  Port-based teleportation (PBT) is a variant of quantum teleportation that,
unlike the canonical protocol by Bennett et al., does not require a correction
operation on the teleported state. Since its introduction by Ishizaka and
Hiroshima in 2008, no efficient implementation of PBT was known. We close this
long-standing gap by building on our recent results on representations of
partially transposed permutation matrix algebras and mixed quantum Schur
transform. We construct efficient quantum algorithms for probabilistic and
deterministic PBT protocols on $n$ ports of arbitrary local dimension, both for
EPR and optimized resource states. We describe two constructions based on
different encodings of the Gelfand-Tsetlin basis for $n$ qudits: a standard
encoding that achieves $\widetilde{O}(n)$ time and $O(n\log(n))$ space
complexity, and a Yamanouchi encoding that achieves $\widetilde{O}(n^2)$ time
and $O(\log(n))$ space complexity, both for constant local dimension and target
error. We also describe efficient circuits for preparing the optimal resource
states.
