In Situ Characterization of Qubit Control Lines: A Qubit as a Vector Network Analyzer

We propose and experimentally realize a technique to measure the transfer function of a control line in the frequency domain using a qubit as a vector network analyzer. Our method requires coupling the line under test to the longitudinal component of the Hamiltonian of the qubit and the ability to induce Rabi oscillations through simultaneous driving of the transverse component. The method can be used to increase the fidelity of entangling gates in a quantum processor. We have demonstrated that by characterizing the “flux” control line of a superconducting transmon qubit in the range from 1 to 450 MHz and using this characterization to improve the fidelity of an entangling cphase gate between two transmon qubits.

Figure 1

A typical dataset for ${\mathrm{\Omega }}_R=19.8\mathrm{MHz}$ and the corresponding trajectories of the Bloch vector in the first rotating frame with only the $x$ drive applied, with $x$ and $z$ drives and with $x$ and $z$ drives in the second rotating frame. (a) State tomography of the qubit with only the $x$ drive applied. The fit yields a rotation vector of $\overrightarrow{\mathrm{\Theta }}=(19.7,-1.9,0.8)\mathrm{MHz}$ and ${\varphi }_x=-0.1$. (b) State tomography of the qubit with both the $x$ and $z$ drives applied. The frequency of $z$ drive is set to ${\omega }_z={\mathrm{\Omega }}_R$. The sampling rate of the experiment is chosen such that the slow oscillations at ${\omega }_R$ can be resolved unambiguously but not necessarily the fast oscillations at ${\mathrm{\Omega }}_R$, which are canceled by $U_3$. The orange curve shows the theoretical dynamics derived from the fit in (c). (c) The dynamics of the qubit in the second rotating frame, transformed from (b). The fit yields $\overrightarrow{\theta }=(0.2,-3.5,0.1)\mathrm{MHz}$, and we can obtain $A_z=3.51\mathrm{MHz}$ and ${\varphi }_z=-1.54$.

Figure 2

Amplitudes $A_z$ and phases ${\varphi }_z$ of the transfer functions of the $z$ control line. Points in blue and green show the response of the flux line for two different configurations (see text for details). Darker points were measured with resonant driving ${\omega }_x={\omega }_0$ by varying $A_x$, lighter points were measured by varying ${\omega }_x-{\omega }_0$ with fixed $A_x$ (off-resonant case).

Figure 3

Amplitude and phase of transmission through a transmission line with a shorted stub resonator. Points in blue were measured with the qubit by comparing $A_z$ and ${\varphi }_z$ with and without the stub in place, the orange line was measured directly with a commercial vector network analyzer.

Figure 4

Vaccum Rabi oscillation between the $|11⟩$ and $|20⟩$ states of two transmon qubits assuming (a) a perfect impulse response, (b) the impulse response measured at room temperature, and (c) the impulse response measured using the qubit.