Controlled phase gate and Grover’s search algorithm on two distant NV-centers assisted by an NAMR

Abstract

We propose a scheme to construct a controlled phase gate on two distant nitrogen-vacancy centers (NV-centers) assisted by a quantized nanomechanical cantilevel resonator (NAMR). Unlike the previous work to complete the gate in the dispersive regime to let NV-centers detune with the NAMR largely, our gate is completed by using resonant operations between NV-centers and the NAMR, and single-qubit operations on the NV-center, which let the gate to be achieved within a short time with a high fidelity. To study the performance of the gate for universal quantum computation, we simulate a two-qubit Grover’s search algorithm on the NV-centers with a fidelity of $98.46\%$.

Appendix: single-qubit gates on the NV-center

Here, we give the details for the construction of the single-qubit gates used in Sect. 4. The gates are achieved by applying the driving field on the NV-center. The Hamiltonian of the NV-center applied with the driving fields can be written as

$$\begin{aligned} H= & {} \omega _{g}\vert g\rangle \langle g\vert \!+\!\omega _{e}\vert e\rangle \langle e\vert +\!\frac{\Omega _{1}}{2}(\vert g\rangle \langle u\vert e^{i\omega _{1}t}\!+\!\vert u\rangle \langle g\vert e^{-i\omega _{1}t} + |e\rangle \langle u|e^{i\omega _{1}t}\!+\!|u\rangle \langle e|e^{-i\omega _{1}t}) \nonumber \\&\quad +\, \frac{\Omega _{2}}{2}(|g\rangle \langle u|e^{i\omega _{2}t}\!+\!|u\rangle \langle g|e^{-i\omega _{2}t} + |e\rangle \langle u|e^{i\omega _{2}t}\!+\!|u\rangle \langle e|e^{-i\omega _{2}t}), \end{aligned}$$

(A1)

where $\Omega _{1}\left( \Omega _{2}\right)$ and $\omega _{1}\left( \omega _{2}\right)$ are the Rabi frequency and the driving frequency of the driving field, respectively. $\omega _{g}\left( \omega _{e}\right)$ is the transition frequency between states $\vert u\rangle$ and $\vert g\rangle \left( \vert e\rangle \right)$ of the+ NV-center.

Transforming H with a rotating frame

$U=\exp \left( -i\omega _{1}t\vert g\rangle \langle g\vert -i\omega _{2}t\vert e\rangle \langle e\vert \right)$, we can obtain the effective Hamiltonian $H_\mathrm{{eff}}$ of the NV-center as

$$\begin{aligned} H_\mathrm{{eff}}= & {} U^{\dagger }HU-iU^{\dagger }\dot{U} \nonumber \\= & {} \Delta _{g1}\vert g\rangle \langle g\vert +\Delta _{e2}\vert e\rangle \langle e\vert \!+\!\frac{\Omega _{1}}{2}(\vert g\rangle \langle u\vert \!+\!\vert u\rangle \langle g\vert \!+\!| e\rangle \langle u| e^{-i\Delta _{12}t}\!+\!| u\rangle \langle e| e^{i\Delta _{12}t}) \nonumber \\&\quad +\, \frac{\Omega _{2}}{2}( |e\rangle \langle u| + | u\rangle \langle e| + | g\rangle \langle u| e^{i\Delta _{12}t} + | u\rangle \langle g| e^{-i\Delta _{12}t}), \end{aligned}$$

(A2)

where $\Delta _{g1}=\omega _{g}-\omega _{1},\Delta _{e2}=\omega _{e}-\omega _{2},\Delta _{12}=\omega _{1}-\omega _{2}$. By defining a ratio $\Omega _{2}/\Omega _{1}=r~\left( r>0\right)$ and taking $\omega _{g}=\omega _{1}\ll \omega _{e}=\omega _{2}$, one can obtain $\frac{\Omega _{1}/2}{\Delta _{12}}\ll 1$ and $\frac{\Omega _{2}/2}{\Delta _{12}}\ll 1$ to meet the rotating-wave approximation to omit the oscillation terms with high frequencies in Eq. (A2). Then, the Hamiltonian $H_\mathrm{{eff}}$ is reduced to

$$\begin{aligned} H_\mathrm{{e}}= & {} \frac{\Omega _{1}}{2}\!\left( |g\rangle \langle u| + | u\rangle \langle g| \right) + \frac{k\Omega _{1}}{2}\left( | e\rangle \langle u| + | u\rangle \langle e|\right) . \end{aligned}$$

(A3)

Using the evolution operator $U=\exp \left( -iH_\mathrm{{e}}t\right)$, the evolution of states $\vert g\rangle$ and $\vert e\rangle$ of the NV-center can be described by

$$\begin{aligned} \exp \!\left( -iH_\mathrm{{e}}t\right) \!\vert g\rangle \!\!= & {} \!\!-\frac{i\sin \left( \frac{1}{2}\Omega _{1}t\sqrt{k^{2}+1}\right) }{\sqrt{k^{2}+1}}\vert u\rangle +\frac{k^{2}\!+\!\cos \left( \frac{1}{2}\Omega _{1}t\sqrt{k^{2}+1}\right) }{k^{2}\!+\!1}\vert g\rangle \nonumber \\&\quad +\, \frac{k\left( -1\!+\!\cos \left( \frac{1}{2}\Omega _{1}t\sqrt{k^{2}+1}\right) \right) }{k^{2}\!+\!1}\vert e\rangle , \nonumber \\ \exp \!\left( \!-iH_\mathrm{{e}}t\right) \!\vert e\rangle \!\!= & {} \!\!-\frac{ik\sin \left( \frac{1}{2}\Omega _{1}t\sqrt{k^{2}+1}\right) }{\sqrt{k^{2}+1}}\vert u\rangle +\frac{k\!\left( \!-1\!\!+\!\cos \!\left( \!\frac{1}{2}\Omega _{1}\!t\sqrt{k^{2}\!+\!1}\right) \!\right) }{k^{2}\!\!+\!1}\vert g\rangle \nonumber \\&\quad +\, \frac{1\!\!+\!k^{2}\!\cos \!\left( \!\frac{1}{2}\Omega _{1}t\sqrt{k^{2}\!+\!1}\right) }{k^{2}\!+\!1}\vert e\rangle . \end{aligned}$$

(A4)

Table 3 Parameters for the constructing of the single-qubit gates on the NV-center. (Unit: GHz)

If one takes $r=\sqrt{2}-1$ and $\frac{1}{2}\Omega _{1}t\sqrt{r^{2}+1}=\pi$, the Eq. (A4) becomes

$$\begin{aligned} \exp \left( -iH_\mathrm{{e}}t\right) \vert g\rangle =-\frac{1}{2}\left( \vert g\rangle +\vert e\rangle \right) , \nonumber \\ \exp \left( -iH_\mathrm{{e}}t\right) \vert e\rangle =-\frac{1}{2}\left( \vert g\rangle -\vert e\rangle \right) , \end{aligned}$$

(A5)

which can be used to construct a H gate on the NV-center.

If one takes $r=1$ and $\frac{1}{2}\Omega _{1}t\sqrt{r^{2}+1}=\pi$, the Eq. (A4) becomes

$$\begin{aligned} \exp \left( -iH_\mathrm{{e}}t\right) \vert g\rangle =-\vert e\rangle , \nonumber \\ \exp \left( -iH_\mathrm{{e}}t\right) \vert e\rangle =-\vert g\rangle , \end{aligned}$$

(A6)

which can be used to construct a X gate on the NV-center.

If one takes $r=\infty$ and $\frac{1}{2}\Omega _{1}t\sqrt{r^{2}+1}=\pi$, the Eq. (A4) becomes

$$\begin{aligned} \exp \left( -iH_\mathrm{{e}}t\right) \vert g\rangle =\vert g\rangle , \nonumber \\ \exp \left( -iH_\mathrm{{e}}t\right) \vert e\rangle =-\vert e\rangle , \end{aligned}$$

(A7)

which can be used to construct a Z gate on the NV-center. Parameters for the construction of the single-qubit gate X, H, Z on the NV-center are shown in Table 3, while the other parameters are the same as the ones used for constructing the c-phase gate.