# An Efficient Quantum Algorithm for Port-based Teleportation

- Sydney Timmerman, Stanford University
- Jiani Fei, Stanford University

In this talk, we will outline an 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 usin NLQC. Similarly, our algorithm provides the first nontrivial efficient algorithm for an approximate universal programmable quantum processor.

The key to our approach is a general quantum algorithm we develop for block diagonalizing so-called generalized induced representations, a novel type of representation that arises from lifting a representation of a subgroup to one for the whole group while relaxing a linear independence condition from the standard definition. Generalized induced representations appear naturally in quantum information, notably in generalizations of Schur-Weyl duality. For the case of port-based teleportation, we apply this framework to develop an efficient twisted Schur transform for transforming to a subgroup-reduced irrep basis of the partially transposed permutation algebra, whose dual is the U⊗n−k ⊗ (U∗) ⊗k representation of the unitary group.

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