Programmability of covariant quantum channels

TL;DR

This paper investigates the programmability of covariant quantum channels, demonstrating that symmetry properties enable finite-dimensional programmable processors and establishing bounds on their resources.

Contribution

It shows how group symmetries allow exact implementation of covariant channels with finite programs and derives minimal program dimensions using representation theory.

Findings

Exact implementation of covariant channels with finite programs via teleportation.

Derived minimal program register dimensions using symmetry group representations.

Provided bounds on program size for approximate implementation of all covariant channels.

Abstract

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.