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

TL;DR

The paper investigates how fixed-depth QAOA performs on the spiked tensor model, a canonical statistical inference problem with a known computational-statistical gap. By analyzing the overlap between QAOA outputs and the planted signal, the authors derive weak-recovery thresholds that match classical tensor power iteration thresholds for 1-step and general p-step QAOA, with a quantum-augmented overlap distribution described by a sine-Gaussian law. They further show that tensor unfolding can reach the classical polynomial-time threshold Θ(n^{(q−2)/4}) if depth grows, and demonstrate potential constant-factor quantum advantages in overlap for certain (p,q) pairs. The work introduces Fourier-transform-based techniques to handle challenging combinatorial sums in QAOA analyses and provides numerical evidence supporting the theoretical predictions, contributing a first rigorous analytical project of QAOA in a planted-inference setting. Overall, it highlights qualitative differences in QAOA outputs and clarifies when quantum approaches may offer practical gains in statistical estimation tasks. The results emphasize that substantial quantum speedups at fixed depth are unlikely to surpass classical thresholds, unless depth scales with problem size, while offering new tools for analyzing quantum algorithms in structured inference problems.

Abstract

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 $Θ(n^{(q-2)/4})$ 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.

Figures (4)

• Figure 1: Different thresholds for the spiked tensor model.
• Figure 2: (a) Example overlap distribution from 1-step QAOA for the spiked matrix model ($q=2$), where simulation data is collected from 40 random generated instances with $n=26$ bits. The signal-to-noise ratio is chosen to be $\lambda_n=n^{1/2}$, and $(\gamma,\beta)=(\sqrt{\ln5/32},\pi/4)$. Dash gray lines connect data from the same instance. (b) Average of squared overlap $\braket{\mathcal{R}^2_\textnormal{QAOA}}_{\gamma,\beta}$ from the QAOA output distribution for 40 random instances generated at various problem dimensions.
• Figure 3: Example overlap distributions from $p$-step QAOA for the spiked tensor model for $1\le p \le 5$. The top row shows data from 40 random 26-bit instances with $q=2$ and $\lambda_n=n^{1/(2p)}$. The bottom row shows data from 40 random 23-bit instances with $q=3$ and $\lambda_n = n^{[1+1/(2^p-1)]/2}$. Different columns correspond to different $p$, using the QAOA parameters (${\boldsymbol{\gamma}},{\boldsymbol{\beta}}$) that optimized $|a_p b_p|^{\varepsilon_p}$ in Table \ref{['tab:norm-enhancement']}. Dash gray lines connect data from the same instance. Blue histograms are the theoretical sine-Gaussian distributions in the $n\to\infty$ limit, where $\mathcal{R}_\textnormal{QAOA} \sim a_p \sin[b_p G^{(q-1)^p}]$ according to Claim \ref{['cl:general-p']}. (Note here $\Lambda=1$.)
• Figure 4: Log-log plots of the difference between observed overlap (averaged over instances and quantum measurements) at various problem dimension $n$ and the predicted value from the sine-Gaussian law in the $n\to\infty$ limit. Different colored lines correspond to different QAOA depth $p$, with parameters $({\boldsymbol{\gamma}},{\boldsymbol{\beta}})$ set to be the same as in Table \ref{['tab:norm-enhancement']}. We choose $\Lambda = 0.2$, $\lambda_n = \Lambda n^{1/(2p)}$ (left), and $\lambda_n = \Lambda n^{[1 + 1/(2^p-1)]/2}$ (right).

Theorems & Definitions (31)

• Theorem 1: Weak recovery threshold and overlap distribution for $1$-step QAOA
• Remark 3.1: Weak recovery threshold
• Remark 3.2: Overlap distribution
• Proposition 3.3: Weak recovery threshold for $1$-step tensor power iteration
• Remark 3.4: Comparing the overlaps
• Remark 3.5: Rounding via $\mathop{\mathrm{sign}}\nolimits(\hat{\boldsymbol{u}})$ will not improve the overlap
• Remark 3.6: Sine-Gaussian law versus sine-arctan-Gaussian law
• Claim 3.7: $p$-step QAOA for weak recovery
• Remark 3.8: Weak recovery threshold
• Proposition 3.9: Corollary of Lemma 3.2 of wu2024sharp