Universal Quantum Computation with ideal Clifford gates and noisy ancillas

TL;DR

The paper analyzes a restricted quantum-computation model in which Clifford gates and |0⟩ preparation are perfect while a single-qubit ancilla ρ is noisy. It shows that universal quantum computation becomes possible if ρ has sufficient overlap with magic states, enabling distillation to high-fidelity |H⟩ or |T⟩ states; with such states, Clifford operations suffice for universality. Two distillation schemes are developed: a T-type protocol based on a 5-qubit code with threshold ε0 ≈ 0.173 (polarization ≈ 0.655) and a H-type protocol based on a 15-qubit CSS code with threshold ε0 ≈ 0.141 (polarization ≈ 0.718), each providing a path to UQC from noisy ancillas. The work also analyzes resource scaling and connects the results to the Gottesman-Knill theorem, while outlining open questions about thresholds and possible code-based improvements for achieving universality at the boundary of stabilizer mixtures.

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 $ρ$, which should be regarded as a parameter of the model. Our goal is to determine for which $ρ$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $ρ$ 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 $ρ$ 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.

Figures (3)

• Figure 1: On the left: The Bloch sphere and the octahedron $O$. On the right: The octahedron $O$ projected on the $x$-$y$ plane. The magic states correspond to the intersections of the symmetry axes of $O$ with the Bloch sphere. The empty and filled circles represent $T$-type and $H$-type magic states, respectively.
• Figure 2: The final error probability $\epsilon_{out}$ and the probability $p_s$ to measure the trivial syndrome as functions of the initial error probability $\epsilon$ for the $T$-type states distillation.
• Figure 3: The final error probability $\epsilon_{out}(\epsilon)$ for the $H$-type states distillation.

Theorems & Definitions (9)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof : Proof of the Lemma