Feedback Control of a Solid-State Qubit Using High-Fidelity Projective Measurement

We demonstrate feedback control of a superconducting transmon qubit using discrete, projective measurement and conditional coherent driving. Feedback realizes a fast and deterministic qubit reset to a target state with 2.4% error averaged over input superposition states, and allows concatenating experiments more than 10 times faster than by passive initialization. This closed-loop qubit control is necessary for measurement-based protocols such as quantum error correction and teleportation.

Figure 1

Transmon equilibration to steady state. Time evolution of the ground-state population $P_{|0⟩}$ starting from states ${\rho }_0$ and ${\rho }_1$ (notation defined in the text). Solid curves are the best fit (including data in Fig. S4 [26]) to Eq. (1), giving the inverse transition rates ${\Gamma }_{01}^{-1}=50\pm 2\mu \mathrm{s}$, ${\Gamma }_{12}^{-1}=20\pm 2\mu \mathrm{s}$, ${\Gamma }_{10}^{-1}=324\pm 32\mu \mathrm{s}$, ${\Gamma }_{21}^{-1}=111\pm 25\mu \mathrm{s}$. From the steady-state solution, we extract residual excitations $P_{|1⟩,\mathrm{ss}}=13.1\pm 0.8\%$, $P_{|2⟩,\mathrm{ss}}=2.4\pm 0.4\%$. Inset: Steady-state population distribution (bars). Markers correspond to a Boltzmann distribution with best-fit temperature 127 mK, significantly higher than the dilution refrigerator base temperature (15 mK).

Figure 2

Reset by measurement and feedback. (a) Before feedback: Histograms of 300 000 shots of $M_1$, with (squares) and without (circles) inverting the qubit population with a $\pi$ pulse. Each shot is obtained by averaging the homodyne voltage over the second half (200 ns) of a readout pulse. $H$ and $L$ denote the two possible outcomes of $M_1$, digitized with the threshold $V_{\mathrm{th}}$, maximizing the contrast, analogously to Ref. 16. Full (empty) dots indicate (no) postselection on $M_0>V_{\mathrm{ps}}$. This protocol [26] is used to prepare ${\rho }_0$ and ${\rho }_1$, which are the input states for the feedback sequences in (b) and (c). (b) After feedback: Histograms of $M_2$ after applying the feedback protocol ${\mathrm{Fb}}_0$, which triggers a $\pi$ pulse when $M_1=L$. Using this feedback, $\sim 99\%$ (92%) of measurements digitize to $H$ for $\theta =0$ ($\pi$), respectively. (c) Feedback with opposite logic ${\mathrm{Fb}}_1$ preparing the excited state. In this case, $\sim 98\%$ (94%) of measurements digitize to $L$ for $\theta =0$ ($\pi$).

Figure 3

Deterministic reset from any qubit state. Ground-state population $P_{|0⟩}$ as a function of the initial state ${\rho }_{\theta }$, prepared by coherent rotation after initialization in ${\rho }_0$, as in Fig. 2. The cases shown are no feedback (circles), ${\mathrm{Fb}}_0$ (squares), ${\mathrm{Fb}}_1$ (diamonds), twice ${\mathrm{Fb}}_0$ (upward triangles), and ${\mathrm{Fb}}_0$ followed by ${\mathrm{Fb}}_1$ (downward triangles). The vertical axis is calibrated with the average measurement outcome for the reference states ${\rho }_0$, ${\rho }_1$, and corrected for imperfect state preparation [26]. The curve with no feedback has a visibility of 99%, equal to the average preparation fidelity. Each experiment is averaged over 300 000 repetitions. Inset: Error probabilities for two rounds of feedback, defined as $1-P_{|t⟩}$, where $|t⟩\in \{0,1\}$ is the target state. The systematic $\sim 0.3\%$ difference between the two cases is attributed to error in the $\pi$ pulse preceding the measurement of $P_{|1⟩}$ following ${\mathrm{Fb}}_1$ [26]. Curves: Model including readout errors and equilibration [26].

Figure 4

Fast qubit reset. Initialization errors as a function of initialization time ${\tau }_{\mathrm{init}}$ under looped execution of a simple experiment leaving the qubit ideally in $|1⟩$ [(a), measurement and $\pi$ pulse] or $|0⟩$ [(b), measurement only]. Empty circles: Initialization by waiting (no feedback). Solid dots: Initialization by feedback, with three rounds of ${\mathrm{Fb}}_0$ and a $\pi$ pulse on the $1\leftrightarrow 2$ transition [26]. Curves correspond to a master equation simulation assuming perfect pulses and either the measured transition rates ${\Gamma }_{ij}$ (dashed line, no feedback; solid line, triple ${\mathrm{Fb}}_0$ with a $\pi$ pulse on $1\leftrightarrow 2$) or the calculated [41] rates for 50 mK (dotted line, no feedback; dot-dashed line, triple ${\mathrm{Fb}}_0$). Feedback reset successfully bounds the otherwise exponential accruement of $P_{\mathrm{err}}$ in case (a) as ${\tau }_{\mathrm{init}}\to 0$. The reduction of $P_{\mathrm{err}}$ in (b) reflects the cooling of the transmon by feedback (see text for details).