  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.
