  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.
