An Introduction to the Quantum Approximate Optimization Algorithm

TL;DR

This paper provides a comprehensive introduction to the Quantum Approximate Optimization Algorithm (QAOA), detailing its principles, implementation, and application to combinatorial optimization problems, emphasizing its properties and potential advantages.

Contribution

It offers a detailed, first-principles tutorial on QAOA, including its Hamiltonian formulation, energy landscape analysis, and extension to higher-order problems, which is novel in its depth and scope.

Findings

QAOA effectively encodes combinatorial problems into quantum circuits.

The energy landscape of QAOA exhibits symmetry and periodicity, enabling parameter space reduction.

Extensions to PUBO problems demonstrate the algorithm's versatility.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quantum algorithm introduced to tackle classically intractable combinatorial optimization problems. This tutorial offers a comprehensive, first-principles introduction to QAOA and its properties, focusing on its application to Quadratic and Polynomial Unconstrained Binary Optimization (QUBO and PUBO) problems. The tutorial begins by outlining variational quantum circuits and QUBO problems, focusing on their key properties and the encoding of problem constraints through quadratic penalty terms. Next, it explores the QAOA in detail, covering its Hamiltonian formulation, gate decomposition, and example applications, along with their implementation and performance results. This is followed by an analysis of the algorithm's energy landscape, where proofs are provided for its symmetry and periodicity, and where a resulting parameter space reduction is proposed. Finally, the tutorial extends these concepts to PUBO problems by generalizing the results to higher-order Hamiltonians and discussing the associated symmetries and circuit construction.