  Deterministic quantum computation with one quantum bit (DQC1) is a model of
quantum computing where the input restricted to containing a single qubit in a
pure state and with all other qubits in a completely-mixed state, with only a
single qubit measurement at the end of the computation [E. Knill and R.
Laflamme, Phys. Rev. Lett. {\bf81}, 5672 (1998)]. While it is known that DQC1
can efficiently solve several problems for which no known classical efficient
algorithms exist, the question of whether DQC1 is really more powerful than
classical computation remains open. In this paper, we introduce a slightly
modified version of DQC1, which we call DQC1$_k$, where $k$ output qubits are
measured, and show that DQC1$_k$ cannot be classically efficiently simulated
for any $k\geq3$ unless the polynomial hierarchy collapses at the third level.
