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Deterministic quantum computation with one quantum bit (DQC1), or the oneclean qubit model, [E. Knill and R. Laflamme, Phys. Rev. Lett. {\bf81}, 5672(1998)] is a model of quantum computing where the input is the tensor productof a single pure qubit and many completely-mixed states, and only the singlequbit is measured at the end of the computation. In spite of its naiveappearance, the DQC1 model can efficiently solve some problems for which noclassical efficient algorithms are known, and therefore it has been conjecturedthat the DQC1 model is more powerful than classical computing (under theassumption of $\mbox{BPP}\subsetneq\mbox{BQP}$). Here we show that the outputprobability distribution of the DQC1 model cannot be classically efficientlyapproximated (exactly within a polynomial bit length or in the fully polynomialrandomized approximation scheme (FPRAS) with at most a constant error) unless$\mbox{BQP}\subseteq\mbox{BPP}$.

Classical simulatability of the one clean qubit model