Quantum Optimization Algorithms

TL;DR

This paper reviews quantum optimization algorithms, focusing on QAOA and VQE, discussing their implementation, constraint handling, and potential for exponential speedups in solving industrially relevant problems.

Contribution

It provides a comprehensive overview of QAOA and VQE, including implementation details, constraint incorporation, and practical demonstrations, advancing understanding of their application in quantum optimization.

Findings

QAOA can be implemented with Hamiltonian simulation techniques.

Constraints can be incorporated into QAOA using Grover mixers.

VQE is a promising approach for NISQ-era quantum computing.

Abstract

Quantum optimization allows for up to exponential quantum speedups for specific, possibly industrially relevant problems. As the key algorithm in this field, we motivate and discuss the Quantum Approximate Optimization Algorithm (QAOA), which can be understood as a slightly generalized version of Quantum Annealing for gate-based quantum computers. We delve into the quantum circuit implementation of the QAOA, including Hamiltonian simulation techniques for higher-order Ising models, and discuss parameter training using the parameter shift rule. An example implementation with Pennylane source code demonstrates practical application for the Maximum Cut problem. Further, we show how constraints can be incorporated into the QAOA using Grover mixers, allowing to restrict the search space to strictly valid solutions for specific problems. Finally, we outline the Variational Quantum Eigensolver (VQE) as a generalization of the QAOA, highlighting its potential in the NISQ era and addressing challenges such as barren plateaus and ansatz design.