Analytical Expressions for the Quantum Approximate Optimization Algorithm and its Variants

TL;DR

This paper provides analytical expressions for the expected performance of various QAOA variants, including product and Grover-type mixers, on weighted graphs and hypergraphs, enhancing theoretical understanding of the algorithm.

Contribution

It derives exact analytical formulas for QAOA with different mixers on weighted graphs and hypergraphs, revealing non-local effects of Grover mixers.

Findings

Analytical expressions for QAOA cost expectation with product mixers.

Analytical expressions for QAOA with Grover-type mixers on hypergraphs.

Grover mixer sensitivity to cycles of all lengths, indicating non-locality.

Abstract

The quantum approximate optimization algorithm (QAOA) is a near-term quantum algorithm aimed at solving combinatorial optimization problems. Since its introduction, various generalizations have emerged, spanning modifications to the initial state, phase unitaries, and mixer unitaries. In this work, we present an analytical study of broad families of QAOA variants. We begin by examining a family of QAOA with product mixers, which includes single-body mixers parametrized by multiple variational angles, and derive exact analytical expressions for the cost expectation on weighted problem graphs in the single-layer ansatz setting. We then analyze a family of QAOA that employs many-body Grover-type mixers, deriving analogous analytical expressions for weighted problem hypergraphs in the setting of arbitrarily many circuit ansatz layers. For both families, we allow individual phase angles for each node and edge (hyperedge) in the problem graph (hypergraph). Our results reveal that, in contrast to product mixers, the Grover mixer is sensitive to contributions from cycles of all lengths in the problem graph, exhibiting a form of non-locality. Our study advances the understanding of QAOA's behavior in general scenarios, providing a foundation for further theoretical exploration.