  A unitary $t$-design is a powerful tool in quantum information science and
fundamental physics. Despite its usefulness, only approximate implementations
were known for general $t$. In this paper, we provide for the first time
quantum circuits that generate exact unitary $t$-designs for any $t$ on an
arbitrary number of qubits. Our construction is inductive and is of practical
use in small systems. We then introduce a $t$-th order generalization of
randomized benchmarking ($t$-RB) as an application of exact $2t$-designs. We
particularly study the $2$-RB in detail and show that it reveals
self-adjointness of quantum noise, a new metric related to the feasibility of
quantum error correction (QEC). We numerically demonstrate that the $2$-RB in
one- and two-qubit systems is feasible, and experimentally characterize
background noise of a superconducting qubit by the $2$-RB. It is shown from the
experiment that interactions with adjacent qubits induce the noise that may
result in an obstacle toward the realization of QEC.
