Universal quantum computation with ideal Clifford gates and noisy ancillas

TLDR

The study examines quantum computation restricted to Clifford unitaries and |0> state preparation, noting its relation to the Gottesman–Knill theorem. The authors aim to identify which ancilla states ρ enable efficient universal quantum computation. They construct purification protocols that, using multiple copies of a mixed ancilla ρ, increase its polarization along specific magic directions, achieving a pure magic state when the polarization exceeds about 65 %. They demonstrate that Clifford operations together with prepared magic states suffice for universal quantum computation.

Abstract

We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state $|0\rangle$ computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state $\rho$, which should be regarded as a parameter of the model. Our goal is to determine for which $\rho$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $\rho$ 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 $\rho$ 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.