# The sum-of-squares hierarchy on the sphere, and applications in quantum information theory

arXiv: https://arxiv.org/abs/1908.05155 · HTML: https://mlsanity.com/paper/1908.05155/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Aug 2019.

## Built on by (most cited first, 3 of 3)

- [An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization](https://mlsanity.com/paper/2105.14033/) (2021)
- [Sum-of-Squares Relaxations for Information Theory and Variational Inference](https://mlsanity.com/paper/2206.13285/) (2022)
- [Approximate Real Symmetric Tensor Rank](https://mlsanity.com/paper/2207.12529/) (2022)

## Abstract (arXiv)

We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of Sum-of-Squares (SOS) relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty & Wehner. Specifically, we show that the rate of convergence is no worse than O(d2/ℓ2) in the regime ℓ ≥ Omega(d) where ℓ is the level of the hierarchy and d the dimension, solving a problem left open in the recent paper by de Klerk & Laurent (arXiv:1904.08828). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the DPS hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascu'es, Owari & Plenio.