Hybrid magic state distillation for universal fault-tolerant quantum computation

A set of stabilizer operations augmented by some special initial states known as “magic states” gives the possibility of universal fault-tolerant quantum computation. However, magic state preparation inevitably involves nonideal operations that introduce noise. The most common method to eliminate the noise is magic state distillation (MSD) by stabilizer operations. Here we propose a hybrid MSD protocol by connecting a four-qubit $H$-type MSD with a five-qubit $T$-type MSD in order to overcome the shortcomings of the previous MSD protocols. The hybrid MSD protocol further integrates distillable ranges of different existing MSD protocols and provides considerable improvement in qubit cost. Moreover, we experimentally demonstrate the four-qubit $H$-type MSD protocol using nuclear magnetic resonance technology, together with the previous five-qubit MSD experiment, to show the feasibility of the hybrid MSD protocol.

Figure 1

Distillable ranges of the five-qubit $T$-type MSD protocol (${\mathcal{A}}_{\text{T}}$, denoted as the gray and orange regions, i.e., 1 and 2 regions) and the seven-qubit $H$-type MSD protocol (${\mathcal{A}}_{\text{H}}$, denoted as the green and gray regions, i.e., 3 and 1 regions). The purple region, i.e., 4 region shows the interior of the stabilizer octahedron ${\mathcal{O}}_{\text{s}}$. (a) One octant of the Bloch sphere. (b) A cross section of one octant of the Bloch sphere through the plane $x=y$.

Figure 2

(a) Flowchart of the hybrid MSD protocol. $D_{\text{T}}\left(\rho \right)=\left(\rho +T\rho T^{†}+T^{†}\rho T\right)/3$ is the $T$-projection operation. (b) The efficiency $\nu$ of the different MSD protocols along their target directions. The efficiency of four-qubit MSD protocol is also plotted along the $T$ direction by transforming the polarization from the $H$ direction to the $T$ direction by $D_{\text{T}}$ for comparison to the five-qubit MSD protocol. (c) The necessary iteration number to distill noisy states for an almost pure magic state ($T$ type or $H$ type) with target fidelity above 0.999. The second horizontal axis shows the corresponding $p_{\text{T}}$ after the operation $D_{\text{T}}$.

Figure 3

(a) Quantum circuit and (b) the corresponding pulse sequence for the four-qubit $H$-type MSD protocol, where $X_{\alpha }=e^{-ia{I}_x},Y_{\alpha }=e^{-ia{I}_y}$, and $Z_{\alpha }=e^{-ia{I}_z}$ denote single-qubit rotations. The $H$-projection operation $D_{\text{H}}\left(\rho \right)=(\rho +H\rho H^{†})/2$ was realized by accumulating signals of two experiments: one with the Hadamard gate implemented by the pulses shown in the dashed box in (b) and the other without it.

Figure 4

(a) Experimental spectra of the four nuclei before and after distillation for $p_{\text{H}}\left({\rho }_{\text{in}}\right)=0.826$. The experimentally measured and simulated spectra are shown as the blue and red curves, respectively. (b) Experimental results of the distillation. Top: polarizations along the $H$ direction before and after distillation. Bottom: the success probability ${\theta }_0$ of distillation vs the input polarization.

Figure 5

The integration between the four-qubit $H$-type MSD and the five-qubit $T$-type MSD.

Figure 6

The success probability of the three individual MSD protocols.

Figure 7

The optimal turning points for different input polarizations with the target polarization above 0.999. The optimal turning points gather around $p_H=0.85$. If the input states' polarizations are higher than 0.86, it is better to distill them directly using the five-qubit $T$-type MSD protocol.

Figure 8

(a) Distillation effect influenced by various $p_a$ and the biggest deviations $\delta$. For example, the point (0.82, 0.08) in the $\delta$ and $p_{\mathrm{in}}$ plane means that we randomly choose the input polarizations from the range $0.82-0.08$ to $0.82+0.08$. The blue perpendicular lines represent the theoretical distillable boundary. (b) Distributions of output polarization corresponding to Gaussian distribution inputs. The solid curves represent the input polarization of Gaussian distributions with centers at 0.778, 0.848, and 0.919. The dashed curves show the distributions of the output.

Figure 9

Characteristics of the four-qubit quantum register. The inset shows the structure, where the four qubits are labeled ${}^{13}\mathrm{C},{\text{F}}_1,{\text{F}}_2,{\text{F}}_3$. The chemical shifts and scalar coupling constants (in Hz) are on and above the diagonal in the table, respectively. The last column shows the transversal relaxation time $T_2$ of each nucleus measured by CPMG sequences. Shown below are spectra of ${}^{13}\mathrm{C}$ obtained by $\pi /2$ readout pulses when the system is prepared in the thermal equilibrium state (black) and PPS ${\rho }_{0000}$ [red (gray)].