Benchmarking a Quantum Teleportation Protocol in Superconducting Circuits Using Tomography and an Entanglement Witness

Teleportation of a quantum state may be used for distributing entanglement between distant qubits in quantum communication and for quantum computation. Here we demonstrate the implementation of a teleportation protocol, up to the single-shot measurement step, with superconducting qubits coupled to a microwave resonator. Using full quantum state tomography and evaluating an entanglement witness, we show that the protocol generates a genuine tripartite entangled state of all three qubits. Calculating the projection of the measured density matrix onto the basis states of two qubits allows us to reconstruct the teleported state. Repeating this procedure for a complete set of input states we find an average output state fidelity of 86%.

Figure 1

(a) Circuit diagram of the standard protocol to teleport the state $|\psi ⟩$ of qubit A to qubit C. Here, $H$ is the Hadamard gate, $Z$ and $X$ are the Pauli matrices ${\sigma }_z$ and ${\sigma }_x$. The CNOT gate is represented by a vertical line between the control qubit (●) and the target qubit ($\oplus$). (b) The circuit implemented in this experiment with controlled phase gates, indicated by vertical lines between the relevant qubits (●), and single-qubit rotations $R_{\widehat{n}}^{\theta }$ of angle $\theta$ about the axis $\widehat{n}$.

Figure 2

(a) Optical microscope image of the sample with three qubits coupled to a coplanar wavequide resonator with individual local microwave and magnetic flux-bias lines for each qubit. (b) and (c) show a close up of qubit B. d) Illustration of the pulse sequence for the implementation of the circuit diagram shown in Fig. 1b. Single and two-qubit operations for the generation of the three-qubit entangled state $|\Phi ⟩$ ($\mathrm{I}+\mathrm{II}$) are carried out with resonant microwave and magnetic flux pulses, respectively, (color code as in Fig. 1b]. The tomography (III) consists of microwave pulses that turn the qubit states to the desired measurement axis and a subsequent microwave pulse applied to the resonator for the joint dispersive qubit readout.

Figure 3

(a) Real and imaginary part of the measured three-qubit density matrix ${\rho }_m$ when applying the circuit shown in 1b to the input state $|\psi ⟩=(|0⟩-i|1⟩)/\sqrt{2}$. (b) Teleported single-qubit state at qubit C found by projecting ${\rho }_m$ from a) onto $|00⟩$ and tracing out qubits A and B. (c) Pauli sets for the state shown in a).

Figure 4

Absolute value of the $\chi$-matrix representation for the teleportation process which transfers the state of qubit A to qubit C for qubit A and B being projected to (a) $|00⟩$, (b) $|01⟩$, (c) $|10⟩$ and (d) $|11⟩$.