Fast controlled unitary protocols using group or quasigroup structures

TL;DR

This paper introduces two fast protocols for implementing controlled unitaries using group and quasigroup algebraic structures, reducing classical communication time and entanglement cost, with applications to quantum information processing.

Contribution

It presents novel fast protocols leveraging group and quasigroup structures for efficient implementation of controlled unitaries, improving over existing methods.

Findings

The first protocol implements certain controlled unitaries exactly with one round of communication.

The second protocol approximates any controlled unitary with lower entanglement cost.

The second protocol can incorporate shared classical randomness to enhance performance.

Abstract

A nonlocal bipartite unitary gate can sometimes be implemented using prior entanglement and only one round of classical communication in which the two parties send messages to each other simultaneously. This cuts the classical communication time by a half compared to the usual protocols, which require back-and-forth classical communication. We introduce such a "fast" protocol that can implement a class of controlled unitaries exactly, where the controlled operators form a subset of a projective representation of a finite group, which may be Abelian or non-Abelian. The entanglement cost is only related to the size of the group and is independent of the dimension of the systems. We also introduce a second fast protocol that can implement any given controlled unitary approximately. This protocol uses the algebraic structure of right quasigroups, which are generalizations of quasigroups, the latter being equivalent to Latin squares. This second protocol could optionally use shared classical randomness as a resource, in addition to using entanglement. When compared with other known fast unitary protocols, the entanglement cost of this second protocol is lower for general controlled unitaries except for some rare cases.