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Variational Quantum Algorithms for Combinatorial Optimization

Daniel F Perez-Ramirez

TL;DR

This work tackles combinatorial optimization on NISQ devices using Variational Quantum Algorithms, focusing on QAOA as a leading COP approach. It outlines the VQA framework, including cost functions, parameterized circuits, and classical optimization, and explains how COP can be mapped to quantum hardware via cost and mixer Hamiltonians. Through experiments on MaxCut with graphs of $n=10$ and $n=20$, the study shows QAOA can reach roughly 70–80% of the classical optimum on average for shallow circuits, with gradient-based optimization offering faster runtimes but not surpassing classical methods in these instances. The paper contributes a public codebase and data, and points to avenues for improving trainability, measurement efficiency, and scalability to strengthen quantum advantages in COP.

Abstract

The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.

Variational Quantum Algorithms for Combinatorial Optimization

TL;DR

This work tackles combinatorial optimization on NISQ devices using Variational Quantum Algorithms, focusing on QAOA as a leading COP approach. It outlines the VQA framework, including cost functions, parameterized circuits, and classical optimization, and explains how COP can be mapped to quantum hardware via cost and mixer Hamiltonians. Through experiments on MaxCut with graphs of and , the study shows QAOA can reach roughly 70–80% of the classical optimum on average for shallow circuits, with gradient-based optimization offering faster runtimes but not surpassing classical methods in these instances. The paper contributes a public codebase and data, and points to avenues for improving trainability, measurement efficiency, and scalability to strengthen quantum advantages in COP.

Abstract

The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.
Paper Structure (14 sections, 3 equations, 3 figures, 1 table)

This paper contains 14 sections, 3 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Variational Quantum Algorithms
  3. Components of a VQA
  4. Application of VQAs
  5. Challenges & Outlook of VQAs
  6. Quantum Approximate Optimization Algorithm (QAOA)
  7. Combinatorial Optimization
  8. Mapping Combinatorial Optimization for Quantum Computation
  9. QAOA
  10. QAOA for Max-Cut Problem
  11. Mapping MaxCut for QAOA
  12. Experimentation
  13. Results
  14. Conclusion

Figures (3)

  • Figure 1: Components of a Variational Quantum Algorithm.
  • Figure 2: When applying QAOA to find the MaxCut for a given graph, a tailored quantum circuit must be build to encode the cost function for that specific graph. Hence, any changes to the graph require a new circuit to be optimized.
  • Figure 3: Overview of results comparing (gradient-, and non-gradient based) QAOA against a classical method for 100 Erdős–Rényi graphs of 10 and 20 nodes. Vertical lines represent the standard deviation.