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Classical simulatability of the one clean qubit model

Deterministic quantum computation with one quantum bit (DQC1), or the one clean 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 product of a single pure qubit and many completely-mixed states, and only the single qubit is measured at the end of the computation. In spite of its naive appearance, the DQC1 model can efficiently solve some problems for which no classical efficient algorithms are known, and therefore it has been conjectured that the DQC1 model is more powerful than classical computing (under the assumption of $\mbox{BPP}\subsetneq\mbox{BQP}$). Here we show that the output probability distribution of the DQC1 model cannot be classically efficiently approximated (exactly within a polynomial bit length or in the fully polynomial randomized approximation scheme (FPRAS) with at most a constant error) unless $\mbox{BQP}\subseteq\mbox{BPP}$.