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Variational and Annealing-Based Approaches to Quantum Combinatorial Optimization

Hala Hawashin, Deep Nath, Marco Alberto Javarone

Abstract

In this work, we review quantum approaches to combinatorial optimization, with the aim of bridging theoretical developments and industrial relevance. We first survey the main families of quantum algorithms, including Quantum Annealing, the Quantum Approximate Optimization Algorithm (QAOA), Quantum Reinforcement Learning (QRL), and Quantum Generative Modeling (QGM). We then examine the problem classes where quantum technologies currently show evidence of quantum advantage, drawing on established benchmarking initiatives such as QOBLIB, QUARK, QASMBench, and QED-C. These problem classes are subsequently mapped to representative industrial domains, including logistics, finance, and telecommunications. Our analysis indicates that quantum annealing currently exhibits the highest level of operational maturity, while QAOA shows promising potential on NISQ-era hardware. In contrast, QRL and QGM emerge as longer-term research directions with significant potential for future industrial impact.

Variational and Annealing-Based Approaches to Quantum Combinatorial Optimization

Abstract

In this work, we review quantum approaches to combinatorial optimization, with the aim of bridging theoretical developments and industrial relevance. We first survey the main families of quantum algorithms, including Quantum Annealing, the Quantum Approximate Optimization Algorithm (QAOA), Quantum Reinforcement Learning (QRL), and Quantum Generative Modeling (QGM). We then examine the problem classes where quantum technologies currently show evidence of quantum advantage, drawing on established benchmarking initiatives such as QOBLIB, QUARK, QASMBench, and QED-C. These problem classes are subsequently mapped to representative industrial domains, including logistics, finance, and telecommunications. Our analysis indicates that quantum annealing currently exhibits the highest level of operational maturity, while QAOA shows promising potential on NISQ-era hardware. In contrast, QRL and QGM emerge as longer-term research directions with significant potential for future industrial impact.
Paper Structure (37 sections, 9 equations, 6 figures, 4 tables)

This paper contains 37 sections, 9 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Quantum Background and Fundamentals
  3. Time-Complexity and quantum advantage over classical
  4. Quantum Methods for Optimization-based Problems
  5. Direct Cost Minimisation
  6. Annealing
  7. QAOA
  8. $H_C(\gamma)$ and $H_M(\beta)$.
  9. $p$-layers
  10. $2p$ parameters: $(\gamma_i,\beta_i)$
  11. Return Maximisation
  12. Likelihood Maximisation
  13. Quantum Variational Models.
  14. Quantum Flow-based Models.
  15. Quantum Diffusion Models.
  16. ...and 22 more sections

Figures (6)

  • Figure 1: Workflow of variational quantum circuit.
  • Figure 2: Time complexity, solution quality associated with different complexity classes of optimization problems. Here, $O(Poly(n))$ signifies polynomial time complexity in input size, $n$.
  • Figure 3: The hierarchical diagram outlines the relationships between quantum optimization methods, which are broadly divided into gate-based and analogue-based approaches. Gate-based methods include fault-tolerant quantum computing (FTQC), shown in grey and excluded as out of scope, and NISQ approaches, which are more practical given current hardware constraints. Analogue-based methods primarily encompass quantum annealing. At the base of each branch is the corresponding objective function adopted by the method. These approaches are categorized by their optimization goals: minimizing a cost function (direct cost minimization), maximizing expected outcomes (return maximization), or increasing the likelihood of observed data (likelihood maximization).
  • Figure 4: Eigen-energy spectrum illustrating quantum annealing dynamics. The ground and excited state energies $E_0(s)$ and $E_1(s)$ evolve as the annealing parameter $s$ varies. The minimum gap $\Delta_{\min}$ marks the point of closest approach between the two lowest eigenstates, which constrains the required annealing rate to maintain adiabatic evolution.
  • Figure 5: Schematic representation of the agent–environment interaction in reinforcement learning.
  • ...and 1 more figures