We give asymptotically converging semidefinite programming hierarchies of
outer bounds on bilinear programs of the form $\mathrm{Tr}\big[M(X\otimes
Y)\big]$, maximized with respect to semidefinite constraints on $X$ and $Y$.
Applied to the problem of quantum error correction this gives hierarchies of
efficiently computable outer bounds on the optimal fidelity for any message
dimension and error model. The first level of our hierarchies corresponds to
the non-signalling assisted fidelity previously studied by [Leung & Matthews,
IEEE Trans.~Inf.~Theory 2015], and positive partial transpose constraints can
be added and used to give a sufficient criterion for the exact convergence at a
given level of the hierarchy. To quantify the worst case convergence speed of
our hierarchies, we derive novel quantum de Finetti theorems that allow
imposing linear constraints on the approximating state. In particular, we give
finite de Finetti theorems for quantum channels, quantifying closeness to the
convex hull of product channels as well as closeness to local operations and
classical forward communication assisted channels. As a special case this
constitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finetti
theorem for quantum channels. Finally, our proof methods also allow us to
answer an open question from [Brand\~ao & Harrow, STOC 2013] by improving the
approximation factor of de Finetti theorems with no symmetry from $O(d^{k/2})$
to $\mathrm{poly}(d,k)$, where $d$ denotes local dimension and $k$ the number
of copies.