Classical and Quantum Bounded Depth Approximation Algorithms

TL;DR

This paper compares classical and quantum bounded-depth approximation algorithms, showing classical algorithms often outperform the quantum approach on certain problems and suggesting classical methods may be more promising for approximate optimization.

Contribution

It introduces a class of local classical algorithms, demonstrating their competitive performance against the single-step QAOA on MAX-3-LIN-2 and MAX-CUT problems.

Findings

Classical local algorithms match single-step QAOA performance on MAX-3-LIN-2.

Classical algorithms outperform QAOA on triangle-free MAX-CUT instances.

Quantum QAOA cannot match classical algorithms' scaling on bounded degree graphs.

Abstract

We consider some classical and quantum approximate optimization algorithms with bounded depth. First, we define a class of "local" classical optimization algorithms and show that a single step version of these algorithms can achieve the same performance as the single step QAOA on MAX-3-LIN-2. Second, we show that this class of classical algorithms generalizes a class previously considered in the literature, and also that a single step of the classical algorithm will outperform the single-step QAOA on all triangle-free MAX-CUT instances. In fact, for all but $4$ choices of degree, existing single-step classical algorithms already outperform the QAOA on these graphs, while for the remaining $4$ choices we show that the generalization here outperforms it. Finally, we consider the QAOA and provide strong evidence that, for any fixed number of steps, its performance on MAX-3-LIN-2 on bounded degree graphs cannot achieve the same scaling as can be done by a class of "global" classical algorithms. These results suggest that such local classical algorithms are likely to be at least as promising as the QAOA for approximate optimization.