Entanglement Purification and Protection in a Superconducting Quantum Network

High-fidelity quantum entanglement is a key resource for quantum communication and distributed quantum computing, enabling quantum state teleportation, dense coding, and quantum encryption. Any sources of decoherence in the communication channel, however, degrade entanglement fidelity, thereby increasing the error rates of entangled state protocols. Entanglement purification provides a method to alleviate these nonidealities by distilling impure states into higher-fidelity entangled states. Here we demonstrate the entanglement purification of Bell pairs shared between two remote superconducting quantum nodes connected by a moderately lossy, 1-meter long superconducting communication cable. We use a purification process to correct the dominant amplitude damping errors caused by transmission through the cable, with fractional increases in fidelity as large as 25%, achieved for higher damping errors. The best final fidelity the purification achieves is $94.09\pm 0.98\%$. In addition, we use both dynamical decoupling and Rabi driving to protect the entangled states from local noise, increasing the effective qubit dephasing time by a factor of 4, from 3 to $12\mu \mathrm{s}$. These methods demonstrate the potential for the generation and preservation of very high-fidelity entanglement in a superconducting quantum communication network.

Figure 1

Device design and vacuum Rabi oscillations. (a) Schematic of the quantum network, comprising two nodes $A$ and $B$, each with three capacitively coupled xmon qubits $Q_i^{A,B}$ ($i=1$, 2, 3). The center qubit $Q_2^{A,B}$ in each node is connected to a 1-meter long superconducting NbTi coaxial cable through a tunable coupler $G^{A,B}$. Qubits $Q_3^{A,B}$ are not used in this experiment. (b) Vacuum Rabi oscillations between $Q_2^A$ and the 5.806 GHz communication mode $C$, measured with coupling strength set to $g^A/2\pi =4.3\mathrm{MHz}$. Inset shows pulse sequence, where after exciting $Q_2^A$, the qubit is tuned into resonance with the communication mode while simultaneously turning on the coupler $G^A$. Blue dashed line is from numerical simulations. The orange and red dashed lines indicate the times for completing a half swap and a full swap of the excitation in $Q_2^A$ to the communication mode.

Figure 2

Deterministic Bell state generation. (a) Inset: Pulse sequence for Bell state generation, including the delay time $t_d$ that the excitation resides in the communication mode $C$. Main plot shows $|e⟩$ state population $P_e$ in $Q_2^A$ (blue), and in $Q_2^B$ for different delay times in the cable $t_d=50$ (orange) and $t_d=200\mathrm{ns}$ (red). (b) Bell state fidelity $\mathcal{F}$ as a function of delay time $t_d$. (c) Bell state tomography for $t_d=10$ and (d) $t_d=400\mathrm{ns}$. Dashed lines in (a) and (b), and dashed outline boxes in (c) and (d), are results from numerical simulations.

Figure 3

Entanglement purification. (a) Circuit schematic. (b) Experimental realization of the purification circuit in (a). The $\mathrm{ST}/2$ process is a half-photon transfer process as in Fig. 2. We prepare the first Bell pair $|Q_2^A{Q}_2^B⟩$ followed by swaps into $|Q_1^A{Q}_1^B⟩$. We then generate the second Bell pair in $|Q_2^A{Q}_2^B⟩$, and purify using these two pairs. (c) Bell state fidelity $\mathcal{F}$ before purification for $|Q_2^A{Q}_2^B⟩$ (blue) and $|Q_1^A{Q}_1^B⟩$ (orange), and after purification for $|Q_2^A{Q}_2^B⟩$ (green), each measured as a function of delay time $t_d$. Gray dashed line is for error-free purification between two identical impure Bell states. (d) Success rate for purification, which is the probability of measuring $|Q_1^A{Q}_1^B⟫$ in $|ee⟩$ with (red) and without (purple) readout measurement correction [49]. (e) State tomography with $t_d=150\mathrm{ns}$ for the pre-purification states $|Q_1^A{Q}_1^B⟩$, representing ${\rho }^{(1)}$, with state fidelity $69.4\pm 2.0\%$, and for (f) $|Q_2^A{Q}_2^B⟩$, representing ${\rho }^{(2)}$, with state fidelity $75.3\pm 1.0\%$. (g) Tomography for the post-purified $|Q_2^A{Q}_2^A⟩$, representing ${\rho }_f$, with state fidelity $86.9\pm 1.8\%$. Dashed lines are simulation results.

Figure 4

Entanglement protection using either dynamical decoupling (DD) or Rabi driving. (a) Pulse sequence for DD and (b) Rabi drive. The $\mathrm{ST}/2$ gate corresponds to the half-photon transfer process as shown in Fig. 2, with cable delay $t_d=10\mathrm{ns}$. (c) Bell state fidelity as a function of time for free evolution (blue), DD (orange), and for Rabi drive strengths $\mathrm{\Omega }/2\pi =5$ (green) and $\mathrm{\Omega }/2\pi =3.3\mathrm{MHz}$ (red). Numerical simulations are for free evolution including amplitude and phase decay (blue dashed line) and for free evolution with only $T_1$ decay (gray dashed line).