Quasi-polynomial time algorithms for free quantum games in bounded dimension

TL;DR

This paper introduces a quasi-polynomial time semidefinite programming hierarchy to approximate quantum correlations in fixed-dimension free games, improving computational efficiency over previous methods.

Contribution

It develops a converging hierarchy with analytical bounds, enabling more efficient approximations of quantum game values in fixed dimensions.

Findings

Polynomial scaling in the number of questions Q

Quasi-polynomial scaling in answers A for fixed dimension T

Improved approximation algorithms with better runtime guarantees

Abstract

We give a converging semidefinite programming hierarchy of outer approximations for the set of quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. In particular, we give a semidefinite program of size $\exp(\mathcal{O}\big(T^{12}(\log^2(AT)+\log(Q)\log(AT))/\epsilon^2\big))$ to compute additive $\epsilon$-approximations on the values of two-player free games with $T\times T$-dimensional quantum assistance, where $A$ and $Q$ denote the numbers of answers and questions of the game, respectively. For fixed dimension $T$, this scales polynomially in $Q$ and quasi-polynomially in $A$, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in $Q$ and $A$. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints. We also derive an informationally complete measurement which minimises the loss in distinguishability relative to the quantum side information - which may be of independent interest.