Experimental Realization of a Deterministic Quantum Router with Superconducting Quantum Circuits

A quantum router is an elementary building block for quantum-information processing. It directs the incoming quantum signal to different routes with phase coherence. To realize scalable quantum routing, a single particle needs to deterministically and coherently route the paths of a flying qubit, which requires nonlinear interaction at the single quantum level and complicated path-data qubit control so far beyond the experimental reach. Here we demonstrate a deterministic quantum router working at microwave frequencies based on superconducting circuits, which can naturally integrate with a leading platform for quantum computing. We encode the quantum signal into a weak coherent pulse and use an entanglement witness to rigorously verify the generated entanglement between the control qubit and the path of the data qubit, thus proving quantum coherence of the routing operation. Our experiment lays the foundation for useful applications of quantum routers in quantum networks and quantum random-access memories.

Figure 1

Experimental setup and quantum router scheme. (a) Schematic of the proposed deterministic quantum router for microwave photons. A superconducting qubit (control qubit) is dispersively coupled with two cavities (red and blue) with different bare frequencies. An incident data qubit ${\left|\psi \right.⟩}_s$ frequency encoded in a single photon state (weak coherent pulse) either reflects back if its two frequency components are resonant with the two cavities, or goes through if off resonant. The control qubit can shift the frequency of the two cavities, and thus determine the output path of the signal. When the control qubit is set to $\lambda {\left|0\right.⟩}_c+\mu {\left|1\right.⟩}_c$, control-path entanglement will be generated after the routing operation and the ideal output state becomes $(\lambda {\left|0\right.⟩}_c{\left|T\right.⟩}_p+\mu {\left|1\right.⟩}_c{\left|R\right.⟩}_p){\left|\psi \right.⟩}_s$. (b) Illustration of the experimental setup. On the chip (brown) two coplanar waveguide resonators (blue and green) are coupled with a transmon qubit (orange). The input signal is incident onto the resonators through a transmission line (red). The transmitted and the reflected (with the help of a circulator) signals are split and interfered by three microwave hybrid couplers (labeled with $T$, $R$ and $T-R$) for the continuous-variable measurements. (c) The transmission and reflection spectra of the two CPW resonators for the control qubit in the ground state ${\left|0\right.⟩}_c$. The two frequency components of the data qubit are resonant with the two cavities when the transmon qubit is in the excited state ${\left|1\right.⟩}_c$ (red and blue arrows), and are shifted from resonance when in the ground state ${\left|0\right.⟩}_c$ (peaks and valleys in the spectra) through dispersive coupling.

Figure 2

Classical routing of weak coherent pulses. Blue and red data points are the output powers from the path $T$ and the path $R$ when rotating the polar angle $\theta$ in the control qubit state $\cos \theta {\left|0\right.⟩}_c+\sin \theta {\left|1\right.⟩}_c$. The quantum state of the input signal is set to (a) the low frequency $f_l$ and (b) the high frequency $f_h$, with an average photon number of 0.35. The curves are corresponding fitting results with a squared sinusoidal model.

Figure 3

Control-path correlation and entanglement. (a) The measured $x$ and $p$ quadratures of the signal field from the path $T-R$ when the control qubit is projected to ${\left|\pm \right.⟩}_c=({\left|0\right.⟩}_c\pm {\left|1\right.⟩}_c)/\sqrt{2}$, which show oscillations when sweeping the azimuthal angle $\varphi$ of the initial control qubit state $({\left|0\right.⟩}_c+e^{i\varphi }{\left|1\right.⟩}_c)/\sqrt{2}$. The solid lines are fitting results with the $\cos \varphi$ model. The dash lines indicate the theoretical amplitude from the calibrated experimental parameters. With any of the four oscillations, $x({\left|+\right.⟩}_c)$, $x({\left|-\right.⟩}_c)$, $p({\left|+\right.⟩}_c)$, and $p({\left|-\right.⟩}_c)$, the fidelity from the theoretical state $\left|\psi \right.⟩=\left({\left|0\right.⟩}_c{\left|{\alpha }_0^T\right.⟩}_T{\left|{\alpha }_0^R\right.⟩}_R+e^{i\varphi }{\left|1\right.⟩}_c{\left|{\alpha }_1^T\right.⟩}_T{\left|{\alpha }_1^R\right.⟩}_R\right)/\sqrt{2}$ can be learnt using the ratio of the measured oscillation amplitude to the corresponding theoretical value. We take the average value as the state fidelity $F$. See Appendix pp4 for more details. Here the input signal is at the low frequency $f_l$ with an average photon number $\overline{n}=0.4$. (b) The blue and the red data points are the fidelity $F$ for the low-frequency $f_l$ and the high-frequency $f_h$ components, respectively. The vertical error bars are given by the minimum and maximum value of the four ratios described in (a). The horizontal error bars are given by the standard deviation in calibrating the average photon number. We use an entanglement witness to verify the existence of quantum entanglement between the path and the control qubits. The solid lines show the entanglement threshold above which the system must be entangled. Here we vary $\overline{n}$ from 0 to 4. For most cases the measured fidelity is well above the entanglement threshold, and thus proves a genuine quantum router. Note that although the protocol requires low average photon number to approximate a single-photon state, we can prove entanglement for a much wider range of $\overline{n}$.

Figure 4

Quantum-process tomography for the data qubit. We virtually project the experimentally calibrated output state [Eq. (1) with phase noise] to the single-photon subspace and perform quantum-process tomography [19, 20]. Real and imaginary parts of the reconstructed process matrix are presented when the control qubit is in ${\left|+\right.⟩}_c=({\left|0\right.⟩}_c+{\left|1\right.⟩}_c)/\sqrt{2}$ for the output signal from path $R$, with an average gate fidelity $\overline{F}=0.973$. In the ideal case the quantum router well preserve the input quantum state, which is equivalent to an identity operation after normalization. The corresponding ideal matrix elements are indicated with the blue hollow caps. The case when the control qubit is in ${\left|+\right.⟩}_c$ (${\left|1\right.⟩}_c$) and the output is from path $T$ ($R$) are shown in Appendix pp5 with higher fidelities.

Figure 5

Schematic of the experimental setup.

Figure 6

Calibration of average photon number through correlation measurement. (a) The measured control qubit population in the ${\left|+\right.⟩}_c$ state as a function of the Bloch azimuthal angle $\varphi$ in Eq. (C8). The value of $\overline{p}$ is read from the output power of the digital-to-analog (DA) converter with arbitrary unit. (b) The oscillations of qubit population in (a) are normalized to [0,1] for comparison. The phase of the oscillation shows clear shifts for different signal power $\overline{p}$. (c) The phase of the oscillation in (a),(b) as a function of the signal power $\overline{p}$ represented by the output of the DA converter. A linear fit is used to compare the experimental result with theory for the calibration of the average photon number $N$.

Figure 7

Experimental pulse sequences to estimate the phase coherence and to witness the control-path entanglement. The state of the control qubit is first prepared in ${\left|0\right.⟩}_c+e^{i\varphi }{\left|1\right.⟩}_c$, then the input signal is sent into the router and the output voltage from path $T-R$ is measured. The state of the control qubit is then projected onto ${\left|\pm \right.⟩}_c$ basis and measured in a single shot. According to state projection result of the control qubit, the measured output signal is recorded and averaged. We scan the azimuthal angle $\varphi$ to get the oscillation curve of the output voltage, as shown in Fig. 3 of the main text.

Figure 8

Comparisons on the conditional average amplitudes obtained from the theory [Eq. (D5)] and measured in the experiment for an average photon number $\overline{n}=2.56$. (a) The real and imaginary part of the conditional amplitudes plotted on the $I/Q$ plane for both of the theoretical results and the experimental data. The theoretical ellipse (dashed line) is tilted to the same orientation as that of the experimental one (scattered points) by adjusting the global phase value [Eq. (D4)] of the conditional amplitude, as indicated by the solid line. (b) Oscillation of $I$ and $Q$ voltages as a function of $\varphi$. The scattered plots are experimental results, which are fitted with sine curves (solid lines). The corresponding theoretical results are plotted as dashed lines.

Figure 9

Real and imaginary parts of the reconstructed process matrices by setting the control qubit in (a) ${\left|1\right.⟩}_c$ state and (b) ${\left|+\right.⟩}_c=({\left|0\right.⟩}_c+{\left|1\right.⟩}_c)/\sqrt{2}$ state. The routed signal is recorded on (a) the reflection path $R$ and (b) the transmission path $T$. The corresponding matrix elements for an ideal routing operation are indicated with the blue hollow caps. The corresponding average gate fidelity $\overline{F}$ are also labeled.