Efficient $Z$ gates for quantum computing

For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary virtual $Z$ gates, which, combined with two $X_{\pi /2}$ gates, can generate any SU(2) gate. Here we show how to best utilize these virtual $Z$ gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and $Z$ gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration $X$ and $Y$ gates. $Z$ gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse-shaping techniques such as derivative removal by adiabatic gate (DRAG). We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and $Z$ gates to correct rotation errors realizes a 13.3 ns $X_{\pi /2}$ gate characterized by low error $[1.95\left(3\right)\times {10}^{-4}]$ and low leakage $[3.1\left(6\right)\times {10}^{-6}]$. Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders of magnitude smaller than pulse errors due to decoherence.

Figure 1

Schematic of the typical experimental setup for generating shaped microwave pulses for driving superconducting qubits.

Figure 2

RB curves for the two basis sets discussed in the main text: $X{Y}_{\frac{\pi }{2}}$ (red circles) and hz (blue squares). Interleaved RB for the $S$ gate using the hz set (green triangles). Each point is the average of 20 random seeds run in five separate experiments and fit to the standard RB exponential decay curve $A{r}^m+B$ where $m$ is the number of Clifford gates and the average error per Clifford is $\frac{1}{2}(1-r)$. The average error per gate is $\frac{1}{2}(1-r^{1/N_g})$ where $N_g$ is the number of gates in the basis set required to implement a Clifford [13].

Figure 3

(a) Bloch-sphere representation of an attempted $\frac{\pi }{2}$ rotation around the $X$ axis starting in the state $|0\rangle$ on resonance [red (light gray)] and with a detuned drive [blue (dark gray)]. For a suitably compensated rotation angle the detuned drive ends up in the $XY$ plane, but with a finite $Z$-rotation error. (b) Correcting the error using a vz gate, i.e., axes rotation.

Figure 4

(a) EPG for the $X{Y}_{\frac{\pi }{2}}$ set (see main text) for three different pulse implementations: Gaussian, DRAG and Gaussian plus vz gate (gz) as a function of the sideband frequency. Dashed lines are theory fits assuming the coherence numbers listed in the main text, LO leakage of $-65$ dBm and an internal AWG filter approximated as a Gaussian filter with a bandwidth of 300 MHz [16]. (b) Sample RB curves for the different pulses for a sideband frequency of 180 MHZ. The DRAG and gz pulses are coherence limited, but the Gaussian pulse is completely dominated by unitary errors.

Figure 5

(a) Single-shot measurements of the $|0\rangle$ (blue, left), $|1\rangle$ (red, top-right) and $|2\rangle$ (green, bottom-right) states in the IQ plane (1000 shots). The colored regions demonstrate the hard thresholding barriers used to bin measurements. For the RB data we use the results from this calibration to construct a POVM and invert the hard-threshold results to compensate for assignment errors. (b) Typical RB measurement for leakage. The $|2\rangle$ state population is multiplied by 20 times for scale. (c) Leakage rate and $|2\rangle$ population after 2901 Clifford gates (averaged over 20 seeds) versus the DRAG parameter $\beta$ (black). This $|2\rangle$ population is used to calibrate $\beta$ for the DRAGZ pulses [red (light gray)].

Figure 6

(a) Leakage per gate versus pulse length for DRAG, gz, DRAGZ and FILTZ. Lines are theory curves assuming the coherences mentioned in the main text, a temperature of 46 mK, LO leakage of $-65$ dBm and an AWG filter bandwidth of 300 MHz (100 MHz for FILTZ). DRAGZ points were only taken to a pulse length of 20 ns; beyond this point calibration of the DRAG parameter (to minimize leakage) was not reliable because the leakage signal was below the noise floor. (b) EPG from the same measurement. The black dashed line is a theory curve for a perfect DRAG pulse (i.e., no mixer or AWG effects) and is representative of the coherence limited error.