Efficient quantum circuits for port-based teleportation

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 describe efficient quantum circuits 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 $O(n\mathrm{log}(n))$ space complexity, and a Yamanouchi encoding that achieves $\widetilde{O}(n^2)$ time and $O(\mathrm{log}(n))$ space complexity, both for constant local dimension and target error. We also describe efficient circuits for preparing the optimal resource states.