Rolling quantum dice with a superconducting qubit

One of the key challenges in quantum information is coherently manipulating the quantum state. However, it is an outstanding question whether control can be realized with low error. Only gates from the Clifford group—containing $\pi$, $\pi /2$, and Hadamard gates—have been characterized with high accuracy. Here, we show how the Platonic solids enable implementing and characterizing larger gate sets. We find that all gates can be implemented with low error. The results fundamentally imply arbitrary manipulation of the quantum state can be realized with high precision, providing practical possibilities for designing efficient quantum algorithms.

Figure 1

The Platonic solids and their rotational groups. The axes of rotation are of the tetra-, octa-, and icosahedral rotational group; the respective Platonic solids are superimposed. The axes are defined by lines intersecting the origin, and a vertex, face center, or midpoint of an edge. The tetrahedral rotational group (orange) is shared among all groups.

Figure 2

Calibrating the angles of rotation. (a) The excited state probability versus $X$ and $Y$ pulse voltage amplitude on the control board. The amplitudes for the required phases of rotations around the $X$ and $Y$ axes are indicated with dotted lines. The data follow a ${sin}^2$ dependence (solid line) on the pulse amplitude, as expected. Data are not corrected for measurement fidelity. (b) The phase of the quantum state as a function of $Z$ pulse voltage amplitude, measured using quantum state tomography. Solid line is a fit to the data. For brevity, only the positive angles are shown. Here $tan\varphi =(1+\sqrt{5})/2$. Insets show the trajectories on the Bloch sphere for the $X$-, $Y$-, and $Z$-axis rotations.

Figure 3

Randomized benchmarking with the (a) tetra-, (b) octa-, and (c) icosahedral rotational groups. The sequence fidelities are plotted as a function of $m$, the number of random rotations or sets of random rotation and interleaved gate. For each $m$, the fidelity is averaged over $k=50$ different, random sequences. From fits to the reference curves (black lines) we extract the average error per group rotation of $r_{\mathrm{ref}},\mathrm{T}=0.0009$, $r_{\mathrm{ref}},\mathrm{C}=0.0010$, and $r_{\mathrm{ref}},\mathrm{I}=0.0019$, consistent with an average physical gate fidelity of $0.9995$. The rotational groups preserve Platonic solids in Bloch space; the respective solids are shown in the insets. The colored lower curves show the data when interleaving four tetrahedral rotations which are shared among all three groups; the rotational axes are shown in the insets; the composed gates are $X_{\pi }(◯)$, $X_{\pi /2}{Y}_{\pi /2}\left(▵\right)$, $X_{-\pi /2}{Y}_{\pi /2}(▿)$, and $Y_{\pi /2}{X}_{\pi /2}\left(⬠\right)$. Here, $X_{\pi /2}{Y}_{\pi /2}$ denotes the unitary $R_Y(\pi /2)R_X(\pi /2)=exp(-i\pi {\sigma }_Y/4)exp(-i\pi {\sigma }_X/4)$. The gate fidelities are tabulated in the figures, extracted from fits to the data (solid lines). Error bars on the data indicate the standard deviation of the mean. The standard deviations of gate fidelities are typically ${10}^{-4}$.

Figure 4

Icosahedral-based randomized benchmarking. We have interleaved three non-Clifford gates whose axes are shown in the inset, the gates rotate around a face center, vertex or edge midpoint of the icosahedron (superimposed). The gates are composed of three, six, and eight elements. Their compositions are: ${\mathrm{Y}}_{\varphi }{\mathrm{X}}_{2\pi /5}{\mathrm{Y}}_{-\varphi }(\diamond )$, ${\mathrm{X}}_{\varphi }{\mathrm{Z}}_{-2\pi /5}{\mathrm{Y}\mathrm{X}}_{2\varphi }{\mathrm{Z}}_{2\pi /5}{\mathrm{X}}_{-\varphi }(⊲)$, and ${\mathrm{X}}_{\varphi }{\mathrm{Z}}_{-2\pi /5}{\mathrm{X}}_{-\varphi }{\mathrm{X}}_{-\pi /2}{\mathrm{Y}}_{-\pi /2}{\mathrm{X}}_{\varphi }{\mathrm{Z}}_{2\pi /5}{\mathrm{X}}_{-\varphi }(⊳)$. The gate fidelities are tabulated in the figure. The average error per physical gate which makes up the interleaved gates is $r=3-4·{10}^{-4}$.