Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement

TL;DR

This paper introduces a recursive greedy initialization method for QAOA that guarantees performance improvement with more layers by analytically constructing transition states and navigating local minima.

Contribution

It provides an analytical construction of transition states for QAOA, enabling guaranteed energy improvements through a recursive greedy initialization strategy.

Findings

Guarantees decreasing energy with increasing layers p.

Constructs 2p+1 transition states connecting local minima.

Performance matches existing heuristics with added guarantees.

Abstract

The quantum approximate optimization algorithm (QAOA) is a variational quantum algorithm, where a quantum computer implements a variational ansatz consisting of $p$ layers of alternating unitary operators and a classical computer is used to optimize the variational parameters. For a random initialization, the optimization typically leads to local minima with poor performance, motivating the search for initialization strategies of QAOA variational parameters. Although numerous heuristic initializations exist, an analytical understanding and performance guarantees for large $p$ remain evasive. We introduce a greedy initialization of QAOA which guarantees improving performance with an increasing number of layers. Our main result is an analytic construction of $2p+1$ transition states - saddle points with a unique negative curvature direction - for QAOA with $p+1$ layers that use the local minimum of QAOA with $p$ layers. Transition states connect to new local minima, which are guaranteed to lower the energy compared to the minimum found for $p$ layers. We use the GREEDY procedure to navigate the exponentially increasing with $p$ number of local minima resulting from the recursive application of our analytic construction. The performance of the GREEDY procedure matches available initialization strategies while providing a guarantee for the minimal energy to decrease with an increasing number of layers $p$.