Realization of a quantum gate using gravitational search algorithm by perturbing three-dimensional harmonic oscillator with an electromagnetic field

Abstract

The aim of this paper is to design a current source obtained as a representation of p information symbols $\{I_k\}$ so that the electromagnetic (EM) field generated interacts with a quantum atomic system producing after a fixed duration T a unitary gate U(T) that is as close as possible to a given unitary gate $U_g$. The design procedure involves calculating the EM field produced by $\{I_k\}$ and hence the perturbing Hamiltonian produced by $\{I_k\}$ finally resulting in the evolution operator produced by $\{I_k\}$ up to cubic order based on the Dyson series expansion. The gate error energy is thus obtained as a cubic polynomial in $\{I_k\}$ which is minimized using gravitational search algorithm. The signal to noise ratio (SNR) in the designed gate is higher as compared to that using quadratic Dyson series expansion. The SNR is calculated as the ratio of the Frobenius norm square of the desired gate to that of the desired gate error.

Appendices

Appendix 1

Consider Eq. (11),

$$\begin{aligned} H(t|I)=H_0+\epsilon \sum _{k=1}^pI_kV_{1k}(t)+\epsilon ^2\sum _{k,m=1}^pI_kI_mV_{2km}(t) \end{aligned}$$

(67)

where $V_{1k}(t)$ is a first-order time-dependent linear partial differential operator in q (that is, the sum of a time-dependent function of q and a time-dependent vector field in q), while $V_{2km}(t)$ is a function of $q \,\text {and} \,t$, that is, a multiplication operator. The operators are defined as follows

$$\begin{aligned} \sum \limits _k I_k V_{1k}(t)=\frac{1}{2}(P,A(t,r_0+q))+ (A(t,r_0+q),P)- \varPhi (t,r_0+q) \end{aligned}$$

(68)

where $P=-i\frac{\partial }{\partial q}$ and $V_{1k}(t)\,\text {and}\, V_{2km}(t)$ is given by

$$\begin{aligned} V_{1k}(t)= & {} -\frac{i}{2}\text {div}\,q \bigg (\frac{\mu }{4\pi |r_0+q|}p_k\bigg (t-\frac{r_0+q}{c}\bigg ){\overrightarrow{d\mathbf{{l}}_k}}\bigg )\nonumber \\&+\, c^2\int \limits _0^t \text {div}\, q \frac{\mu }{4\pi |r_0+q|}p_k\bigg (t-\frac{r_0+q}{c}\bigg ){\overrightarrow{d\mathbf{{l}}_k}}\nonumber \\= & {} - \frac{i \mu }{8\pi }\bigg ({\overrightarrow{d\mathbf{{l}}_k}}, \nabla _q \frac{p_k\big (t-\frac{|r_0+q|}{c}\big )}{|r_0+q|}\bigg )+ \frac{\mu c^2}{4\pi }\int \limits _0^t\bigg ({\overrightarrow{d\mathbf{{l}}_k}}, \nabla _q \frac{p_k\big (t-\frac{|r_0+q|}{c}\big )}{|r_0+q|}\bigg )\hbox {d}t\nonumber \\ \end{aligned}$$

(69)

Now, consider

$$\begin{aligned} \sum \limits _{k,m}I_k I_m V_{2km}(t)= \frac{1}{2}A^2(t, r_0+q) \end{aligned}$$

(70)

where

$$\begin{aligned} V_{2km}(t)= \bigg (\frac{\mu }{4\pi |r_0+q|}\bigg )^2 ({\overrightarrow{d\mathbf{{l}}_k}}, {\overrightarrow{d\mathbf{{l}}_m}}) p_k\bigg (t -\frac{r_0+q}{c}\bigg )p_m\bigg (t -\frac{r_0+q}{c}\bigg ) \end{aligned}$$

(71)

The pulses $p_k$ have their meaning in Eq. (3).

Appendix 2

Consider Eq. (13),

$$\begin{aligned} U(T|I)=U_0+\epsilon \sum _kI_kU_{1k}+\epsilon ^2\sum _{k,m}I_kI_mU_{2km}+\epsilon ^3\sum _{kml}I_kI_mI_lU_{3kml}+O(\epsilon ^4) \end{aligned}$$

(72)

where $U_0,U_{1k},U_{2km}, U_{3kml}$ are operators that do not depend on the current amplitudes $I_1,..., I_p$. In other words, these operators are expressible entirely in terms of the operators $H_0,V_{1k}(t),V_{2km}(t)$ with the operators $V_{1k}(t), V_{2k}(t)$ completely independent of $I_1,..., I_p$, that is, expressible entirely in terms of the current pulses $p_k(t)$ and the length vectors $d\mathbf{{l}}_k. U_{1k}, U_{2km}$ and $U_{3kml}$ are defined as follows.

$$\begin{aligned} U_{1k}&= U_0(T)W_{1k}\end{aligned}$$

(73)

$$\begin{aligned} U_{2km}&=U_0(T)W_{2km} \end{aligned}$$

(74)

and

$$\begin{aligned} U_{3kml}=U_0(T)W_{3kml} \end{aligned}$$

(75)

Now $W_{1k}, W_{2km}\,\text {and}\, W_{3kml}$ are defined, respectively, as

$$\begin{aligned} W_{1k}=-i \int \limits _0^t\widetilde{V}_{1k}(t)\hbox {d}t \end{aligned}$$

(76)

where $\widetilde{V}_{1k}(t)= U_0(-t)V_{1k}(t)U_0(t)$ and similarly

$$\begin{aligned} W_{2km}= - \int \limits _{0<t_2<t_1<T}\widetilde{V}_1k(t_1)\widetilde{V}_{1m}(t_2)\hbox {d}t_1\hbox {d}t_2- i\int \limits _0^\mathrm{T} \widetilde{V}_{2km}(t)\hbox {d}t \end{aligned}$$

(77)

where $\widetilde{V}_{2km}(t)= U_0(-t) V_{2km}U_0(t)$ and

$$\begin{aligned} W_{3kml}&= i\int _{0<t_3<t_2<t_1<T}\widetilde{V}_{1k}(t_1)\widetilde{V}_{1m}(t_2)\widetilde{V}_{1l}(t_3)\hbox {d}t_1\hbox {d}t_2\hbox {d}t_3\nonumber \\&\quad -\int _{0<t_2<t_1<T}\widetilde{V}_{1k}(t_1)\widetilde{V}_{2ml}(t_2)\hbox {d}t_1\hbox {d}t_2-\int _{0<t_2<t_1<T}\widetilde{V}_{2km}(t_1)\widetilde{V}_{1l}(t_2)\hbox {d}t_1\hbox {d}t_2 \end{aligned}$$

(78)

Appendix 3

Consider Eqs. (30) and (31),

$$\begin{aligned} W(T)&= I -i\epsilon \int _0^t \widetilde{V}_1(t_1)\hbox {d}t_1-i\epsilon ^2 \int _0^t \widetilde{V}_2(t_1)\hbox {d}t_1\nonumber \\&\quad - \epsilon ^2\int _{0<t_2<t_1<t} \widetilde{V}_1(t_1)\widetilde{V}_1(t_2)\hbox {d}t_1\hbox {d}t_2\nonumber \\&\quad +i\epsilon ^3 \int _{0<t_3<t_2<t_1<t}\widetilde{V}_1(t_1)\widetilde{V}_1(t_2)\widetilde{V}_1(t_3)\hbox {d}t_1\hbox {d}t_2\hbox {d}t_3 \nonumber \\&\quad -\epsilon ^3\int _{0<t_2<t_1<t}\bigg (\widetilde{V}_1(t_1)\widetilde{V}_2(t_2)+ \widetilde{V}_2(t_1)\widetilde{V}_1(t_2) \bigg ) \hbox {d}t_1\hbox {d}t_2 +O(\epsilon ^4)\end{aligned}$$

(79)

$$\begin{aligned} W(T)&=I+\epsilon F_1+\epsilon ^2F_2+\epsilon ^3F_3+O(\epsilon ^4) \end{aligned}$$

(80)

where

$$\begin{aligned} F_1&=-i\int _0^\mathrm{T}\widetilde{V}_1(t_1)\hbox {d}t_1,\end{aligned}$$

(81)

$$\begin{aligned} F_2&=-i\int _0^\mathrm{T} \widetilde{V}_2(t_1)\hbox {d}t_1 -\int _{0<t_2<t_1<T}\widetilde{V}_1(t_1)\widetilde{V}_1(t_2)\hbox {d}t_1\hbox {d}t_2 \end{aligned}$$

(82)

and

$$\begin{aligned} F_3&=i\int _{0<t_3<t_2<t_1<T}\widetilde{V}_1(t_1)\widetilde{V}_1(t_2) \widetilde{V}_1(t_3)\hbox {d}t_1\hbox {d}t_2\hbox {d}t_3\nonumber \\&\quad -\int _{0<t_2<t_1<T}\bigg (\widetilde{V}_1(t_1)\widetilde{V}_2(t_2) +\widetilde{V}_2(t_1)\widetilde{V}_1(t_2)\bigg )\hbox {d}t_1\hbox {d}t_2 \end{aligned}$$

(83)

with $\widetilde{V}_k(t)=U_0^*(t)V_k(t)U_0(t)=U_0(-t)V_k(t)U_0(t), k=1,2$. Let $|n\rangle ,\quad n=1,2,\ldots$ denote the energy eigenstates of $H_0$ (so far the discussion is general, it is applicable to any non-relativistic charged particle moving in a time-invariant potential and excited by an electromagnetic field). Then, if $H_0 |n\rangle = E_n |n\rangle , n=1,2$ it follows that

$$\begin{aligned} F_1[n,m]&=\langle n|F_1|m\rangle = -i\int _0^\mathrm{T} \exp (i(E_n-E_m)t_1)\langle n| V_1(t_1)| m\rangle \hbox {d}t_1\end{aligned}$$

(84)

$$\begin{aligned} F_2[n,m]&=\langle n|F_2| m\rangle = -i \int _0^\mathrm{T} \exp (i(E_n-E_m)t_1) \langle n|V_2(t_1)|m\rangle \hbox {d}t_1 \nonumber \\&\quad - \sum _p \int _{0<t_2<t_1<T}\exp (i(E_n-E_p)t_1\nonumber \\&\quad +(E_p-E_m)t_2) \langle n| V_1(t_1)|p \rangle \langle p| V_1(t_2)|m\rangle \hbox {d}t_1\hbox {d}t_2\end{aligned}$$

(85)

$$\begin{aligned} F_3[n,m]&= \langle n|F_3|m\rangle = i\sum _{p_1,p_2}\int _{0<t_3<t_2<t_1<T}\nonumber \\&\quad \times \,\exp \bigg (i(E_n-E_{p_1})t_1+ (E_{p_1}-E_{p_2})t_2+ (E_{p_2}-E_m)t_3\bigg )\nonumber \\&\quad \langle n|V_1(t_1)|p_1\rangle \langle p_1|V_1(t_2)|p_2\rangle \langle p_2|V_1(t_3)|p_3 \rangle \hbox {d}t_1\hbox {d}t_2\hbox {d}t_3\nonumber \\&\quad -\sum _p\int _{0<t_2<t_1<T} \exp (i (E_n-E_p)t_1+(E_p-E_m)t_2) \nonumber \\&\quad \times \bigg (\langle n| V_1(t_1)|p \rangle \langle p|V_2(t_2)|m \rangle + \langle n|V_2(t_1)|p\rangle \langle p| V_1(t_2)|m \rangle \bigg ) \hbox {d}t_1\hbox {d}t_2 \end{aligned}$$

(86)

Thus $F_1[n,m]= \sum _k I_k G_{1k}[n,m]$, where

$$\begin{aligned} G_{1k}[n,m]= -i \int _0^\mathrm{T} \exp (i E(n,m)t_1)\langle n| V_{1k}(t_1)|m\rangle \hbox {d}t_1 \end{aligned}$$

(87)

similarly, we can write

$$\begin{aligned} F_2[n,m]=\sum \limits _{k,r}I_kI_rG_{2kr}[n,m] \end{aligned}$$

(88)

where

$$\begin{aligned} G_{2kr}[n,m]&= -i \int _0^\mathrm{T} \exp (i(E(n,m)t_1)\langle n| V_{2kr}(t_1)|m \rangle )\hbox {d}t_1\nonumber \\&\quad -\sum _p\int _{0<t_2<t_1<T}\exp (i(E(n,p)t_1)\nonumber \\&\quad +E(p,m)t_2) \langle n|V_{1k} (t_1)|p \rangle \langle p| V_{1r}(t_2)|m\rangle \hbox {d}t_1\hbox {d}t_2 \end{aligned}$$

(89)

Finally

$$\begin{aligned} F_3[n,m]=\sum \limits _{krs} I_k I_r I_s G_{3krs}[n,m] \end{aligned}$$

(90)

where

$$\begin{aligned}&G_{3krs}[n,m]\nonumber \\&\quad = i\sum \limits _{p_1,p_2}\int _{0<t_3<t_2<t_1<T} \exp \bigg \{i \bigg (E(n,p_1)t_1+E(p_1,p_2)t_2+E(p_2,m)t_3\bigg )\bigg \} \nonumber \\&\quad \quad \times \langle n| V_{1k}(t_1)|p_1 \rangle \langle p_1| V_{1r}(t_2)|p_2\rangle \langle p_2 |V_{1s}(t_3)|m\rangle \hbox {d}t_1\hbox {d}t_2\hbox {d}t_3\nonumber \\&\quad \quad - \sum \limits _p \int _{0<t_2<t_1<T} \exp \bigg \{i\bigg (E(n,p)t_1+E(p,m)t_2\bigg )\bigg \}\nonumber \\&\quad \quad \times \bigg (\langle n|V_{1k}(t_1)|p\rangle \langle p|V_{2rs}(t_2)|m\rangle + \langle n|V_{2rs}(t_1)|p\rangle \langle p|V_{1k}(t_2)|m\rangle \bigg )\hbox {d}t_1\hbox {d}t_2 \end{aligned}$$

(91)

The notation E(n, p) can be used in this way $E(n,p)=E_n-E_p$. Equivalently we can write

$$\begin{aligned} F_1&=F_1[n,m]=\sum \limits _kI_k G_{1k}\end{aligned}$$

(92)

$$\begin{aligned} F_2&= \sum \limits _{k,r}I_kI_rG_{2kr}\end{aligned}$$

(93)

$$\begin{aligned} F_3&=\sum \limits _{k,r,s}I_kI_rI_s G_{3krs} \end{aligned}$$

(94)

It should be noted that $F_k[n,m]$ are, respectively, the matrix elements of the operator $F_k$ defined in Eqs. (81), (82) and (83), where $G_{1k}, G_{2kr}, G_{3krs}$ are the matrix coefficients of $I_k, I_kI_r, I_kI_rI_s$ in $F_1, F_2, F_3$, respectively.