A unified approach to quantum de Finetti theorems and SoS rounding via geometric quantization

TL;DR

This paper establishes a unified framework connecting quantum de Finetti theorems and SoS hierarchy rounding through geometric quantization, leading to new proofs, applications, and bounds in quantum information and optimization.

Contribution

It introduces a novel approach linking quantum de Finetti theorems with sum-of-squares rounding via geometric quantization, providing new proofs and applications.

Findings

Recasting HSoS algorithms as quantization and eigenvector extraction.

Recovering quantum de Finetti theorems through Husimi distribution and quantization.

New proofs of de Finetti theorems and applications to quantum Max-$d$-Cut.

Abstract

The sum-of-squares hierarchy of semidefinite programs has become a common tool for algorithm design in theoretical computer science, including problems in quantum information. In this work we study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem, and geometric quantization of compact K\"ahler manifolds (such as complex projective space $\mathbb{C}P^{d}$, the set of all pure states in a $(d + 1)$-dimensional Hilbert space). We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function to obtain a finite-dimensional matrix, finding its top eigenvector, and then (possibly nonconstructively) rounding it by using a version of the Husimi quasiprobability distribution. Dually, we recover most known quantum de Finetti theorems by doing the same steps in the reverse order: a quantum state is first approximated by its Husimi distribution, and then quantized to obtain a separable state approximating the original one. In cases when there is a transitive group action on the manifold we give some new proofs of existing de Finetti theorems, as well as some applications including a new version of Renner's exponential de Finetti theorem proven using the Borel--Weil--Bott theorem, and hardness of approximation results and optimal degree-2 integrality gaps for the basic SDP relaxation of \textsc{Quantum Max-$d$-Cut} (for arbitrary $d$). We also describe how versions of these results can be proven when there is no transitive group action. In these cases we can deduce some error bounds for the HSoS hierarchy on complex projective varieties which are smooth.