Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics

Based on a microwave-photon quantum processor with two superconducting resonators coupled to one transmon qutrit, we construct the controlled-phase (c-phase) gate on microwave-photon-resonator qudits, by combination of the photon-number-dependent frequency-shift effect on the transmon qutrit by the first resonator and the resonant operation between the qutrit and the second resonator. This distinct feature provides us a useful way to achieve the c-phase gate on the two resonator qudits with a higher fidelity and a shorter operation time, compared with the previous proposals. The fidelity of our c-phase gate can reach $99.51\%$ within 93 ns. Moreover, our device can be extended easily to construct the three-qudit gates on three resonator qudits, far different from the existing proposals. Our controlled-controlled-phase gate on three resonator qudits is accomplished with the assistance of a transmon qutrit and its fidelity can reach $92.92\%$ within 124.64 ns.

Figure 1

Schematic diagram for our c-phase gate on two microwave-photon-resonator qudits, by combination of the number-state-dependent interaction between the transmon qutrit and the left resonator ($r_1$) and the simple resonant operation between the transmon qutrit and the right resonator ($r_2$). The two resonators are capacitively coupled to the qutrit whose transition frequency can be tuned by an external flux.

Figure 2

The expectation values of the probability distributions of the quantum Rabi oscillations ROT${}_0^{g,e}$, ROT${}_0^{e,f}$, ROT${}_1^{g,e}$, and ROT${}_1^{e,f}$. ROT${}_0^{g,e}$ and ROT${}_0^{e,f}$ represent the oscillations of ${\left({\left|0\right.〉}_{r_1}{\left|g\right.〉}_q\right)}_{\mathrm{dress}}\leftrightarrow {\left({\left|0\right.〉}_{r_1}{\left|e\right.〉}_q\right)}_{\mathrm{dress}}$ and ${\left({\left|0\right.〉}_{r_1}{\left|e\right.〉}_q\right)}_{\mathrm{dress}}\leftrightarrow {\left({\left|0\right.〉}_{r_1}{\left|f\right.〉}_q\right)}_{\mathrm{dress}}$, respectively. ROT${}_1^{g,e}$ and ROT${}_1^{e,f}$ represent the oscillations of ${\left({\left|1\right.〉}_{r_1}{\left|g\right.〉}_q\right)}_{\mathrm{dress}}\leftrightarrow {\left({\left|1\right.〉}_{r_1}{\left|e\right.〉}_q\right)}_{\mathrm{dress}}$ and ${\left({\left|1\right.〉}_{r_1}{\left|e\right.〉}_q\right)}_{\mathrm{dress}}\leftrightarrow {\left({\left|1\right.〉}_{r_1}{\left|f\right.〉}_q\right)}_{\mathrm{dress}}$, respectively.

Figure 3

(a) The density operator (${\rho }_0$) of the initial state $\left|{\psi }_0\right.〉$ of the system composed of the two resonators and the qutrit in our c-phase gate. Panels (b) and (c) are the real part (Real[${\rho }_3$]) and the imaginary part (Imag[${\rho }_3$]) of the final state $|{\psi }_f\rangle$ of the system, respectively.

Figure 4

(a) The schematic diagram for our cc-phase gate on a three-qudit microwave-photon system. (b) The schematic diagram for the coupling between a transmon qutrit and a microwave-photon resonator. Here $q$ represents a transmon qutrit. $r_1$, $r_2$, and $r_3$ are three microwave-photon resonators which have the same structure as those shown in Fig. 1.

Figure 5

The real part (a) and the imaginary part (b) of the final state $|{\Psi }_f\rangle$ of the system composed of the three microwave-photon resonators and the superconducting qutrit in our cc-phase gate, respectively.