Evidence that the Quantum Approximate Optimization Algorithm Optimizes the Sherrington-Kirkpatrick Model Efficiently in the Average Case

• URL: http://arxiv.org/abs/2505.07929v1
• Date: Mon, 12 May 2025 18:00:01 GMT
• Title: Evidence that the Quantum Approximate Optimization Algorithm Optimizes the Sherrington-Kirkpatrick Model Efficiently in the Average Case
• Authors: Sami Boulebnane, Abid Khan, Minzhao Liu, Jeffrey Larson, Dylan Herman, Ruslan Shaydulin, Marco Pistoia,
• Abstract summary: Sherrington-Kirkpatrick (SK) model serves as a foundational framework for understanding disordered systems.<n>Quantum Approximate Optimization Algorithm (QAOA) is a quantum optimization algorithm whose performance monotonically improves with its depth $p$.<n>We analyze QAOA applied to the SK model in the infinite-size limit and provide numerical evidence that it obtains a $(1-epsilon)$ approximation to the optimal energy with circuit depth $mathcalO(n/epsilon1.13)$ in the average case.
• Score: 3.4872784636892047
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Sherrington-Kirkpatrick (SK) model serves as a foundational framework for understanding disordered systems. The Quantum Approximate Optimization Algorithm (QAOA) is a quantum optimization algorithm whose performance monotonically improves with its depth $p$. We analyze QAOA applied to the SK model in the infinite-size limit and provide numerical evidence that it obtains a $(1-\epsilon)$ approximation to the optimal energy with circuit depth $\mathcal{O}(n/\epsilon^{1.13})$ in the average case. Our results are enabled by mapping the task of evaluating QAOA energy onto the task of simulating a spin-boson system, which we perform with modest cost using matrix product states. We optimize QAOA parameters and observe that QAOA achieves $\varepsilon\lesssim2.2\%$ at $p=160$ in the infinite-size limit. We then use these optimized QAOA parameters to evaluate the QAOA energy for finite-sized instances with up to $30$ qubits and find convergence to the ground state consistent with the infinite-size limit prediction. Our results provide strong numerical evidence that QAOA can efficiently approximate the ground state of the SK model in the average case.

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