High-Fidelity Gates in a Single Josephson Qubit

We demonstrate new experimental procedures for measuring small errors in a superconducting quantum bit (qubit). By carefully separating out gate and measurement errors, we construct a complete error budget and demonstrate single qubit gate fidelities of 0.98, limited by energy relaxation. We also introduce a new metrology tool—a Ramsey interference error filter—that can measure the occupation probability of the state $|2⟩$, which is outside the computational basis, down to ${10}^{-4}$, thereby confirming that our quantum system stays within the qubit manifold during single qubit logic operations.

Figure 1

Qubit operation and state measurement. (a) The potential energy $U$ of a Josephson phase qubit versus junction phase $\delta$. The qubit is formed from the two lowest eigenstates $|0⟩$ and $|1⟩$, with a transition frequency ${\omega }_{10}/2\pi \simeq 6.75\mathrm{GHz}$ that can be adjusted by varying the dc bias $I_{\varphi }=I_{dc}+I_z$. (b) A measurement pulse lowers the energy barrier $\Delta U$, increasing the $|1⟩$ state tunneling probability. (c) Tunneling probability versus measurement amplitude $I_z$ for the qubit in the states $|0⟩$ (open circles) and $|1⟩$ (closed circles) at qubit frequency ${\omega }_{10}/2\pi =7.22\mathrm{GHz}$. Fits are shown by the solid lines. (d) Data as for $C$ but with a larger current bias, $I_{dc}$ giving a smaller qubit transition frequency ${\omega }_{10}/2\pi =6.75\mathrm{GHz}$. The visibility between states $|0⟩$ and $|1⟩$ is 0.85 and 0.895 for data in (c) and (d), respectively. The difference is directly attributed to coupling to a TLS located at 7.05 GHz, as observed with spectroscopy. The inset illustrates the pulse sequence. The $|1⟩$ state is prepared by applying a shaped microwave pulse for $\tau =8\mathrm{ns}$, with amplitude chosen to generate a $\pi$ rotation. For the $|0⟩$ state we apply no microwaves. After state preparation, the current $I_z$ is pulsed to measure the qubit state.

Figure 2

Measurement of a high-fidelity gate. (a) The pulse sequence consists of two 8 ns Gaussian-shaped $\pi$-pulses, separated in time by $t_{\mathrm{sep}}$, followed by a measure pulse $I_z$. The first $\pi$-pulse defines the rotation axes; by convention this is the $x$ axis. For the second pulse, which is delayed by $t_{\mathrm{sep}}$, we sweep the rotation axis angle $\Theta$ by changing the phase of the microwaves. By adding a radio frequency signal (0–50 MHz) to the carrier wave we shift the microwaves away from the qubit frequency (a technique known as sideband mixing) [28]. This sequence ideally returns the qubit to the $|0⟩$ state. (b) Gray scale plot of measured $|1⟩$ state probability $P_1$ versus detuning $\Delta$ and phase $\Theta$ with $t_{\mathrm{sep}}=12\mathrm{ns}$ and (c) quantum simulation. On resonance, the phase $\Theta$ does not change $P_1$, as expected. (d) Plot of $P_1$ versus $t_{\mathrm{sep}}$. Measurement error of the $|0⟩$ state is 0.034, as obtained by performing the experiment with no microwaves. The difference between the data and this stray tunneling is 0.04 at $t_{\mathrm{sep}}=12\mathrm{ns}$, corresponding to an error of magnitude 0.02 for each $\pi$-pulse, and a single qubit gate fidelity of 0.98.

Figure 3

Ramsey interference error filter. (a) For a singe-pulse sequence, plot of tunneling probability $P_{\mathrm{tunnel}}$ versus $I_z$ for $|0⟩$ and $|1⟩$ state (gray) for $\tau =4$, 5, and 8 ns FWHM Gaussian-shaped $X_{\pi }$-pulses. (b) For two-pulse sequence plot of $|2⟩$ state probability $P_2$ vs $t_{\mathrm{sep}}$ for $\tau =5\mathrm{ns}$. The two $X_{\pi }$-pulses are followed by a measure pulse with an amplitude calibrated to tunnel only the $|2⟩$ state. During the first $X_{\pi }$-pulse both of the states $|1⟩$ and $|2⟩$ are excited. The second $X_{\pi }$-pulse causes the coherent beating of the $|2⟩$ state. The amplitude of the oscillation is 4 times the error probability, whereas the beat frequency $1/T=1/(5\mathrm{ns})$ corresponds to the qubit nonlinearity $({\omega }_{10}-{\omega }_{21})/2\pi$. The insets in (a)  and (b) illustrate the microwave control pulses; (b) also depicts the three-level system and the unwanted transitions to the $|2⟩$ state.

Figure 4

Error from $|2⟩$ state occupation, measured at a magnitude of ${10}^{-4}$. (a) Plot of $|2⟩$ state error versus Gaussian pulse width for both single $\pi$-pulses (black circles) and Ramsey error (gray circles) data. The 8 ns FWHM Gaussian produces a $|2⟩$ probability of ${10}^{-4}$. The solid line is the quantum prediction obtained from numerical simulation. Spectrum analyzer data for the Gaussian-shaped pulses (asterisks) are also plotted with the Fourier-transform theory curve (solid gray line). The inset illustrates that a 4 ns pulse produces a significant amount of spectral power at ${\omega }_{21}/2\pi$.