Nonlinear parity readout with a microwave photodetector

Robust high-fidelity parity measurement is an important operation in many applications of quantum computing. In this work we show how in a circuit QED architecture, one can measure parity in a single shot at very high contrast by taking advantage of the nonlinear behavior of a strongly driven microwave cavity coupled to one or multiple qubits. We work in a nonlinear dispersive regime treated in an exact dispersive transformation. We show that appropriate tuning of experimental parameters leads to very high contrast in the cavity and therefore to a high-efficiency parity readout with a microwave photon counter or another amplitude detector. These tuning conditions are based on nonlinearity and are hence more robust than previously described linear tuning schemes. In the first part of the paper we show in detail how to achieve this for two-qubit parity measurements and extend this to $N$ qubits in the second part of the paper. We also study the quantum nondemolition character of the protocol.

Figure 1

System schematic. Two qubits are coupled to a driven microwave cavity. The existence of photons in the cavity is read out by a microwave photon counter.

Figure 2

Photon amplitude (top) and effective cavity frequency (bottom) depending on the drive strength $\epsilon$ for the four different qubit states. The parameters here are ${({\omega }_{10},{\omega }_{21},g_1)}^{\left(1\right)}/2\pi =(4.297,4.071,0.12)$ GHz, ${({\omega }_{10},{\omega }_{21},g_1)}^{\left(2\right)}/2\pi =(4.094,3.868,0.12)$ GHz, ${\omega }_c=5.005$ GHz, and ${\delta }_c=0$. For every state a specific drive strength ${\epsilon }_{\mathrm{crit}}$ exists where the frequency rapidly jumps back to the bare cavity frequency and one observes a strong enhancement in the cavity occupation.

Figure 3

Critical point ${\epsilon }_{\mathrm{crit}}={\epsilon }_2$ depending on the cavity drive detuning ${\delta }_c$ for the same parameters as in Fig. 2. The dotted vertical lines indicate the value of ${\delta }_c$ where the condition $\mathrm{\Delta }\omega \chi <0$ is no longer fulfilled such that no bifurcation occurs if we increase ${\delta }_c$ to higher negative values and the system stays in the low-amplitude attractor for all drive strengths. For a parity measurement we have to choose ${\delta }_c$ such that we are in the area where the transition to the high-amplitude attractor occurs first for $|01\rangle$ and $|10\rangle$ (here ${\delta }_c\approx -0.02$). In this region $|11\rangle$ does not show a bifurcation and stays in the low-amplitude attractor such that for the corresponding critical drive strength ${\epsilon }_{\mathrm{crit}}$ the photon number for $|11\rangle$ is about 0.5 (see the inset). Additionally, we have to drive with ${\epsilon }_2^{01}<\epsilon <{\epsilon }_2^{00}$ such that we do not reach the bifurcation point for $|00\rangle$, which ensures that we also have a small photon amplitude if the system is in $|00\rangle$.

Figure 4

System schematic of the general case. Here $N$ qubits are coupled to a driven microwave cavity. We take into account $M$ levels.

Figure 5

(a) Photon amplitude and (b) effective cavity frequency depending on the drive strength $\epsilon$. The plot is for four identical qubits including ten energy levels. The parameters here are ${({\omega }_{10},{\omega }_{21},g_1)}^{\left(j\right)}/2\pi =(4.297,4.071,0.12)$ GHz, ${\omega }_c=5.005$ GHz, and ${\delta }_c=0$. For every set of states with the same number of excitations there exists a specific drive strength where the frequency rapidly jumps back to the bare cavity frequency and one observes a strong enhancement in the cavity occupation.

Figure 6

Value of ${\epsilon }_{\mathrm{crit}}=\epsilon$ depending on the detuning between the drive and cavity frequency for four identical qubits with the same parameters as in Fig. 5. The dotted vertical lines indicate the border where the instable behavior disappears for larger detunings and the dynamics is just described by the low-amplitude attractor. To measure parity we need to drive the system with two different frequencies, one such that ${\delta }_c$ lies in between the dotted vertical lines of $|1111\rangle$ and $|0111\rangle$ and the other one such that ${\delta }_c$ lies in between the dotted vertical lines corresponding to $|0011\rangle$ and $|0001\rangle$.

Figure 7

Comparison between the results of the effective cavity frequency when including three or ten energy levels. We see that the results are absolutely identical such that it seems to be reasonable to only include the lowest nonoccupied energy level in the calculations to get correct results. The parameters are the same as in Fig. 2.