On the explicit constructions of certain unitary t-designs

Unitary t-designs are 'good' finite subsets of the unitary group that approximate the whole unitary group 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 exist for any t and d, the unitary t-groups do not exist for if , as it is shown by Guralnick–Tiep (Guralnick and Tiep 2005 Represent. Theory 9 138–208) and Bannai–Navarro–Rizo–Tiep (Bannai et al 2018 J. Math. Soc. Japan (accepted)). Explicit constructions of exact unitary t-designs in are not easy in general. In particular, explicit constructions of unitary 4-designs in have been an open problem in quantum information theory. We prove that some exact unitary -designs in the unitary group are constructed from unitary t-groups in that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary 3-designs in from the unitary 2-group in and also unitary 4-designs in from the unitary 3-group in numerically. We also discuss some related problems and a few applications to physics problems. (paper)