  We present a construction method for complete sets of cyclic mutually
unbiased bases (MUBs) in Hilbert spaces of even prime power dimensions. In
comparison to usual complete sets of MUBs, complete cyclic sets possess the
additional property of being generated by a single unitary operator. The
construction method is based on the idea of obtaining a partition of
multi-qubit Pauli operators into maximal commuting sets of orthogonal operators
with the help of a suitable element of the Clifford group. As a consequence, we
explicitly obtain complete sets of cyclic MUBs generated by a single element of
the Clifford group in dimensions $2^m$ for $m=1,2,...,24$.
