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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 ≥ 4 if d ≥ 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.

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