Moment-sos and spectral hierarchies for polynomial optimization on the sphere and quantum de Finetti theorems

We revisit the convergence analysis of two approximation hierarchies for polynomial optimization on the unit sphere. The first one is based on the moment-sos approach and gives semidefinite bounds for which Fang and Fawzi (2021) showed an analysis in $O(1/r^2)$ for the r-th level bound, using the polynomial kernel method. The second hierarchy was recently proposed by Lovitz and Johnston (2023) and gives spectral bounds for which they show a convergence rate in $O(1/r)$, using a quantum de Finetti theorem of Christandl et al. (2007) that applies to complex Hermitian matrices with a “double” symmetry. We investigate links between these approaches, in particular, via duality of moments and sums of squares. Our main results include showing that the spectral bounds cannot have a convergence rate better than $O(1/r^2)$ and that they do not enjoy generic finite convergence. In addition, we propose alternative performance analyses that involve explicit constants depending on intrinsic parameters of the optimization problem. For this we develop a novel “banded” real de Finetti theorem that applies to real matrices with “double” symmetry. We also show how to use the polynomial kernel method to obtain a de Finetti type result in $O(1/r^2)$ for real maximally symmetric matrices, improving an earlier result in $O(1/r)$ of Doherty and Wehner (2012).

💡 Research Summary

The paper studies two hierarchical approximation schemes for minimizing a homogeneous polynomial $p(x)$ of degree $2d$ over the unit sphere $S^{n-1}$: the classical moment‑SOS hierarchy and the recently introduced spectral hierarchy. The moment‑SOS hierarchy replaces the non‑negativity condition on $p-\lambda$ with a sum‑of‑squares (SOS) representation of degree $2r$, leading to a semidefinite program (SDP). Fang and Fawzi (2021) showed that the gap $p_{\min}-\mathrm{sos}_r(p)$ decays as $O(1/r^{2})$ by employing an optimized polynomial kernel $K_r(x,y)=(x^{\top}y)^{2r}$ and constructing a feasible SOS polynomial directly from the kernel operator.

The spectral hierarchy, proposed by Lovitz and Johnston (2023), rewrites $p-\lambda$ as a matrix that is doubly symmetric (invariant under permutations of rows and columns). The smallest eigenvalue of this matrix yields a lower bound $\mathrm{spr}r(p)$. Their analysis relies on a finite quantum de Finetti theorem for complex Hermitian matrices with double symmetry (Christandl et al., 2007), which approximates the partial trace of such a matrix by a convex combination of product states with error $O(1/r)$. Consequently they obtain $p{\min}-\mathrm{spr}_r(p)=O(1/r)$.

The authors investigate the relationship between these two approaches. First, they prove that the $O(1/r)$ rate for the spectral hierarchy cannot be improved in general: by constructing a specific 5‑variable quartic form derived from the Choi‑Lam example (known to be non‑SOS but non‑negative), they show a lower bound $p_{\min}-\mathrm{spr}_r(p)=\Omega(1/r^{2})$. This demonstrates that the spectral hierarchy’s convergence is fundamentally limited to at best $O(1/r^{2})$ and that the previously known $O(1/r)$ bound is not tight. Second, they show that the spectral hierarchy does not enjoy generic finite convergence; for almost all polynomials the hierarchy never reaches the exact optimum in a finite number of steps (Theorem 5.14).

A major contribution is a new “banded” real quantum de Finetti theorem for real doubly symmetric positive semidefinite matrices. This result interpolates between the complex de Finetti theorem (applicable to Hermitian matrices) and the real maximally symmetric de Finetti theorem (requiring a stronger symmetry). The banded theorem approximates the symmetrized partial trace of a real doubly symmetric matrix by a scaled separable matrix with error $O(1/r)$. Using this theorem, the authors re‑derive the $O(1/r)$ convergence of the spectral hierarchy without passing to the complex sphere, and they obtain explicit error constants that depend on the polynomial’s value range $p_{\max}-p_{\min}$ and on the dimensions $n$ and degree $d$, rather than on the opaque condition number $\kappa(M)$ used in the original analysis.

Furthermore, they improve the real quantum de Finetti theorem of Doherty and Wehner (2012) from $O(1/r)$ to $O(1/r^{2})$ by applying the polynomial kernel method on the dual (moment) side. This yields tighter bounds for the SOS hierarchy as well.

In summary, the paper establishes a duality link between the moment‑SOS and spectral hierarchies, clarifies the limitations of the spectral approach, introduces a novel banded de Finetti theorem for real matrices, and provides more transparent convergence analyses with explicit constants. These results deepen the theoretical connection between polynomial optimization on the sphere, semidefinite relaxations, and quantum de Finetti theorems, and they open avenues for further research on optimal convergence rates and lower bounds for both hierarchies.