Rapid High-Fidelity Single-Shot Dispersive Readout of Superconducting Qubits

The speed of quantum gates and measurements is a decisive factor for the overall fidelity of quantum protocols when performed on physical qubits with a finite coherence time. Reducing the time required to distinguish qubit states with high fidelity is, therefore, a critical goal in quantum-information science. The state-of-the-art readout of superconducting qubits is based on the dispersive interaction with a readout resonator. Here, we bring this technique to its current limit and demonstrate how the careful design of system parameters leads to fast and high-fidelity measurements without affecting qubit coherence. We achieve this result by increasing the dispersive-interaction strength, by choosing an optimal linewidth of the readout resonator, by employing a Purcell filter, and by utilizing phase-sensitive parametric amplification. In our experiment, we measure 98.25% readout fidelity in only 48 ns, when minimizing readout time, and 99.2% in 88 ns, when maximizing the fidelity, limited predominantly by the qubit lifetime of $7.6\mu \mathrm{s}$. The presented scheme is also expected to be suitable for integration into a multiplexed readout architecture.

Figure 1

(a) False-color micrograph of the experimental sample (see the text for details). (b) Pulse scheme including the preselection procedure, the gated (light purple) and the two-step measurement pulse (dark and light purple); see the text for details. (c) A characteristic ground (blue) and excited (red) single-shot trajectory plotted together with the mean of all ground and excited trajectories (light-blue and light-red solid lines, respectively) with their respective standard deviations (blue and orange shaded regions). The black dashed line shows the theoretically expected dynamics. The gray region depicts the typical integration time $\tau$ of the experiment. QB denotes qubit.

Figure 2

(a)–(d) Histograms of the integrated quadrature amplitude $q_{\tau }$ at selected integration times between 24 and 136 ns after preparing the qubit in the ground (blue) or excited (red) state. The single shots are performed with a gated measurement pulse and optimal drive power. The solid lines are fits to a double-Gaussian model whose individual components are indicated by dashed lines. The green areas depict the overlap error. The dashed gray line indicates the qubit-state threshold.

Figure 3

(a) Errors $\epsilon$ of the gated single-shot experiment at optimal drive strength $n_{\text{drive}}=2.5$ as a function of integration time $\tau$. The solid black circles represent the infidelity extracted from the data directly. The open circles are excited-state (red), ground-state (blue), and overlap errors (green) extracted from the fits. The solid green line is the numerical solution of the full model for the overlap error [Eq. (d1)] with the independently measured system parameters, including the finite amplifier bandwidth, while the black dashed curve assumes $\eta =1$. (b) Results of the overlap error ${\epsilon }_0$ at 56 ns extracted from Gaussian fits (the green circles) with the gated single-shot experiments versus measurement drive power $n_{\text{drive}}$. The green solid line is the theoretical prediction using the full overlap model [Eq. (d1)], while the black dots are the measured infidelity $1-F$. (c) Experimental results of infidelity versus integration time for the gated (black), two-step (purple), and high-amplifier-gain (orange) measurements.

Figure 4

(a) Ratio $\chi /\kappa$ that maximizes $s(\tau )$ as a function of integration time $\tau$. The solid black line represents the quasi-steady-state solution, while the dashed lines are the numerical solutions with our circuit parameters with the full model [27], which includes the dynamics of the Purcell filter. (b) Measurement signal $S(t)$ as a function of time $t$ for the indicated values of $\chi$. The solid lines indicate a ratio $\chi /\kappa =0.5$, while the dashed line is for our experimentally used ratio of 0.2, chosen for being optimal for short integration times below 50 ns. (c) Integrated signal $s(\tau )$, as a function of integration time $\tau$, for the indicated values of $\chi$ and $\chi /\kappa$. The gray dashed line represents the typical integration time used in our experiment. For both $S(t)$ and $s(\tau )$, we use $\alpha /2\pi =-340\mathrm{MHz}$, $\mathrm{\Delta }/2\pi =1562\mathrm{MHz}$, and $n_{\text{drive}}=n_{\mathrm{crit}}/5$.

Figure 5

(a) Schematic of the experimental setup. (b) Measured lifetime $T_1$, Ramsey decay time $T_2^{\star }$, and spin-echo time $T_2^{\mathrm{echo}}$ as a function of the qubit frequency. The black and orange dashed lines are the theoretical Purcell lifetimes with and without the Purcell filter. (c) Measured transmission of readout line with the qubit in its ground (blue) and excited (red) state. The solid black lines are fits to the input-output model of the circuit used to extract the relevant parameters of the system.

Figure 6

(a) Qubit frequency ${\nu }_{\mathrm{qb}}$ (the black dots) as a function of measured power $P_{\mathrm{out}}$ with a linear fit (the purple line). The calibrated photon number is shown on the bottom axis and the gray dashed line indicates the optimal drive power used in the experiment. (b) Measured output power as a function of input power $P_{\mathrm{rf}}$. The purple line is a linear fit to the low-power data, while the gray dashed line indicates the optimal drive power. (c) Mean value $q_p$ of the last 50 ns of the preselection pulse (the blue points) for $n_{\text{drive}}=2.5$ with a Gaussian fit (the blue line). The gray dashed line is the threshold used to remove thermally excited states from further analysis set at the 99% cumulative distribution function of the Gaussian fit. (d) Percentage of removed data after preselection analysis as a function of drive power. The gray dashed line indicates the minimum percent of traces which could be removed due to our choice of the preselection-state discrimination threshold.

Figure 7

(a) Measurement of the gain $G_0$ of the amplifier versus the signal frequency ${\nu }_{\mathrm{s}}$. The red line is a Lorentzian fit to extract the peak gain of approximately 19.7 dB and a 3-dB bandwidth of 27 MHz. (b) Micrograph of the Josephson parametric dimer used in the experiment made from etched niobium (black) on sapphire (yellow). The circuit consists of two capacitively coupled lumped-element nonlinear resonators, each with a large finger capacitor and an array of 45 SQUIDs (shadow-evaporated aluminum, gray in the picture) acting as a nonlinear inductor element. The left resonator is capacitively coupled to the output line of the measurement setup. (c) Measurement of the gain $G_0$ versus signal power $P_s$ to extract the 1-dB compression point of the amplifier at the experimentally used bias point. The red dashed line is the optimal readout power used in the experiment, while the gray dashed line depicts the 1-dB compression power. (d) Power spectral density (PSD) of the amplified quadrature versus signal frequency detuning from the pump tone with the parametric amplifier on (blue) and off (yellow) used to calibrate the measurement efficiency of the output line. The $y$ axis is scaled such that the distance between the peak of the parametric amplifier-on spectrum (blue) and the amplifier-off noise spectrum (yellow) is equal to the vacuum noise times the gain, i.e., $1/4\times 4G_0$.