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

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 $t\ge 4$ if $d\ge 3$, 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.

arXiv comment17 pages, 1 figure

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Cited by in corpus (8)

• Quantum circuits for exact unitary $t$-designs and applications to higher-order randomized benchmarking
• Efficient Unitary T-designs from Random Sums
• Efficient unitary designs and pseudorandom unitaries from permutations
• Quantum Codes from Twisted Unitary $t$-groups
• Unitary $t$-designs from $relaxed$ seeds
• Randomized measurements for multi-parameter quantum metrology
• A quantum tug of war between randomness and symmetries on homogeneous spaces
• Quantum Codes and Irreducible Products of Characters