# On the explicit constructions of certain unitary $t$-designs

## TL;DR

This paper presents explicit constructions of certain unitary t-designs in quantum information science, including new constructions of 3-designs in U(3) and 4-designs in U(4), addressing open problems in the field.

## Contribution

The authors establish a method to construct higher-order unitary t-designs from specific unitary t-groups, providing explicit examples in U(3) and U(4).

## Key findings

- Constructed exact 3-designs in U(3) from the unitary 2-group SL(3,2).
- Constructed exact 4-designs in U(4) from the unitary 3-group Sp(4,3).
- Provided numerical evidence and discussed related open problems.

## Abstract

Unitary $t$-designs are `good' finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $t\geq 4$ if $d\geq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.