  The one-clean qubit model (or the DQC1 model) is a restricted model of
quantum computing where only a single qubit of the initial state is pure and
others are maximally mixed. Although the model is not universal, it can
efficiently solve several problems whose classical efficient solutions are not
known. Furthermore, it was recently shown that if the one-clean qubit model is
classically efficiently simulated, the polynomial hierarchy collapses to the
second level. A disadvantage of the one-clean qubit model is, however, that the
clean qubit is too clean: for example, in realistic NMR experiments,
polarizations are not enough high to have the perfectly pure qubit. In this
paper, we consider a more realistic one-clean qubit model, where the clean
qubit is not clean, but depolarized. We first show that, for any polarization,
a multiplicative-error calculation of the output probability distribution of
the model is possible in a classical polynomial time if we take an
appropriately large multiplicative error. The result is in a strong contrast to
that of the ideal one-clean qubit model where the classical efficient
multiplicative-error calculation (or even the sampling) with the same amount of
error causes the collapse of the polynomial hierarchy. We next show that, for
any polarization lower-bounded by an inverse polynomial, a classical efficient
sampling (in terms of a sufficiently small multiplicative error or an
exponentially-small additive error) of the output probability distribution of
the model is impossible unless BQP is contained in the second level of the
polynomial hierarchy, which suggests the hardness of the classical efficient
simulation of the one non-clean qubit model.
