Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm

• URL: http://arxiv.org/abs/2402.19456v1
• Date: Thu, 29 Feb 2024 18:50:28 GMT
• Title: Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm
• Authors: Leo Zhou, Joao Basso, Song Mei
• Abstract summary: The quantum approximate optimization algorithm (QAOA) is a general-purpose algorithm for optimization. We prove that the weak recovery threshold of $1$-step QAOA matches that of $1$-step tensor power iteration. For some $p$ and $q$, the QAOA attains an overlap that is larger by a constant factor than the tensor power overlap.
• Score: 17.955614278088238
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The quantum approximate optimization algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization. In this paper, we analyze the performance of the QAOA on a statistical estimation problem, namely, the spiked tensor model, which exhibits a statistical-computational gap classically. We prove that the weak recovery threshold of $1$-step QAOA matches that of $1$-step tensor power iteration. Additional heuristic calculations suggest that the weak recovery threshold of $p$-step QAOA matches that of $p$-step tensor power iteration when $p$ is a fixed constant. This further implies that multi-step QAOA with tensor unfolding could achieve, but not surpass, the classical computation threshold $\Theta(n^{(q-2)/4})$ for spiked $q$-tensors. Meanwhile, we characterize the asymptotic overlap distribution for $p$-step QAOA, finding an intriguing sine-Gaussian law verified through simulations. For some $p$ and $q$, the QAOA attains an overlap that is larger by a constant factor than the tensor power iteration overlap. Of independent interest, our proof techniques employ the Fourier transform to handle difficult combinatorial sums, a novel approach differing from prior QAOA analyses on spin-glass models without planted structure.

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