High-fidelity gates in a Josephson qubit

References

I Supplementary Material

High-power spectroscopy reveals the transition frequencies between states $|0\rangle$, $|1\rangle$, and $|2\rangle$ and directly measures the nonlinearity of the qubit. The probability of tunneling versus frequency is plotted in the Fig. 5A. The peak at $6.25\ \textrm{GHz}$ corresponds to the qubit \mbox{|0\rangle}\rightarrow\mbox{|1\rangle} transition. The \mbox{|1\rangle}\rightarrow\mbox{|2\rangle} transition is $200\ \textrm{MHz}$ lower in frequency, a value equal to the Ramsey error frequency. For this peak, the $|1\rangle$ state is populated by off-resonant excitation of the \mbox{|0\rangle}\rightarrow\mbox{|1\rangle} transition due to the high power. A two-photon \mbox{|0\rangle}\rightarrow\mbox{|2\rangle} transition is also observed centered between these two resonances.

The Ramsey error filter data was taken for 4, 5, 6, 6.5, 7, 7.5, and 8 ns FWHM Gaussian pulses. For the longest length pulses, the experiment was repeated $10^{6}$ times.

Qubit spectroscopy is shown in Fig. 5B, where the probability of tunneling is plotted in grayscale for qubit frequency and qubit bias Cooper2004. A two-level state (TLS) gives a resonance at $7.05\ \textrm{GHz}$ that couples to the qubit with splitting size $50\ \textrm{MHz}$ The qubit was operated above ($7.22\ \textrm{GHz}$) and below ($6.75\ \textrm{GHz}$) the TLS resonance.

Shown in Fig. 5C is an example of a Gaussian-shaped microwave pulse taken with a high-speed sampling oscilloscope. These pulses have nearly ideal spectral quality, and are significantly improved compared to those used in previous experiments Katz2006. They are created with a continuous microwave source controlled by an IQ mixer fed by dual 1 GHz digital to analog converters (DAC). The microwave source drives in saturation the local oscillator input of the mixer at frequency $f_{0}$. The DAC channels are generated in a custom board using AD9736 chips that have 14 bit resolution. They drive the I and Q ports through $250\,{\rm MHz}$ ($-3\,{\rm dB}$ frequency) dissipative Gaussian lowpass filters and low distortion differential amplifiers. The microwave output of the mixer is filtered by a 7 pole Chebyshev lowpass filter at $8.5\,{\rm GHz}$ to suppress harmonics of $f_{0}$. The large bandwidth of the control signal allows for sideband mixing. By applying sine and cosine waves at $f_{\rm sb}$ to the I and Q ports, the mixer generates an output signal at frequency $f_{0}+f_{\rm sb}$. Sideband mixing allows for very high on/off ratios of qubit control since the (small) carrier leakage at $f_{0}$ is off resonance with the qubit. The digital control allows imperfections of the DAC chain and the IQ mixer to be corrected by first measuring its response function and then correcting it with deconvolution. The relative amplitudes and phases of the I and Q mixer channels are calibrated by minimizing the power at the opposite sideband $f_{0}-f_{\rm sb}$. This is done at enough sideband frequencies so that all Fourier component of an arbitrary digital input signal can be corrected. In total, we obtain accurate pulse shapes with greater than $60\,{\rm dB}$ suppression of spurious frequencies and harmonics.

For the gate fidelity measurements, the shape of the control pulses were Slepian Slepian1978. These pulses have similar envelopes to Gaussians, but have tails that are strictly set to zero.