Rydberg-atom-based scheme of nonadiabatic geometric quantum computation

Nonadiabatic geometric quantum computation provides a means to perform fast and robust quantum gates. It has been implemented in various physical systems, such as trapped ions, nuclear magnetic resonance, and superconducting circuits. Another system being adequate for implementation of nonadiabatic geometric quantum computation may be Rydberg atoms, since their internal states have very long coherence time and the Rydberg-mediated interaction facilitates the implementation of a two-qubit gate. Here, we propose a scheme of nonadiabatic geometric quantum computation based on Rydberg atoms, which combines the robustness of nonadiabatic geometric gates with the merits of Rydberg atoms.

Figure 1

Setup for two-level Rydberg atom. A stable ground states $|g\rangle$ is coupled to Rydberg state $|r\rangle$ by resonant laser pulse with Rabi frequency $\mathrm{\Omega }\left(t\right)$.

Figure 2

Bloch sphere representation for the evolution of state $|d\rangle$. The state $|d\rangle$ undergoes cyclic evolution, such that $|d\rangle \stackrel{e^{-i{\int }_0^t{H}_1\left(t\right)dt}}{\to }|g\rangle \stackrel{e^{-i{\int }_{{\tau }_1}^t{H}_2\left(t\right)dt}}{\to }{e}^{i(\gamma +\phi )}|r\rangle \stackrel{e^{-i{\int }_{{\tau }_2}^t{H}_1\left(t\right)dt}}{\to }{e}^{i\gamma }|d\rangle$, and acquires a geometric phase $\gamma$. In this process, the dynamical phase is always zero, i.e., $\langle d\left(t\right)|H_1\left(t\right)|d\left(t\right)\rangle =0(0\le t\le {\tau }_1),\langle d\left(t\right)|H_2\left(t\right)|d\left(t\right)\rangle =0({\tau }_1<t\le {\tau }_2)$, and $\langle d\left(t\right)|H_1\left(t\right)|d\left(t\right)\rangle =0({\tau }_2<t\le \tau )$, where $\left|d\right(t)\rangle =U(t\left)\right|d\rangle$ and $U\left(t\right)$ is defined in Eq. (5).

Figure 3

Setup for two identical Rydberg atoms with Rydberg-mediated interaction. The interaction between Rydberg atoms is dipole-dipole or van der Waals force with strength $V$. For the individual Rydberg atoms, the transition ${|g\rangle }_{\alpha }\to {|r\rangle }_{\alpha }$ is facilitated by resonant laser pulse with Rabi frequency ${\mathrm{\Omega }}_{\alpha }\left(t\right)$.