  The deterministic quantum computing with one qubit (DQC1) is a mixed-state
quantum computation algorithm that evaluates the normalized trace of a unitary
matrix and is more powerful than the classical counterpart. We find that the
normalized trace of the unitary matrix can be directly described by the
entangling power of the quantum circuit of the DQC1, so the nontrivial DQC1 is
always accompanied with the non-vanishing entangling power. In addition, it is
shown that the entangling power also determines the intrinsic complexity of
this quantum computation algorithm, i.e., the larger entangling power
corresponds to higher complexity. Besides, it is also shown that the
non-vanishing entangling power does always exist in other similar tasks of
DQC1.
