Universal quantum computation with ideal Clifford gates and noisy ancillas

We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state 0⟩, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state, which should be regarded as a parameter of the model. Our goal is to determine for which universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain ``magic'' directions. If the polarization of along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed.