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

arXiv: https://arxiv.org/abs/2402.19456 · HTML: https://mlsanity.com/paper/2402.19456/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Feb 2024.

## Built on by (most cited first, 1 of 1)

- [Towards Quantum Tensor Decomposition in Biomedical Applications](https://mlsanity.com/paper/2502.13140/) (2025)

## Abstract (arXiv)

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.