[Paper Review] A Review on Quantum Approximate Optimization Algorithm and its Variants

Kostas Blekos, Dean Brand+5 more|arXiv (Cornell University)|Jun 15, 2023

Quantum Computing Algorithms and Architecture261 references12 citations

TL;DR

This paper surveys the Quantum Approximate Optimization Algorithm (QAOA) and its variants, analyzing performance, hardware challenges, and practical guidance for use on NISQ devices.

ABSTRACT

The Quantum Approximate Optimization Algorithm (QAOA) is a highly promising variational quantum algorithm that aims to solve combinatorial optimization problems that are classically intractable. This comprehensive review offers an overview of the current state of QAOA, encompassing its performance analysis in diverse scenarios, its applicability across various problem instances, and considerations of hardware-specific challenges such as error susceptibility and noise resilience. Additionally, we conduct a comparative study of selected QAOA extensions and variants, while exploring future prospects and directions for the algorithm. We aim to provide insights into key questions about the algorithm, such as whether it can outperform classical algorithms and under what circumstances it should be used. Towards this goal, we offer specific practical points in a form of a short guide. Keywords: Quantum Approximate Optimization Algorithm (QAOA), Variational Quantum Algorithms (VQAs), Quantum Optimization, Combinatorial Optimization Problems, NISQ Algorithms

Motivation & Objective

• Assess the current state of QAOA across problem instances and hardware.
• Compare selected QAOA extensions and variants.
• Analyze factors affecting performance, error resilience, and resource needs.
• Provide practical guidance on when and how to use QAOA for combinatorial optimization.

Proposed method

• Perform an extensive literature survey of QAOA and its variants.
• Conduct a comparative study of selected QAOA extensions on MaxCut.
• Analyze parameter optimization, noise effects, and hardware considerations.
• Synthesize experimental results and practical recommendations.
• Discuss open questions and future directions for QAOA.

Figure 1: Left: A problem graph with 6 vertices and 11 equal-weight edges. Right: The solution to the MaxCut problem, where the vertices are partitioned into two groups (red and blue) such that the number of edges crossed by the cut (black curve) is maximized, which is 8.

Experimental results

Research questions

• RQ1 Under what circumstances can QAOA outperform classical algorithms for combinatorial optimization?
• RQ2 Which QAOA variants or ansatz structures are most effective for a given problem class (e.g., MaxCut) and problem size?
• RQ3 How do noise, hardware constraints, and barren plateaus affect the potential quantum advantage of QAOA?
• RQ4 What practical guidelines optimize parameter selection and implementation on NISQ devices?

Key findings

• QAOA variants exist that adapt the ansatz and optimization strategies for improved performance.
• The algorithm's advantage depends on problem instance characteristics and hardware quality, with evidence subject to noise and limitations.
• Parameter optimization, barren plateaus, and reusability of parameters are central challenges across variants.
• Hardware-specific approaches and noise mitigation techniques are essential for achieving any practical quantum advantage.
• The paper provides a practical guide answering which variant to use and how to optimize parameters for MaxCut and related problems.

Figure 2: Implementation of the elements of mixer (left) and cost (right) layers based on the cost and mixer Hamiltonians, $\hat{H}_{C}$ and $\hat{H}_{M}$ . By $\big{(}e^{-i\beta_{k}\hat{H}_{M}}\big{)}_{v_{i}}\eqqcolon\big{(}\hat{U}_{M}(\beta_{k})\big{)}_{v_{i}}$ we mean the element of $\hat{U}_{M}(