Closing a quantum feedback loop inside a cryostat: Autonomous state preparation and long-time memory of a superconducting qubit

We propose to use a nonlinear resonator for projective readout, classical memory, and feedback for a superconducting qubit. Keeping the classical controller at cryogenic temperatures sidesteps many of the inefficiencies inherent in two-way communication between temperature stages in typical systems with room-temperature controllers, and avoids increasing the cryogenic heat load. This controller may find a broad range of uses in multiqubit systems, but here we analyze two specific demonstrative cases in single qubit control. In the first case, the nonlinear controller is used to initialize the qubit in a definite eigenstate. And in the second case, the qubit's state is read into the controller's classical memory and used to reinstate the measured state after the qubit has decayed. We analyze the properties of this system and we show simulations of the time evolution for the full system dynamics.

Figure 1

A qubit-cavity system is connected to a Kerr-resonator through a tee junction. The qubit with a transition frequency ${\mathrm{\Omega }}_{qb}$ is embedded in the 3D cavity with a resonance at ${\omega }_b$ and the coupling port for the 3D cavity is placed a distance $d_{\mathrm{cav}}$ from the tee junction. The Kerr resonator, which consists of an array of superconducting quantum interference devices that allows for a tunable frequency ${\omega }_a$, is placed a distance $d_K$ from the tee junction. The Kerr resonator is driven by a strong drive ${\alpha }_d$, while both the Kerr resonator and the 3D cavity are driven by the field ${\alpha }_{\mathrm{in}}$. Both drives are operated at the same frequency ${\omega }_d$.

Figure 2

(a) A simplified schematic of Fig. 1, introducing symbols used in panels (b)–(d) and in Fig. 3. The rectangle with a rectangular cutout represents the 3D cavity, the circle the qubit (in state $|\psi \rangle$), and the rectangle with a triangle cutout the Kerr resonator. Object colors are coarse representations of the center frequency of each component. (b) An empty 3D cavity is driven by a resonant microwave drive (left-facing blue arrow), microwave power is built up in the 3D cavity (cutout is filled in blue), and the drive is reflected (right-facing arrow). When the same cavity is driven by a slightly off- (light blue arrows) or far off-resonant drive (green arrows), less or no power is built up inside the cavity. The reflected signal will now have a phase that depends on the sign of the detuned incident field (indicated by an either dotted or dashed left-facing arrow). (c) When the total power incident on the Kerr resonator is sufficiently low the power built up is relatively low (lightly filled triangle) but increases disproportionally when a certain pair of critical powers, $|{\alpha }_{c\pm }{|}^2$, are exceeded as detailed in the text (fully filled triangle). (d) The qubit inside the 3D cavity will shift the center frequency of the cavity by a small but significant amount, as described in the text, and yields a qubit-state dependent phase shift of the reflected signal. When the power is built up in the 3D cavity, the center frequency of the qubit shifts proportionally (yellow circle accompanying the filled-in cutout).

Figure 3

Protocol for measurement, classical memory storage, and feedback to a qubit. (a) With the qubit in an initial state $|\psi \rangle$, the Kerr resonator and drive are tuned to be coresonant with the 3D cavity (all three are colored blue). (b) Measurement operation as described in the text. (Note: the signal emitted by the Kerr resonator is typically phase shifted as well, which we disregard in this section for simplicity.) (c) Stabilizing the classical memory state by tuning the Kerr resonator and drive, and increasing drive power. (d) Using the classical memory state to apply a conditional $\pi$ pulse to the qubit, stabilizing it in the $|1\rangle$ state.

Figure 4

Illustration of the external controls used in the feedback scheme. The upper panel sketches the power of the input drives, while the lower panel illustrates the detuning between the Kerr-resonator frequency and the 3D cavity, ${\mathrm{\Delta }}_a^{\prime }={\omega }_a-{\omega }_b$ (solid blue line) and between the drive and the 3D cavity ${\mathrm{\Delta }}_d^{\prime }={\omega }_d-{\omega }_b$ (dashed red line). During the first time step $T_M$, we apply a measurement driven by ${\alpha }_d$ (solid red line) and a drive at the 3D cavity, ${\alpha }_p$ (dashed gray line), which is followed by a stabilization time $T_s$, where the drive and Kerr-resonator parameters are changed to further stabilize the Kerr-resonator states by placing the drive far away from the critical input powers indicated by ${\alpha }_{c\pm }$ (thinly dotted light red line). Simultaneously the Kerr resonator and the input drive are tuned away from the 3D cavity by ${\mathrm{\Delta }}_t$. After a waiting time $T_{\mathrm{wait}}$ in which no parameters are changed, we tune the Kerr resonator back into resonance with the 3D cavity. During the second $T_{\mathrm{\Delta }}$ a classical input drive ${\alpha }_{\mathrm{in}}$ (dashed blue line) is applied to the upper port and will interfere with the field from the Kerr resonator. Finally, the $\pi$ pulse (dotted orange line) of duration $T_{\pi }$ excites the qubit state depending on the Kerr-resonator field strength. An optional $\pi$ pulse (dotted green line) at the end of $T_{\mathrm{wait}}$ restores the qubit to its measured state, as described in the final paragraph of Sec. 2.

Figure 5

Solution of the classical field equation (1) with drive amplitudes slightly above and slightly below the critical drive, ${\alpha }_{c\pm }$. The inset shows how the two applied drive amplitudes linearly approach the critical value at time $20/({\kappa }_a+{\kappa }_d)$ indicated by the vertical dotted (green) line. A constant detuning ${\mathrm{\Delta }}_a=1.75{\mathrm{\Delta }}_{a,c}$ is assumed between the drive and the Kerr resonator and the nonlinearity is set to $K=-0.012{\mathrm{\Delta }}_a$. For the slightly higher drive power, we obtain a steady state with $\sim 110$ photons, which is much larger than the $\sim 35$ photons obtained for the only slightly weaker initial drive.

Figure 6

Numerical simulations of the state preparation with parameters described in the text without qubit decay. In (a) we start with the qubit in $|0\rangle$ and bring the state into $|1\rangle$ with 95.7% fidelity while in (b) we start with the qubit in the $|1\rangle$ state, where it remains with 98.5% fidelity. The upper panel of each figure displays the mean value of ${\sigma }_z$, the middle panel displays the photon population in the 3D cavity, and the lower panel the population in the Kerr resonator. In each panel we plot both the mean values for 50 trajectories (thin lines) and the average over 200 trajectories (thick line). The vertical dotted lines indicate the different time intervals similar to Fig. 4 and explained in the text.

Figure 7

Stabilization of the qubit by running the same scheme as in Fig. 6 but 20 times after each other. The parameters are all listed in the text. The dashed (red) line is the average of 400 trajectories, while the dotted (light red) line is an exponential Purcell-enhanced decay with the decay rate of the qubit, $1/\gamma =10 \mu s$. The solid (blue) curve shows the same but with a decay rate of $1/\gamma =30 \mu s$ achieved with an additional Purcell filter as explained in the text and the densely dotted (light blue) curve shows the corresponding exponential decay.

Figure 8

Demonstration of the memory recovery scheme with parameters described in the text. The main figure shows the mean value of the qubit operator ${\sigma }_z$ averaged over 200 trajectories for each. The solid light blue line assumes $|1\rangle$ as the initial state, while the dashed red line starts in $|0\rangle$. The $\pi$ pulse is applied after 78 $\mu s$, where we obtain a fidelity with the original state of 98.6% for $|1\rangle$ and 96.0% for $|0\rangle$. The upper inset shows Kerr-resonator excitations for 20 randomly chosen realizations and we see two clearly distinct stable states. The lower inset displays a zoom of the region around the recovery of the state.

Figure 9

Schematics to stabilize a Bell state. The circuit components are the same as in Fig. 1, but instead of 3D cavities, we use waveguides on a 2D chip. The state of interest is encoded into the data qubits, Q1 and Q2, while M1 and M2 are measurement qubits and they serve the same purpose as the qubit in Fig. 1. To perform the stabilization of Q1 and Q2, we apply quantum gates between these and M1 and M2. This can be achieved through the two quantum buses (in yellow). Control channels from the Kerr resonators to each quantum bus allow the feedback to correct errors in the data qubits.

Figure 10

A simulation of the Kerr resonator with the similar parameters as in the main text, but with ${\kappa }_b=0$. The solid lines are the semiclassical approach described in Sec. 3, while the dashed lines are the same but using the Monte Carlo wave functions of the Appendix. In both types of simulation we have averaged over 100 trajectories. The legend indicates the value of drive expressed as ${\alpha }_d/\sqrt{{\kappa }_d}$.