Initialization by Measurement of a Superconducting Quantum Bit Circuit

We demonstrate initialization by joint measurement of two transmon qubits in 3D circuit quantum electrodynamics. Homodyne detection of cavity transmission is enhanced by Josephson parametric amplification to discriminate the two-qubit ground state from single-qubit excitations nondestructively and with 98.1% fidelity. Measurement and postselection of a steady-state mixture with 4.7% residual excitation per qubit achieve 98.8% fidelity to the ground state, thus outperforming passive initialization.

Figure 1

JPA-backed dispersive transmon readout. (a) Simplified diagram of the experimental setup, showing the input path for the readout signal carrying the information on the qubit state (RF) and the stronger, degenerate tone (Pump) biasing the JPA. Both microwave tones are combined at the JPA and their sum is reflected with a phase dependent on the total power (b) amplifying the small signal. An additional tone (Null) is used to cancel any pump leakage into the cavity. The JPA is operated at the low-signal gain of $\sim 25\mathrm{dB}$ and $2\mathrm{MHz}$ bandwidth. (c) Scatter plot in the $I\mathrm{\text{−}}Q$ plane for sets of 500 single-shot measurements. Light dots: readout signal obtained with a RF tone probing the cavity for qubits in $|00⟩$ and $|01⟩$, respectively. Dark dots: the Pump tone is added to the RF. (d) Spectroscopy of the cavity fundamental mode for qubits in $|00⟩$ and $|01⟩$. The RF frequency is chosen halfway between the two resonance peaks, giving the maximum phase contrast (163°; see inset on the right).

Figure 2

Ground-state initialization by measurement. (a) Pulse sequence used to distinguish between the qubit states ($M_1$), upon conditioning on the result of an initialization measurement $M_0$. The sequence is repeated every $250\mu \mathrm{s}$. (b) Histograms of 500 000 shots of $M_1$, without (dots) and with (squares) inverting the population of ${\mathrm{Q}}_{\mathrm{A}}$ with a $\pi$ pulse. (c) Histograms of $M_0$, with $V_{\mathrm{th}}$ indicating the threshold voltage used to digitize the result. (d) $M_1$ conditioned on $M_0=H$ to initialize the system in the ground state, suppressing the residual steady-state excitation. The conditioning threshold, selecting 91% of the shots, matches the value for optimum discrimination of the state of ${\mathrm{Q}}_{\mathrm{A}}$.

Figure 3

Analysis of readout fidelity. (a) Cumulative histograms for $M_1$ without and with conditioning on $M_0=H$, obtained from data in Figs. 2c, 2d. The optimum threshold maximizing the contrast between the two prepared states is the same in both cases. Deviations of the outcome from the intended prepared state are: 8.9% (1.3%) for the ground state, 6.2% (2.1%) for the excited state without (with) conditioning. Therefore, initialization by measurement and postselection increases the readout contrast from 84.9% to 96.6%. (b) Schematics of the readout error model, including the qubit populations in the steady state and at $\tau =2.4\mu \mathrm{s}$ after $M_0$. Only the arrows corresponding to readout errors are shown. (c) Rabi oscillations of ${\mathrm{Q}}_{\mathrm{A}}$ without (empty) and with (solid dots) initialization by measurement and postselection. In each case, data are taken by first digitizing 10 000 single shots of $M_1$ into $H$ or $L$, then averaging the results. Error bars on the average values are estimated from a subset of 175 measurements per point. For each angle, 7 randomly chosen single-shot outcomes are also plotted (black dots at 0 or 1). The visibility of the averaged signal increases upon conditioning $M_1$ on $M_0=H$.

Figure 4

Projectiveness of the measurement. (a) Conditional probabilities for two consecutive measurements $M_1$ and $M_2$, separated by $\tau =2.4\mu \mathrm{s}$. Following an initial measurement pulse $M_0$ used for initialization into $|00⟩$ by the method described, a Rabi pulse with variable amplitude rotates ${\mathrm{Q}}_{\mathrm{A}}$ by an angle $\theta$ along the $x$-axis of the Bloch sphere, preparing a state with $P_{|01⟩}={sin}^2(\theta /2)$. Dots (squares): probability to measure $M_2=H$ $(L)$ conditioned on having obtained the same result in $M_1$, as a function of the initial excitation of ${\mathrm{Q}}_{\mathrm{A}}$. Error bars are the standard error obtained from 40 repetitions of the experiment, each one having a minimum of 250 postselected shots per point. Deviations from an ideal projective measurement are due to the finite readout fidelity, and to partial recovery after $M_1$ [33]. The latter effect is shown in (b) where the conditional probabilities approach the unconditioned values, $P_H=0.91$ and $P_L=0.09$ for $\tau \gg T_1$, in agreement with Fig. 2, taking into account relaxation between the $\pi$ pulse and $M_2$. Error bars are smaller than the dot size.