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.