Efficient Quantum Algorithm for Port-based Teleportation

Abstract: In this paper, we provide the first efficient algorithm for port-based teleportation, a unitarily equivariant version of teleportation useful for constructing programmable quantum processors and performing instantaneous nonlocal computation (NLQC). The latter connection is important in AdS/CFT, where bulk computations are realized as boundary NLQC. Our algorithm yields an exponential improvement to the known relationship between the amount of entanglement available and the complexity of the nonlocal part of any unitary that can be implemented using NLQC. Similarly, our algorithm provides the first nontrivial efficient algorithm for an approximate universal programmable quantum processor. The key to our approach is a generalization of Schur-Weyl duality we call twisted Schur-Weyl duality, as well as an efficient algorithm we develop for the twisted Schur transform, which transforms to a subgroup-reduced irrep basis of the partially transposed permutation algebra, whose dual is the $U^{\otimes n-k} \otimes (U^*)^{\otimes k}$ representation of the unitary group.