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

29 pages; v2: improved presentation, minor errors corrected, references updated, conclusions extended; v3: minor errors corrected, template changed; v4: title changed, appendices updated, 2-design construction simplified, figure removed, new theorems added