Near-Optimal Parameter Tuning of Level-1 QAOA for Ising Models

2025-01-27

AI Summary

For QAOA₁ applied to Ising models or QUBO problems, the two-dimensional (γ, β) cost landscape exhibits oscillatory behavior, causing gradient-based parameter optimization to frequently converge to spurious local minima. Method: We propose a γ-dominant one-dimensional analytical dimensionality reduction: leveraging a closed-form expression for the optimal β as a function of γ, we rigorously prove that, on regular graphs, the optimal γ* concentrates near zero—enabling efficient gradient-based optimization in lieu of costly brute-force grid search—and derive a theoretical upper bound on the maximum sampling period ensuring distortion-free landscape reconstruction. Contribution/Results: Our polynomial-time parameter estimation algorithm, integrated into the Recursive QAOA (RQAOA) framework, consistently outperforms both standard RQAOA and semidefinite programming relaxations across diverse QUBO instances, delivering substantially improved solution quality and robustness. Empirical results validate both the tightness of our theoretical bound and the practical efficacy of the proposed method.

Technology Category

• Search and Optimization: Local Search
• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Quantum Approximate Optimisation Algorithm (QAOA) is a hybrid quantum-classical algorithm for solving combinatorial optimisation problems. QAOA encodes solutions into the ground state of a Hamiltonian, approximated by a $p$-level parameterised quantum circuit composed of problem and mixer Hamiltonians, with parameters optimised classically. While deeper QAOA circuits can offer greater accuracy, practical applications are constrained by complex parameter optimisation and physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement errors, limiting implementations to shallow depths. This work focuses on QAOA$_1$ (QAOA at $p=1$) for QUBO problems, represented as Ising models. Despite QAOA$_1$ having only two parameters, $(gamma, eta)$, we show that their optimisation is challenging due to a highly oscillatory landscape, with oscillation rates increasing with the problem size, density, and weight. This behaviour necessitates high-resolution grid searches to avoid distortion of cost landscapes that may result in inaccurate minima. We propose an efficient optimisation strategy that reduces the two-dimensional $(gamma, eta)$ search to a one-dimensional search over $gamma$, with $eta^*$ computed analytically. We establish the maximum permissible sampling period required to accurately map the $gamma$ landscape and provide an algorithm to estimate the optimal parameters in polynomial time. Furthermore, we rigorously prove that for regular graphs on average, the globally optimal $gamma^* in mathbb{R}^+$ values are concentrated very close to zero and coincide with the first local optimum, enabling gradient descent to replace exhaustive line searches. This approach is validated using Recursive QAOA (RQAOA), where it consistently outperforms both coarsely optimised RQAOA and semidefinite programs across all tested QUBO instances.

Problem

• Ising Model
• QAOA_1
• Parameter Optimization

Innovation

• QAOA Optimization
• Ising Model
• Parameter Simplification