Efficient Quantum Algorithm for Port-based Teleportation

TL;DR

This paper introduces an efficient quantum algorithm for port-based teleportation, significantly improving the relationship between entanglement and nonlocal unitary implementation, with implications for quantum processing and AdS/CFT.

Contribution

It presents the first efficient algorithm for port-based teleportation using twisted Schur-Weyl duality and the twisted Schur transform, advancing quantum teleportation and programmable processors.

Findings

Exponential improvement in entanglement vs. complexity relationship.

First efficient approximate universal programmable quantum processor.

Enhanced capabilities for nonlocal quantum computation.

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.