Appendix A: Logical six-particle entangled state \(|\Lambda _\mathrm{dp}\rangle _{ABCDEF}\)
A six-particle entangled state \(|\Lambda \rangle _{ABCDEF}\) (see Eq. 1) will become a logical six-particle entangled state \(|\Lambda _\mathrm{dp}\rangle _{ABCDEF}\) if its particles C, D, E, F are replaced with the corresponding logical qubits \(|0_\mathrm{dp}\rangle \) or \(|1_\mathrm{dp}\rangle \), respectively. That is, the logical six-particle entangled state \(|\Lambda _\mathrm{dp}\rangle _{ABCDEF}\) is as follows:
$$\begin{aligned} |\Lambda _\mathrm{dp}\rangle _{ABCDEF}= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|00\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|00\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|00\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|00\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|00\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\ \end{aligned}$$
$$\begin{aligned}&+\,|10\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|10\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F}) \\&-\,(|01\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\ \end{aligned}$$
$$\begin{aligned}&+\,|10\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|10\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|00\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\, |00\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{dp}\rangle _{C}|1_\mathrm{dp}\rangle _{D}|1_\mathrm{dp}\rangle _{E}|0_\mathrm{dp}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{dp}\rangle _{C}|0_\mathrm{dp}\rangle _{D}|0_\mathrm{dp}\rangle _{E}|1_\mathrm{dp}\rangle _{F})] \\ \end{aligned}$$
$$\begin{aligned}= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}) \nonumber \\&-\,(|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|10\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|10\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\, |00\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|10\rangle _{C_{1}C_{2}}|10\rangle _{D_{1}D_{2}}|10\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|10\rangle _{F_{1}F_{2}})] \end{aligned}$$
(10)
Appendix B: The formula derivation of Eq. (4)
The CNOT operation satisfies \(U^{C_{1}C_{2}}_\mathrm{CNOT}|00\rangle _{C_{1}C_{2}}=|00\rangle _{C_{1}C_{2}}\), \(U^{C_{1}C_{2}}_\mathrm{CNOT}|01\rangle _{C_{1}C_{2}}=|01\rangle _{C_{1}C_{2}}\), \(U^{C_{1}C_{2}}_\mathrm{CNOT}|10\rangle _{C_{1}C_{2}}=|11\rangle _{C_{1}C_{2}}\) and \(U^{C_{1}C_{2}}_\mathrm{CNOT}|11\rangle _{C_{1}C_{2}}=|10\rangle _{C_{1}C_{2}}\). After four CNOT operations with the particles \(C_{1}\), \(D_{1}\), \(E_{1}\), \(F_{1}\) used as the control qubits and the particles \(C_{2}\), \(D_{2}\), \(E_{2}\), \(F_{2}\) used as the target qubits, respectively, each logical quantum state \(|\Lambda _\mathrm{dp}\rangle _{ABCDEF}\) becomes
$$\begin{aligned} |\Lambda ^{(1)}_\mathrm{dp}\rangle _{ABCDEF}= & {} U^{C_{1}C_{2}}_\mathrm{CNOT}\otimes U_\mathrm{CNOT}^{D_{1}D_{2}}\otimes U_\mathrm{CNOT}^{E_{1}E_{2}}\otimes U_\mathrm{CNOT}^{F_{1}F_{2}}\otimes |\Lambda _\mathrm{dp}\rangle _{ABCDEF}\nonumber \\= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|00\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}})\nonumber \\&-\,(|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|01\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|11\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\, |00\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|11\rangle _{C_{1}C_{2}}|11\rangle _{D_{1}D_{2}}|11\rangle _{E_{1}E_{2}}|01\rangle _{F_{1}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|01\rangle _{C_{1}C_{2}}|01\rangle _{D_{1}D_{2}}|01\rangle _{E_{1}E_{2}}|11\rangle _{F_{1}F_{2}})]\nonumber \\ \end{aligned}$$
$$\begin{aligned}= & {} \frac{1}{4\sqrt{2}}[(|000000\rangle +|111111\rangle +|000011\rangle +|111100\rangle \nonumber \\&+\,|000101\rangle +|111010\rangle +|000110\rangle +|111001\rangle \nonumber \\&+\,|001001\rangle +|110110\rangle +|001111\rangle +|110000\rangle \nonumber \\&+\,|010001\rangle +|101110\rangle +|010010\rangle +|101101\rangle \nonumber \\&+\,|011000\rangle +|100111\rangle +|011101\rangle +|100010\rangle )\nonumber \\&-\,(|010100\rangle +|101011\rangle +|010111\rangle +|101000\rangle \nonumber \\&+\,|011011\rangle +|100100\rangle +|001010\rangle +|110101\rangle \nonumber \\&+\,|001100\rangle +|110011\rangle +|011110\rangle +|100001\rangle )]_{ABC_{1}D_{1}E_{1}F_{1}}\nonumber \\&\otimes |1111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\= & {} |\Lambda \rangle _{ABC_{1}D_{1}E_{1}F_{1}}\otimes |1111\rangle _{C_{2}D_{2}E_{2}F_{2}} \end{aligned}$$
(11)
Appendix C: Logical six-particle entangled state \(|\Lambda _\mathrm{r}\rangle _{ABCDEF}\)
If the particles C, D, E, F of the entangled state \(|\Lambda \rangle _{ABCDEF}\) (see Eq. 1) are replaced with the corresponding logical qubits \(|0_\mathrm{r}\rangle =|\phi ^{+}\rangle \) or \(|1_\mathrm{r}\rangle =|\psi ^{-}\rangle \), respectively, the entangled state \(|\Lambda \rangle _{ABCDEF}\) becomes the logical six-particle entangled state \(|\Lambda _\mathrm{r}\rangle _{ABCDEF}\) as follows:
$$\begin{aligned} |\Lambda _\mathrm{r}\rangle _{ABCDEF}= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|00\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|00\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|00\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|00\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}{} \mathbf \rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|00\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F}) \\ \end{aligned}$$
$$\begin{aligned}&-\,(|01\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|00\rangle _{AB}|1_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\, |00\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|11\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F} \\&+\,|01\rangle _{AB}|1_\mathrm{r}\rangle _{C}|1_\mathrm{r}\rangle _{D}|1_\mathrm{r}\rangle _{E}|0_\mathrm{r}\rangle _{F} \\&+\,|10\rangle _{AB}|0_\mathrm{r}\rangle _{C}|0_\mathrm{r}\rangle _{D}|0_\mathrm{r}\rangle _{E}|1_\mathrm{r}\rangle _{F})] \\= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \\&+\,|11\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \\&+\,|00\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \\&+\,|11\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \\&+\,|00\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \\&+\,|11\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \\&+\,|00\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \\&+\,|11\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \\&+\,|00\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \\&+\,|11\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \\&+\,|00\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \\&+\,|11\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \\ \end{aligned}$$
$$\begin{aligned}&+\,|01\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}}) \nonumber \\&-\,(|01\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|00\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\, |00\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|11\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|01\rangle _{AB}|\psi ^{-}\rangle _{C_{1}C_{2}}|\psi ^{-}\rangle _{D_{1}D_{2}}|\psi ^{-}\rangle _{E_{1}E_{2}}|\phi ^{+}\rangle _{F_{1}F_{2}} \nonumber \\&+\,|10\rangle _{AB}|\phi ^{+}\rangle _{C_{1}C_{2}}|\phi ^{+}\rangle _{D_{1}D_{2}}|\phi ^{+}\rangle _{E_{1}E_{2}}|\psi ^{-}\rangle _{F_{1}F_{2}})] \end{aligned}$$
(12)
It needs to be emphasized that the second equation of Eq. (12) is obtain by replacing \(|0_\mathrm{r}\rangle \) and \(|1_\mathrm{r}\rangle \) with \(|\phi ^{+}\rangle \) and \(|\psi ^{-}\rangle \), respectively.
Appendix D: The formula derivation of Eq. (5)
The CNOT operation satisfies
$$\begin{aligned} U_\mathrm{CNOT}^{C_{1}C_{2}}|\phi ^{+}\rangle _{C_{1}C_{2}}= & {} U_\mathrm{CNOT}^{C_{1}C_{2}} \otimes \frac{(|00\rangle _{C_{1}C_{2}}+|11\rangle _{C_{1}C_{2}})}{\sqrt{2}} =\frac{(|00\rangle _{C_{1}C_{2}}+|10\rangle _{C_{1}C_{2}})}{\sqrt{2}}\\= & {} \frac{(|0\rangle _{C_{1}}+|1\rangle _{C_{1}})}{\sqrt{2}}\otimes |0\rangle _{C_{2}}=|+\rangle _{C_{1}}|0\rangle _{C_{2}}, \end{aligned}$$
and
$$\begin{aligned} U_\mathrm{CNOT}^{C_{1}C_{2}}|\psi ^{-}\rangle _{C_{1}C_{2}}= & {} U_\mathrm{CNOT}^{C_{1}C_{2}} \otimes \frac{(|01\rangle _{C_{1}C_{2}}-|10\rangle _{C_{1}C_{2}})}{\sqrt{2}} =\frac{(|01\rangle _{C_{1}C_{2}}-|11\rangle _{C_{1}C_{2}})}{\sqrt{2}}\\= & {} \frac{(|0\rangle _{C_{1}}-|1\rangle _{C_{1}})}{\sqrt{2}}\otimes |1\rangle _{C_{2}}=|-\rangle _{C_{1}}|1\rangle _{C_{2}}. \end{aligned}$$
If the particles \(C_{1}\), \(D_{1}\), \(E_{1}\), \(F_{1}\) are used as the control qubits and the particles \(C_{2}\), \(D_{2}\), \(E_{2}\), \(F_{2}\) are used as the target qubits, respectively, the four CNOT operations make each logical quantum state \(|\Lambda _\mathrm{r}\rangle _{ABCDEF}\) become as follows:
$$\begin{aligned} |\Lambda ^{(1)}_\mathrm{r}\rangle _{ABCDEF}= & {} U_\mathrm{CNOT}^{C_{1}C_{2}}\otimes U_\mathrm{CNOT}^{D_{1}D_{2}}\otimes U_\mathrm{CNOT}^{E_{1}E_{2}} \otimes U_\mathrm{CNOT}^{F_{1}F_{2}}\otimes |\Lambda _\mathrm{r}\rangle _{ABCDEF}\nonumber \\= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1010\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1001\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1001\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|01\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0001\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0010\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0010\rangle _{C_{2}D_{2}E_{2}F_{2}})\nonumber \\&-\,(|01\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|10\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |1011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |0100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|-\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1010\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|+\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\, |00\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|+\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|-\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|-\rangle _{C_{1}}|-\rangle _{D_{1}}|-\rangle _{E_{1}}|+\rangle _{F_{1}}\otimes |1110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|+\rangle _{C_{1}}|+\rangle _{D_{1}}|+\rangle _{E_{1}}|-\rangle _{F_{1}}\otimes |0001\rangle _{C_{2}D_{2}E_{2}F_{2}})] \end{aligned}$$
(13)
Appendix E: The formula derivation of Eq. (6)
Since the Hadamard gates satisfy \(H_{C_{1}}|+\rangle _{C_{1}}=|0\rangle _{C_{1}}\) and \(H_{C_{1}}|-\rangle _{C_{1}}=|1\rangle _{C_{1}}\), then the Hadamard gates on particles \(C_{1}\), \(D_{1}\), \(E_{1}\) and \(F_{1}\) make each quantum state \(|\Lambda ^{(1)}_\mathrm{r}\rangle _{ABCDEF}\) become as follows:
$$\begin{aligned} |\Lambda ^{(2)}_\mathrm{r}\rangle _{ABCDEF}= & {} H_{C_{1}}\otimes H_{D_{1}}\otimes H_{E_{1}} \otimes H_{F_{1}}\otimes |\Lambda ^{(1)}_\mathrm{r}\rangle _{ABCDEF}\nonumber \\= & {} \frac{1}{4\sqrt{2}}[(|00\rangle _{AB}|0000\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|1111\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|0011\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|1100\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|0101\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|1010\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1010\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|0110\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|1001\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1001\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|1001\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1001\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|0110\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|1111\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\ \end{aligned}$$
$$\begin{aligned}&+\,|11\rangle _{AB}|0000\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|0001\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0001\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|1110\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|0010\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0010\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|1101\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|1000\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|0111\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|1101\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|0010\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0010\rangle _{C_{2}D_{2}E_{2}F_{2}})\nonumber \\ \end{aligned}$$
$$\begin{aligned}&-\,(|01\rangle _{AB}|0100\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|1011\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|0111\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0111\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|1000\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|1011\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|0100\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|00\rangle _{AB}|1010\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1010\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|0101\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0101\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\, |00\rangle _{AB}|1100\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1100\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|11\rangle _{AB}|0011\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0011\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|01\rangle _{AB}|1110\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |1110\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\&+\,|10\rangle _{AB}|0001\rangle _{C_{1}D_{1}E_{1}F_{1}}\otimes |0001\rangle _{C_{2}D_{2}E_{2}F_{2}})] \end{aligned}$$
(14)
Appendix F: The formula derivation of Eq. (7)
According to Eq. (14) and the definition of CNOT operation, we obtain the following results after four CNOT operations are performed.
$$\begin{aligned} |\Lambda ^{(3)}_\mathrm{r}\rangle _{ABCDEF}= & {} U_\mathrm{CNOT}^{C_{1}C_{2}}\otimes U_\mathrm{CNOT}^{D_{1}D_{2}}\otimes U_\mathrm{CNOT}^{E_{1}E_{2}} \otimes U_\mathrm{CNOT}^{F_{1}F_{2}}\otimes |\Lambda ^{(2)}_\mathrm{r}\rangle _{ABCDEF}\nonumber \\= & {} \frac{1}{4\sqrt{2}}[(|000000\rangle +|111111\rangle +|000011\rangle +|111100\rangle \nonumber \\&+\,|000101\rangle +|111010\rangle +|000110\rangle +|111001\rangle \nonumber \\&+\,|001001\rangle +|110110\rangle +|001111\rangle +|110000\rangle \nonumber \\&+\,|010001\rangle +|101110\rangle +|010010\rangle +|101101\rangle \nonumber \\&+\,|011000\rangle +|100111\rangle +|011101\rangle +|100010\rangle )\nonumber \\&-\,(|010100\rangle +|101011\rangle +|010111\rangle +|101000\rangle \nonumber \\&+\,|011011\rangle +|100100\rangle +|001010\rangle +|110101\rangle \nonumber \\&+\,|001100\rangle +|110011\rangle +|011110\rangle +|100001\rangle )]_{ABC_{1}D_{1}E_{1}F_{1}}\nonumber \\&\otimes |0000\rangle _{C_{2}D_{2}E_{2}F_{2}}\nonumber \\= & {} |\Lambda \rangle _{ABC_{1}D_{1}E_{1}F_{1}}\otimes |0000\rangle _{C_{2}D_{2}E_{2}F_{2}} \end{aligned}$$
(15)
