Implementation of nonadiabatic holonomic quantum computation via two blockaded Rydberg atoms

Abstract

Nonadiabatic holonomic quantum computing has received great attention due to its advantages of avoiding long-term evolution of the system and maintaining the robustness of holonomic gates to control errors. In this paper, we study a scheme for implementing nonadiabatic holonomic computation using two blockaded Rydberg atoms. With the presence of the Rydberg blockade effect, we realized controlled-phase gates, which play an irreplaceable role in quantum computation and quantum information processing. Therefore, the current scheme may open up new prospects for the design of new types of effective Rydberg quantum gate devices.

Appendix A: Derivation of the effective Hamiltonian

According to the Hamiltonian shown in Eq. (2), the system evolves in Hilbert space spanned by:$\vert {\varphi _{1}}\rangle =\vert {BB}\rangle _{12}$, $\vert {\varphi _{2}}\rangle =\vert {BD}\rangle _{12}$, $\vert {\varphi _{3}}\rangle =\vert {Br}\rangle _{12}$, $\vert {\varphi _{4}}\rangle =\vert {DB}\rangle _{12}$, $\vert {\varphi _{5}}\rangle =\vert {DD}\rangle _{12}$, $\vert {\varphi _{6}}\rangle =\vert {Dr}\rangle _{12}$, $\vert {\varphi _{7}}\rangle =\vert {rB}\rangle _{12}$, $\vert {\varphi _{8}}\rangle =\vert {rD}\rangle _{12}$, $\vert {\varphi _{9}}\rangle =\vert {rr}\rangle _{12}$, Thus, the Hamiltonian shown in Eq. (2) can be rewritten as

$$\begin{aligned} H_{I}(t)&=(\Omega _{1}(t)e^{i\Delta t}+\Omega _{1}(t)e^{-i\Delta t})(\vert {\varphi _{1}}\rangle \langle {\varphi _{7}}\vert +\vert {\varphi _{2}}\rangle \langle {\varphi _{8}}\vert \nonumber \\&\quad +\vert {\varphi _{3}}\rangle \langle {\varphi _{9}}\vert )+(\Omega _{2}(t)e^{i\Delta t}+\Omega _{2}(t)e^{-i\Delta t})(\vert {\varphi _{1}}\rangle \langle {\varphi _{3}}\vert \nonumber \\&\quad +\vert {\varphi _{4}}\rangle \langle {\varphi _{7}}\vert +\vert {\varphi _{7}}\rangle \langle {\varphi _{9}}\vert )+h.c+V\vert {rr}\rangle _{12}\langle {rr}\vert . \end{aligned}$$

(6)

By introducing a picture transformation with unitary operator $U(t)=e^{(-iV\vert {rr}\rangle _{12}\langle {rr}\vert t)}$ [29], one can move to another picture where the Hamiltonian is

$$\begin{aligned} \begin{aligned} H^{'}_{I}(t)&=U^{\dagger }(t)H_{I}(t)U(t)-iU^{\dagger }(t){\dot{U}}(t)\\&=\sum ^{2}_{N=1}P^{\dagger }_{N}(t)e^{i\Delta t}+P_{N}(t)e^{-i\Delta t}.\\ \end{aligned} \end{aligned}$$

(7)

Here,

$$\begin{aligned} P_{1}= & {} \Omega _{1}(t)(\vert {\varphi _{7}}\rangle \langle {\varphi _{1}}\vert +\vert {\varphi _{8}}\rangle \langle {\varphi _{2}}\vert +\vert {\varphi _{9}}\rangle \langle {\varphi _{3}}\vert )+h.c \end{aligned}$$

(8)

$$\begin{aligned} P_{2}= & {} \Omega _{2}(t)(\vert {\varphi _{3}}\rangle \langle {\varphi _{1}}\vert +\vert {\varphi _{7}}\rangle \langle {\varphi _{4}}\vert +\vert {\varphi _{9}}\rangle \langle {\varphi _{7}}\vert )+h.c. \end{aligned}$$

(9)

And we assume $M_{1}=\Delta$ and $M_{2}=\Delta +V$. According to Ref. [34, 40], we can get the mathematical expression of effective Hamiltonian

$$\begin{aligned} H_{\mathrm{eff}}(t)=\sum ^{2}_{m,N=1}\frac{1}{M^{'}_{mN}}[P^{\dagger }_{m}(t),P_{N}(t)]e^{i(M_{m}-M_{N})t}. \end{aligned}$$

(10)

Here, $\frac{1}{M^{'}_{mN}}=\frac{1}{2}\left( \frac{1}{M_{m}}+\frac{1}{M_{N}}\right)$. Using the above calculation formula of effective Hamiltonian, we can get the simplified equation of effective Hamiltonian

$$\begin{aligned} H_{\mathrm{eff}}(t)&=\frac{2\Omega ^{2}_{1}(t)}{\Delta }(\vert {\varphi _{1}}\rangle \langle {\varphi _{7}}\vert +\vert {\varphi _{7}}\rangle \langle {\varphi _{1}}\vert +\vert {\varphi _{2}}\rangle \langle {\varphi _{8}}\vert \nonumber \\&\quad +\vert {\varphi _{8}}\rangle \langle {\varphi _{2}}\vert )+\frac{2\Omega ^{2}_{2}(t)}{\Delta }(\vert {\varphi _{1}}\rangle \langle {\varphi _{3}}\vert \nonumber \\&\quad +\vert {\varphi _{3}}\rangle \langle {\varphi _{1}}\vert +\vert {\varphi _{4}}\rangle \langle {\varphi _{6}}\vert +\vert {\varphi _{6}}\rangle \langle {\varphi _{4}}\vert ). \end{aligned}$$

(11)

Because $\vert {\varphi _{1}}\rangle =\vert {BB}\rangle _{12}$, $\vert {\varphi _{2}}\rangle =\vert {BD}\rangle _{12}$, $\vert {\varphi _{3}}\rangle =\vert {Br}\rangle _{12}$, $\vert {\varphi _{4}}\rangle =\vert {DB}\rangle _{12}$, $\vert {\varphi _{5}}\rangle =\vert {DD}\rangle _{12}$, $\vert {\varphi _{6}}\rangle =\vert {Dr}\rangle _{12}$, $\vert {\varphi _{7}}\rangle =\vert {rB}\rangle _{12}$, $\vert {\varphi _{8}}\rangle =\vert {rD}\rangle _{12}$, $\vert {\varphi _{9}}\rangle =\vert {rr}\rangle _{12}$. Therefore, the effective Hamiltonian can also be written as

$$\begin{aligned} H_{\mathrm{eff}}(t)&=\frac{2\Omega ^{2}_{1}(t)}{\Delta }(\vert {rB}\rangle _{12}\langle {BB}\vert +\vert {BD}\rangle _{12}\langle {rD}\vert )\nonumber \\&\quad +\frac{2\Omega ^{2}_{2}(t)}{\Delta }(\vert {Br}\rangle _{12}\langle {BB}\vert +\vert {DB}\rangle _{12}\langle {Dr}\vert )+H.c. \end{aligned}$$

(12)