@inproceedings{
zhou2024statistical,
title={Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm},
author={Leo Zhou and Joao Basso and Song Mei},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=WTLvXdzhmP}
}
The quantum approximate optimization algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization that has been a promising avenue for near-term quantum advantage. 
In this paper, we analyze the performance of the QAOA on the spiked tensor model, a statistical estimation problem that exhibits a large computational-statistical 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 asymptotic classical computation threshold $\Theta(n^{(q-2)/4})$ for spiked $q$-tensors. 
Meanwhile, we characterize the asymptotic overlap distribution for $p$-step QAOA, discovering an intriguing sine-Gaussian law verified through simulations. For some $p$ and $q$, the QAOA has an effective recovery threshold that is a constant factor better than tensor power iteration.
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.
Statistical Estimation in the Spiked Tensor Model via the Quantum Approximate Optimization Algorithm
Leo Zhou Joao Basso Song Mei