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The Better Solution Probability Metric: Optimizing QAOA to Outperform its Warm-Start Solution

Sean Feeney, Reuben Tate, Stephan Eidenbenz

TL;DR

The paper investigates Warm-Start QAOA for 3-regular Max-Cut, revealing that practical performance can exceed worst-case theoretical bounds and that standard parameter optimization often fails to beat the initial classical cut at $p=1$. It introduces Better Solution Probability (BSP) as a practical objective to optimize toward better-than-warm-start solutions, showing BSP advantages at non-trivial tilt angles and larger problem sizes. Through basin-hopping, local-max optimization, and parameter-region analyses, the study demonstrates both the potential and limitations of single-round Warm-Start QAOA, and highlights BSP as a promising direction for achieving quantum-assisted improvements in combinatorial optimization. The results motivate extending to higher circuit depths and broader problem classes to better quantify quantum advantage in near-term devices.

Abstract

This paper presents a numerical simulation investigation of the Warm-Start Quantum Approximate Optimization Algorithm (QAOA) as proposed by Tate et al. [1], focusing on its application to 3-regular Max-Cut problems. Our study demonstrates that Warm-Start QAOA consistently outperforms theoretical lower bounds on approximation ratios across various tilt angles, highlighting its potential in practical scenarios beyond worst-case predictions. Despite these improvements, Warm-Start QAOA with traditional parameters optimized for expectation value does not exceed the performance of the initial classical solution. To address this, we introduce an alternative parameter optimization objective, the Better Solution Probability (BSP) metric. Our results show that BSP-optimized Warm-Start QAOA identifies solutions at non-trivial tilt angles that are better than even the best classically found warm-start solutions with non-vanishing probabilities. These findings underscore the importance of both theoretical and empirical analyses in refining QAOA and exploring its potential for quantum advantage.

The Better Solution Probability Metric: Optimizing QAOA to Outperform its Warm-Start Solution

TL;DR

The paper investigates Warm-Start QAOA for 3-regular Max-Cut, revealing that practical performance can exceed worst-case theoretical bounds and that standard parameter optimization often fails to beat the initial classical cut at . It introduces Better Solution Probability (BSP) as a practical objective to optimize toward better-than-warm-start solutions, showing BSP advantages at non-trivial tilt angles and larger problem sizes. Through basin-hopping, local-max optimization, and parameter-region analyses, the study demonstrates both the potential and limitations of single-round Warm-Start QAOA, and highlights BSP as a promising direction for achieving quantum-assisted improvements in combinatorial optimization. The results motivate extending to higher circuit depths and broader problem classes to better quantify quantum advantage in near-term devices.

Abstract

This paper presents a numerical simulation investigation of the Warm-Start Quantum Approximate Optimization Algorithm (QAOA) as proposed by Tate et al. [1], focusing on its application to 3-regular Max-Cut problems. Our study demonstrates that Warm-Start QAOA consistently outperforms theoretical lower bounds on approximation ratios across various tilt angles, highlighting its potential in practical scenarios beyond worst-case predictions. Despite these improvements, Warm-Start QAOA with traditional parameters optimized for expectation value does not exceed the performance of the initial classical solution. To address this, we introduce an alternative parameter optimization objective, the Better Solution Probability (BSP) metric. Our results show that BSP-optimized Warm-Start QAOA identifies solutions at non-trivial tilt angles that are better than even the best classically found warm-start solutions with non-vanishing probabilities. These findings underscore the importance of both theoretical and empirical analyses in refining QAOA and exploring its potential for quantum advantage.
Paper Structure (17 sections, 9 equations, 10 figures)

This paper contains 17 sections, 9 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. QAOA
  5. Classical Max-Cut
  6. Approximation Ratio
  7. Warm-Start QAOA
  8. Experimental Setup
  9. Parameter Optimization
  10. Improving Warm-Start Parameter Initialization and Optimization
  11. A Novel Metric: Optimizing for Better Solution Probabilities
  12. Discussion and Conclusion
  13. Future Work
  14. Acknowledgments
  15. Appendix
  16. ...and 2 more sections

Figures (10)

  • Figure 1: The states $\ket{0_\theta}$ and $\ket{1_\theta}$ geometrically depicted on the Bloch sphere. The green half-circle, $\textbf{Arc}$, in the $xz$-plane denotes all the possible positions for $\ket{0_\theta}$ and $\ket{1_\theta}$ as $\theta$ varies from $0$ to $\pi$.
  • Figure 2:
  • Figure 3: Approximation ratio for a single instance of a randomly generated 3-regular graph, where 3 of the randomly generated bitstrings converge to the same locally optimized bitstring.
  • Figure 4: Average approximation ratio achieved with basin-hopping (red curve) vs numerical results for Tate's lower bounds with no further optimizations (green curve), and after local maximum optimization (blue curve) for each tilt angle $\theta$.
  • Figure 5:
  • ...and 5 more figures