Qubit Measurement by Multichannel Driving

We theoretically propose and experimentally implement a method of measuring a qubit by driving it close to the frequency of a dispersively coupled bosonic mode. The separation of the bosonic states corresponding to different qubit states begins essentially immediately at maximum rate, leading to a speedup in the measurement protocol. Also the bosonic mode can be simultaneously driven to optimize measurement speed and fidelity. We experimentally test this measurement protocol using a superconducting qubit coupled to a resonator mode. For a certain measurement time, we observe that the conventional dispersive readout yields close to 100% higher average measurement error than our protocol. Finally, we use an additional resonator drive to leave the resonator state to vacuum if the qubit is in the ground state during the measurement protocol. This suggests that the proposed measurement technique may become useful in unconditionally resetting the resonator to a vacuum state after the measurement pulse.

Figure 1

(a) Schematic presentation of the readout scheme where the qubit is driven (blue color) at the frequency of a dispersively coupled bosonic mode. A compensation tone (brown color) on the resonator may be used to optimize the result. We consider the case where the detuning $\mathrm{\Delta }={\omega }_q-{\omega }_r$ is much greater than the qubit-mode coupling strength $g$. (b) Evolution of the mean of the resonator state in phase space for the conventional dispersive readout starting from vacuum (cross) provided that the qubit was prepared in $|g⟩$ (blue) or $|e⟩$ (red). (c) As (b) but the readout pulse is applied directly to the qubit. Thus, the resonator states start to rotate about a new virtual origin ${\alpha }_{\mathrm{vo}}$ (circle) leading to a faster separation. After measurement, we may reverse the sign of the virtual origin and wait for the resonator states corresponding to different qubit states to fully overlap (faint colors). A subsequent shift (brown color) finalizes an unconditional reset of the resonator.

Figure 2

(a) Simplified measurement setup and scanning electron micrographs of the experimental sample realizing the theoretical scheme. We employ an Xmon qubit [59], the frequency of which may be tuned by applying an external magnetic flux to the accompanying dc SQUID. The qubit is capacitively coupled to a voltage drive line and to a coplanar waveguide resonator which is, in turn, coupled to a transmission line. After amplification, we measure the two quadratures $\mathrm{Re}\widehat{a}$ and $\mathrm{Im}\widehat{a}$ of the resonator field. (b) Evolution of the amplitudes of the coherent states corresponding to the ground (blue line) and excited (red line) states of the qubit during a 420-ns conventional dispersive readout. Corresponding results of single-shot measurements are shown by dots (see text for details). The dotted line indicates the threshold for assigning the measurement outcome. (c) As (b), but the drive tone is applied to both the qubit and the resonator with an optimized relative phase. The single-shot measurement fidelities are 96.4% and 96.6% for (b) and (c), respectively.

Figure 3

Average measurement error ${\epsilon }_{\text{total}}-{\epsilon }_{\text{prep}}$ as a function of integration time for the conventional readout (blue markers), qubit driving (red markers), and multichannel driving (yellow markers). Each data point shows the average and the standard deviation of 10 measurement runs consisting of $10^4$ single-shot measurements. The stars indicate the time for which the lowest error is obtained for each method. The inset shows the relative increase in the measurement error when the readout method is changed from the multichannel scheme to the conventional readout. When we drive the resonator only, the experimental parameters are identical to those in Fig. 2 and with multichannel driving to those in Fig. 2. For the multichannel readout, the drive powers to the resonator and to the qubit are decreased by 2 and 1 dB compared with single-channel driving, respectively.

Figure 4

(a) Measured means of the steady states corresponding to $|g⟩$ (${\alpha }_g$) and $|e⟩$ (${\alpha }_e$) in the multichannel readout as functions of the phase ${\varphi }_q$ of the qubit drive pulse, as indicated by the different colors of the markers. The black arrows denote phase-independent contributions to the steady state owing to the resonator drive. With a particular choice of ${\varphi }_q$, indicated by the gray arrows, one of the resonator states returns to vacuum during the measurement. (b) Evolution of the amplitude of the coherent state (solid lines) for qubit ground (blue color) and excited (red color) states in the partial reset scheme. The phase ${\varphi }_q$ is chosen such that the steady state for $|g⟩$ lies at the origin.