Fig. 1

Illustration for the one-qubit superconducting circuit where the LC resonator (red) coupled to the CPB (blue) through capacitor $C_{c}$ (green)"

Fig. 2

Energy spectrum of Hamiltonian ${\mathcal{H}_1}$ and ${\mathcal{H}_{QRM}}$ as a function of coupling strength ${\widetilde{g}/\omega_{r}}$"

Fig. 3

Relationship of coupling ratio ${\mathcal{R}}$ as a function of ${E_{J}/E_{C}}$"

Fig. 4

Relationship of coupling ratio ${\mathcal{R}}$ as a function of ${E_{J}/\hbar}$ for different oscillator impedance"

Table 1

Circuit parameters and the Hamiltonian parameters/ratios of the one-qubit superconducting model"

Fig. 5

Schematic diagram of the two-qubit superconducting circuit where the LC resonator coupled to two CPB"

Table 2

Circuit parameters and the Hamiltonian parameters/ratios of the two-qubit super-conducting model"

Fig. 6

Illustration of the QST protocol"

Fig. 7

Populations between the state $|1\rangle|\psi_0\rangle|0\rangle$ and the state $|0\rangle|\psi_0\rangle|1\rangle$ ($\omega_{{q},1}=\omega_{{q},2}=\omega_{r}=2\pi\times6$ GHz)"

Fig. 8

Populations between the state $\rho_{1}=|{1}\rangle\langle{1}|\otimes\rho^{Th}\otimes|{0}\rangle\langle{0}|$ and the state $\rho_{2}=|{0}\rangle\langle{0}|\otimes\rho^{Th}\otimes|{1}\rangle\langle{1}|$ ($\omega_{{q},{1(2)}}=\omega_{r}=2\pi\times6$ GHz,$\lambda_{1(2)}=0.02 \omega_{r})$ by considering dissipative system"

Fig. 9

Fidelity of the QST process between the state ${\rho_{2}}$ and the state ${\rho(t)}$"