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We give asymptotically converging semidefinite programming hierarchies ofouter bounds on bilinear programs of the form $\mathrm{Tr}\big[M(X\otimesY)\big]$, maximized with respect to semidefinite constraints on $X$ and $Y$.Applied to the problem of quantum error correction this gives hierarchies ofefficiently computable outer bounds on the optimal fidelity for any messagedimension and error model. The first level of our hierarchies corresponds tothe non-signalling assisted fidelity previously studied by [Leung & Matthews,IEEE Trans.~Inf.~Theory 2015], and positive partial transpose constraints canbe added and used to give a sufficient criterion for the exact convergence at agiven level of the hierarchy. To quantify the worst case convergence speed ofour hierarchies, we derive novel quantum de Finetti theorems that allowimposing linear constraints on the approximating state. In particular, we givefinite de Finetti theorems for quantum channels, quantifying closeness to theconvex hull of product channels as well as closeness to local operations andclassical forward communication assisted channels. As a special case thisconstitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finettitheorem for quantum channels. Finally, our proof methods also allow us toanswer an open question from [Brand\~ao & Harrow, STOC 2013] by improving theapproximation 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 numberof copies.

Semidefinite programming hierarchies for constrained bilinear optimization