  Randomized benchmarking is a useful scheme for evaluation the average
fidelity of a noisy quantum circuit. However, it is insensitive to the unitary
error. Here, we propose a method of randomized benchmarking in which a unitary
t-design is applied and by which the unitary error estimation can be converted
to analysis of pseudo-randomness on a set of unitary operators. We give a bound
on the number of randomized benchmarking sequences, when performing a unitary
t-design on n-qubit d-dimensional system. By applying local random unitary
operators, a decomposition of a unitary t-design, the bound is more practical
than the previous bound for multi-qubit circuit. We also give a rigorous bound
of a diamond norm between arbitrary and uniform distributions of a set of
unitary operators to form an \epsilon-approximate unitary t-design. It can be
used to quantitatively analyze the corresponding average fidelity and errors in
a large-scale quantum circuit.
