Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm
<|reference_start|>Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm: 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.<|reference_end|>
@article{zhou2024statistical,
    title={Statistical Estimation in the Spiked Tensor Model via the Quantum
  Approximate Optimization Algorithm},
    author={Leo Zhou, Joao Basso, Song Mei},
    journal={arXiv preprint arXiv:2402.19456},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.19456},
    primaryClass={quant-ph cs.DS math.PR math.ST stat.ML stat.TH}
}