Symmetric bidirectional quantum teleportation using a six-qubit cluster state as a quantum channel

Abstract

Bidirectional quantum teleportation is a fundamental protocol for exchanging quantum information between two quantum nodes. Till now, all bidirectional quantum teleportation protocols have achieved a maximum efficiency of $40\%$. Here, we propose a new scheme for a symmetric bidirectional quantum teleportation using a six-qubit cluster state as the quantum channel, for symmetric ($3\leftrightarrow 3$) qubit bidirectional quantum teleportation of a special three-qubit entangled state. The novelty of our scheme lies in its generalisation for ($N\leftrightarrow N$) qubit bidirectional quantum teleportation employing a 2N-qubit cluster state. The efficiency of the proposed protocol is remarkably increased to $50\%$ which is the highest till now. Interestingly, only GHZ-state measurements and four Toffoli-gate operations are necessary which is independent of the number (N) of qubits to be teleported.

Appendix A. Construction of six-qubit entangled channel on IBM quantum composer platform

In our proposed BQT scheme, utilising the six-qubit cluster state as the quantum channel, bilateral transmission of a special three-qubit state is achieved. Here, the construction of the six-qubit entangled quantum channel is presented using two H gates and four CNOT gates from single-qubit states. The corresponding quantum circuit as shown in figure 3 is developed on IBM quantum composer making it practically realisable.

The six-qubit channel is constructed from the joint state of six single-qubit states as an input in (3).

$$\begin{aligned} {|\psi \rangle _{0}=|0\rangle _{1}\otimes |0\rangle _{2}\otimes |0\rangle _{3}\otimes |0\rangle _{4}\otimes |0\rangle _{5}\otimes |0\rangle _{6}}. \end{aligned}$$

(A.1)

The construction of the quantum channel is as follows:

1. Hadamard gate (H) is applied on qubits 1 and 4, and the corresponding state is

$$\begin{aligned} |{\psi ^{\prime }\rangle }= & {} \frac{1}{2} \big ((|0\rangle +|{1}\rangle )_{1}\otimes |0\rangle _{2}\otimes |0\rangle _{3} \nonumber \\{} & {} \quad \otimes (|0\rangle +|{1}\rangle )_{4}\otimes |0\rangle _{5}\otimes |0\rangle _{6} \big ). \end{aligned}$$

(A.2)

2. Four CNOT gates are applied on qubit pairs (1,2), (2,3), (4,5) and (5,6) with the first qubit being the control and the second target to get the required six qubit entangled state.

$$\begin{aligned} |\phi \rangle _{123456}{} & {} =\frac{1}{2}(|{000000}\rangle +|{000111}\rangle \nonumber \\ {}{} & {} \quad +|{111000}\rangle + |{111111}\rangle ). \end{aligned}$$

(A.3)

This six-qubit quantum channel constructed on the IBM quantum composer platform (eq. (A.3)) can also be represented by the amplitude of the corresponding computational basis states as shown in figure 4.