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 $\ensuremath{\mid}0⟩$, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state $\ensuremath{\rho}$, which should be regarded as a parameter of the model. Our goal is to determine for which $\ensuremath{\rho}$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $\ensuremath{\rho}$ 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 $\ensuremath{\rho}$ 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.

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