Implementation of a generalized controlled-NOT gate between fixed-frequency transmons

We have embedded two fixed-frequency ${\mathrm{Al}/\mathrm{AlO}}_x$/Al transmons, with ground-to-excited transition frequencies at 6.0714 and 6.7543 GHz, in a single three-dimensional Al cavity with a fundamental mode at 7.7463 GHz. Strong coupling between the cavity and each transmon results in an effective qubit-qubit coupling strength of 26 MHz and a $-1$ MHz dispersive shift in each qubit's transition frequency, depending on the state of the other qubit. Using the all-microwave SWIPHT (speeding up wave forms by inducing phases to harmful transitions) technique [Economou and Barnes, Phys. Rev. B 91, 161405 (2015)], we demonstrate the operation of a generalized controlled-not gate between the two qubits, with a gate time of ${\tau }_g=907$ ns optimized for this device. Using quantum process tomography we find that the gate fidelity is 83–84%, somewhat less than the 87% fidelity expected from relaxation and dephasing in the transmons during the gate time.

Figure 1

Computer-generated rendering of the transmon qubit chips (L and H), and the two sections of the 3D Al cavity.

Figure 2

Plot of the analytically derived envelope $\mathrm{\Omega }$ of the SWIPHT pulse versus time $t$. The peak amplitude was ${\mathrm{\Omega }}_{\text{max}}/2\pi =913$ kHz and the duration was ${\tau }_g=907$ ns.

Figure 3

Plot of pulsed cavity transmission measurements $\mathrm{\Delta }V/V$, averaged over 1000 shots, versus cavity bias power $P_{\text{bias}}$ for different initial two-qubit states. The measurement was performed at the bare cavity frequency ${\omega }_{\text{R}}/2\pi =7.680438$ GHz. Dashed lines show powers $P_1$–$P_7$ chosen for the joint-qubit readout. Each power $P_i$ has four associated symbols, corresponding to the system being prepared in $|gg\rangle$ (black), $|eg\rangle$ (red), $|ge\rangle$ (blue), or $|ee\rangle$ (green), and then mapped to a higher level in some cases (see Table 1) before being read out.

Figure 4

Pulse sequence for performing the joint readout (see Table 1) (not to scale).

Figure 5

(a) Pulse sequence used to perform QST (not to scale). The two pulses before and after the SWIPHT pulse are used for initializing the qubits and to apply tomographic gates, respectively. (b) Measured populations in $|gg\rangle$ (blue circles), $|ge\rangle$ (red triangles), $|eg\rangle$ (orange rectangles), and $|ee\rangle$ (green stars) from QST, after initializing the system in $|gg\rangle$ and executing the SWIPHT protocol with qubit L being the control qubit. The solid curves are from master equation simulations of the two-qubit system. (c) Measured population during the SWIPHT protocol after preparing the system in $|eg\rangle$. Note that time $t=0$ is at the start of the SWIPHT pulse.

Figure 6

Plot of $\mathrm{Im}\left({\rho }_{\text{ge,eg}}\right)$ versus the phase ${\varphi }_{\text{d}}$ of the SWIPHT pulse, for four different initial superposition states of qubit L $\left(R_{x,y}^{\pm \pi /2}\otimes I\right)|gg\rangle$. Points are from QST measurements and the solid curves are from master equation simulations of $\rho$ with a fixed added phase offset of ${245}^{\circ }$.

Figure 7

(a) Real and imaginary parts of experimental ($\tilde{\chi }$), (b) simulated (${\chi }_{s}im$), and (c) ideal (${\chi }_{\text{SW}}$) process matrices $\chi$ for the SWIPHT gate with qubit L as the control. The theoretical maximum magnitude for any element is 0.25.