Generalized group designs: constructing novel unitary 2-, 3- and 4-designs

TL;DR

This paper introduces new methods for constructing exact generalized group unitary designs, surpassing previous limitations and enabling the creation of 2-, 3-, and 4-designs in arbitrary dimensions, which are crucial for quantum information protocols.

Contribution

It presents novel construction techniques for exact generalized group unitary designs, overcoming the 4-design barrier and enabling designs in any dimension.

Findings

New construction methods for generalized group 4-designs

Construction of generalized group 2-designs in arbitrary dimensions

Overcoming previous dimensional limitations of group designs

Abstract

Unitary designs are essential tools in several quantum information protocols. Similarly to other design concepts, unitary designs are mainly used to facilitate averaging over a relevant space, in this case, the unitary group $\mathrm{U}(d)$. While it is known that exact unitary $t$-designs exist for any degree $t$ and dimension $d$, the most appealing type of designs, group designs (in which the elements of the design form a group), can provide at most $3$-designs. Moreover, even group $2$-designs can exist only in limited dimensions. In this paper, we present novel construction methods for creating exact generalized group designs based on the representation theory of the unitary group and its finite subgroups that overcome the $4$-design-barrier of unitary group designs. Furthermore, a construction is presented for creating generalized group $2$-designs in arbitrary dimensions.