence of false traps. First, we present a complete framework for analyzing critical features of optimization landscapes, by deriving necessary and sufficient conditions to identify all critical points and to classify them as local maxima, minima, or saddles, under some assumptions. Then, we show that false traps can still emerge on landscapes even with sufficient parameters, implying their appearance cannot be solely attributed to parameter insufficiency. Moreover, a close connection between landscape topology and quantum distinguishability is revealed that the emergence of false traps is linked to the loss of distinguishability among states or operators in the objective function. Finally, implications of our results are noted. Our work not only provides a deeper understanding of the intrinsic complexity of quantum-classical optimization, but also provides practical guidance for solving quantum-classical optimization problems, thus significantly aiding the progress in witnessing quantum advantages of the underlying quantum information processing tasks.