  Quantum teleportation is one of the fundamental building blocks of quantum
Shannon theory. While ordinary teleportation is simple and efficient,
port-based teleportation (PBT) enables applications such as universal
programmable quantum processors, instantaneous non-local quantum computation
and attacks on position-based quantum cryptography. In this work, we determine
the fundamental limit on the performance of PBT: for arbitrary fixed input
dimension and a large number $N$ of ports, the error of the optimal protocol is
proportional to the inverse square of $N$. We prove this by deriving an
achievability bound, obtained by relating the corresponding optimization
problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered
simplex. We also give an improved converse bound of matching order in the
number of ports. In addition, we determine the leading-order asymptotics of PBT
variants defined in terms of maximally entangled resource states. The proofs of
these results rely on connecting recently-derived representation-theoretic
formulas to random matrix theory. Along the way, we refine a convergence result
for the fluctuations of the Schur-Weyl distribution by Johansson, which might
be of independent interest.
