---
abstract: 'We propose a non-deterministic CNOT gate based on a quantum cloner, a quantum switch based on all optical routing of single photon by single photon, a quantum-dot spin in a double-sided optical microcavity with two photonic qubits, delay lines and other linear optical photonic devices. Our CNOT provides a fidelity of 78% with directly useful outputs for a quantum computing circuit and requires no ancillary qubits or electron spin measurements.'
author:
- '[Amor Gueddana$^{1,\,2}$ , Peyman Gholami$^{1}$ and Vasudevan Lakshminarayanan$^{1,\,3}$]{}'
bibliography:
- 'Quantum\_Bib\_2019.bib'
title: '[Can A Universal Quantum Cloner Be Used to Design an Experimentally Feasible Near-Deterministic CNOT Gate ?]{}'
---

1\. Theoretical & Experimental Epistemology Lab, TEEL, School of Optometry and Vision Science, University of Waterloo 200 University Avenue West, Waterloo, Ontario N2l 3G1, Canada.

2\. Green & Smart Communication Systems Lab, Gres’Com, Engineering School of Communication of Tunis, Sup’Com, University of Carthage, Ghazela Technopark, 2083, Ariana, Tunisia.

3\. Department of Physics, Department of Electrical and Computer Engineering and Department of Systems Design Engineering , University of Waterloo 200 University Avenue West, Waterloo, Ontario N2l 3G1, Canada.

Introduction
============

Physical implementation of the photonic quantum computer and secure optical quantum communication systems are based on photonic quantum gates [@Kok_2007]. The gate which is universal for building all quantum circuits is the Controlled-NOT gate (CNOT) [@Shende_2006; @Maslov_2008]. Experimentally realizable photonic CNOT gates are those based on linear optical devices [@Pittman_2001; @Ralph_2002; @OBrien_2003; @Pittman_2003; @Gasparoni_2004; @Pittman_2004; @Bao_2007; @Clark_2009; @Gueddana_2013a; @Gueddana_2015]. These gates have success probabilities less than 1/4. Further improvement in the success probability of this CNOT model is not possible because the best success probability for the CNOT functioning is 3/4 when using linear optical devices [@Knill_2003]. This is a major hurdle for realization of all complex versions of the quantum computing circuits, since the success probability of such circuits may be very low due to multiple combinations of many non-deterministic CNOTs . To overcome this inefficiency, other techniques such as superconducting qubits [@Plantenberg_2007] have to be employed. Other work using non-linearities have been proposed in order to achieve non-deterministic CNOTs with high success probability [@Luo_2016; @Li_2013]. Other designs of CNOTs are based on spin of electron trapped in Quantum Dot (QD) and confined in a double sided optical micro-cavity [@Wei_2013; @Wei_2014; @Wang_2013; @Luo_2014; @Bonato_2010]. Although the fidelity values of these CNOTs seems to be near unity, in practice all these designs have major drawbacks. This is because of physical constraints that make them less effective in serial or parallel combinations. For the special CNOT model of Wang *et al* [@Wang_2013], we presented comments on the parametric values taken for the simulation and showed that the proposed CNOT gate is valid only in the strong coupling regime [@Gueddana_2018d]. In this paper, we refer to this same CNOT and show that it provides a fidelity of only 47% in a realistic implementation (while it is 93.7% in the theoretical case), then we propose an optimized design based on a quantum cloner.

Several theoretical and experimental work has addressed quantum cloning machines providing optimal polarization cloning of single photons when using either Parametric Down Conversion (PDC) or photon bunching on Beam Splitter (BS). Fasel *et al [@Fasel_2002]* proposed close-to optimal quantum cloning of the polarization state of light using standard Erbium dopped fiber for amplification and provided a fidelity $F_{cloner}$ equal to 0.82. Linares *et al* [@Lamas_Linares_2002] proposed a cloning technique, based on stimulated emission in PDC using non linear $\beta$-Barium borate (BBo) crystal, and obtained experimental results with $F_{cloner}=0.81$. Martini *et al* [@Martini_2004] used for cloning a BBo crystal slab cut for type II phase matching for implementing optimal cloning and NOT gate. Bartuskova *et al* [@Bartifmmodemathringuelserufiifmmodecheckselsevsfikova_2007] addressed in their work a phase covariant $1\rightarrow2$ qubit cloner based on two single mode optical fiber, non linear crystal, attenuator and phase modulator, providing a fidelity of $F_{cloner}=0.854$, which slightly surpasses the theoretical optimal value of the Universal Cloner (UC), denoted $F_{UC}=5/6$. The question is: can these cloners be used for designing photonic CNOT gates? The main objective of this paper is to answer this question. Therefore, this paper is composed of five sections: Section 2 describes the several photonic components used for the CNOT design in the imperfect case. Section 3 presents the concept and modelling of the photonic CNOT using a quantum cloner and two quantum switches. In section 4, we present the simulation results and the corresponding experimental realization challenges. Finally, a conclusion is given in section 5.

Imperfect photonic devices
==========================

Let us first consider the basic photonic components in the imperfect case as illustrated by figure \[fig:1\].

Figure \[fig:1a\] illustrates a Half Wave Plate (HWP) with arbitrary error[ $\xi$]{}. For inputs right-circularly polarized single photon denoted[$\left|R\right\rangle $ ]{}and left-circularly polarized single photon denoted[$\left|L\right\rangle $,]{} the HWP behaves as follows:

[ $$\begin{array}{c}
\left|R\right\rangle \rightarrow\sqrt{\frac{1-\xi}{2}}\left|R\right\rangle +\sqrt{\frac{1+\xi}{2}}\left|L\right\rangle ;\,\,\,\left|L\right\rangle \rightarrow\sqrt{\frac{1-\xi}{2}}\left|R\right\rangle -\sqrt{\frac{1+\xi}{2}}\left|L\right\rangle \end{array}\label{eq:1}$$ ]{}

An ideal Circular Polarizing Beam Splitter (CPBS) transmits[ $\left|R\right\rangle $]{} and totally reflects[ ]{}[$\left|L\right\rangle $.]{} When considering a CPBS with arbitrary errors[ $\tau_{R}$]{} and [$\tau_{L}$ ]{}on [$\left|R\right\rangle $]{} and [$\left|L\right\rangle $,]{} and for an arbitrary incident input state[ $\alpha\left|R\right\rangle +\beta\left|L\right\rangle $]{} (figure \[fig:1b\]), the CPBS transmits the state[ $\sqrt{\alpha-\tau_{R}}\left|R\right\rangle +\sqrt{\tau_{L}}\left|L\right\rangle $ ]{}and reflects the state [$\sqrt{\tau_{R}}\left|R\right\rangle +\sqrt{\beta-\tau_{L}}\left|L\right\rangle $.]{}

A quantum switch (SW) has two inputs $I_{1}$ and $I_{2}$, and two outputs $O_{1}$ and $O_{2}$. The transmittance ( solid arrows in figure\[fig:1c\]) and reflectance ( dashed arrows in figure\[fig:1c\]) coefficients from $I_{1}$ and $I_{2}$ to $O_{1}$ and $O_{2}$, are denoted by $T_{1,2}$, $T_{2,1}$, $R_{1,1}$ and $R_{2,2}$. Several works addressed physical implementation of the quantum SW [@OShea_2013; @Smart_2014; @Sun_2016; @Volz_2012]. The switching process considered in this work is based on all optical routing of single photon by single photon, without any additional control field [@Shomroni_2014]. This switch is based on three-level atomic $\Lambda$-configuration, with two different transitions denoted $\sigma^{+}$ and $\sigma^{-}$, and representing transitions coupled only to the right or to the left photonic mode propagation, respectively. Each of the inputs $I_{1}$ or $I_{2}$, may be either transmitted or reflected into $O_{1}$ or $O_{2}$, depending on the atom’s state, being either on the $m_{F}=+1$ state or $m_{F}=-1$. When the atom is in this latter state, incoming photon on $I_{1}$ in the state $\sigma^{+}$ are reflected to $O_{1}$ and their state becomes $\sigma^{-}$, which toggles the atom’s state to $m_{F}=+1$. When the atom’s state is $m_{F}=+1$, it doesn’t interact with $\sigma^{+}$ photons from $I_{1}$ and they are totally transmitted to $O_{2}$. The whole process is symmetric for $I_{2}$.

Other photonic devices used in our optimized model are the Beam Splitter (BS), the Quarter Wave Plate (QWP) and Delay Line (DL). In this work, we consider these devices only in the ideal case.

Optimized CNOT gate model based on a quantum cloner 
====================================================

With the previously mentioned components , we propose a CNOT architecture using the quantum universal cloner as illustrated by figure \[fig:2\].

The central part of the CNOT shaded grey is the proposed CNOT [@Wang_2013] under optimization. This CNOT is based on spin of electron in a QD trapped in a double sided optical microcavity which behaves like a BS [@Hu_2009]. Inside the grey background, we consider two input photons 1 and 2, being the control and target photons, with initial states$\left|\Psi_{ph}^{1}\right\rangle $ and $\left|\Psi_{ph}^{2}\right\rangle $, respectively.

![\[fig:2\] CNOT gate optimized model. ](Fig2){width="0.55\columnwidth"}

The electron spin state inside the QD is denoted $\left|\Psi_{s}\right\rangle $. Consider the following initial states:

[ $$\begin{array}{c}
\left|\Psi_{ph}^{1}\right\rangle =\alpha\left|R_{1}\right\rangle +\beta\left|L_{1}\right\rangle \\
\Psi_{ph}^{2}=\delta\left|R_{2}\right\rangle +\gamma\left|L_{2}\right\rangle \\
\left|\Psi_{s}\right\rangle =\left(\left|\uparrow_{s}\right\rangle -\left|\downarrow_{s}\right\rangle \right)/\sqrt{2}
\end{array}\label{eq:2}$$ ]{}

The two photons come successively to interact with the optical micro-cavity. For the coupled cavity, when considering equal the frequencies of the input photon, cavity mode and the spin-dependent optical transition, the reflection and transmission coefficients of the double sided optical micro-cavity system used in the CNOT design, are denoted $r\left(\omega\right)$ and $t\left(\omega\right)$, respectively, and are given by [@Wang_2013; @Bonato_2010; @Hu_2009]:

[ $$t\left(\omega\right)=-\frac{2\gamma\kappa}{\gamma\left(2\kappa+\kappa_{s}\right)+4g^{2}};\,\,\,\,\,r\left(\omega\right)=1+t\left(\omega\right)\label{eq:3}$$ ]{}

where $g$ is the coupling strength, $\kappa$ and $\kappa_{s}/2$ are the cavity field decay rate into the input/output modes and the leaky modes, respectively, and $\gamma/2$ is the $X^{-}$ dipole decay rate. For the uncoupled cavity, the reflection and transmission coefficients are denoted $r_{0}\left(\omega\right)$ and $t_{0}\left(\omega\right)$ , and they are directly obtained from equation \[eq:3\] for $g=0$. For a realistic spin cavity unit, the side leakage and cavity loss can not be neglected. In this case, $t\left(\omega\right)$ in the coupled cavity and $r_{0}\left(\omega\right)$ in the uncoupled cavity will introduce bit-flip errors. Relevant energy levels and optical selection rules for exciton $X^{-}$ inside the single charged GaAs/ InAs QD, have been well detailed in [@Wang_2013; @Wei_2014], and the dynamics of the interaction of the QD spin in a double sided optical micro-cavity, for $r_{0}=\left|r_{0}\left(\omega\right)\right|$, $t_{0}=\left|t_{0}\left(\omega\right)\right|$, $r_{1}=\left|r\left(\omega\right)\right|$ and $t_{1}=\left|t\left(\omega\right)\right|$, are given as follows:

[ $$\begin{array}{c}
\left|R^{\downarrow},\,\uparrow_{s}\right\rangle \rightarrow-t_{0}\left|R^{\downarrow},\,\uparrow_{s}\right\rangle -r_{0}\left|L^{\uparrow},\,\uparrow_{s}\right\rangle \\
\left|R^{\downarrow},\,\downarrow_{s}\right\rangle \rightarrow r_{1}\left|L^{\uparrow},\,\downarrow_{s}\right\rangle +t_{1}\left|R^{\downarrow},\,\downarrow_{s}\right\rangle \\
\left|R^{\uparrow},\,\uparrow_{s}\right\rangle \rightarrow r_{1}\left|L^{\downarrow},\,\uparrow_{s}\right\rangle +t_{1}\left|R^{\uparrow},\,\uparrow_{s}\right\rangle \\
\left|R^{\uparrow},\,\downarrow_{s}\right\rangle \rightarrow-t_{0}\left|R^{\uparrow},\,\downarrow_{s}\right\rangle -r_{0}\left|L^{\downarrow},\,\downarrow_{s}\right\rangle \\
\left|L^{\downarrow},\,\uparrow_{s}\right\rangle \rightarrow r_{1}\left|R^{\uparrow},\,\uparrow_{s}\right\rangle +t_{1}\left|L^{\downarrow},\,\uparrow_{s}\right\rangle \\
\left|L^{\downarrow},\,\downarrow_{s}\right\rangle \rightarrow-t_{0}\left|L^{\downarrow},\,\downarrow_{s}\right\rangle -r_{0}\left|R^{\uparrow},\,\downarrow_{s}\right\rangle \\
\left|L^{\uparrow},\,\uparrow_{s}\right\rangle \rightarrow-t_{0}\left|L^{\uparrow},\,\uparrow_{s}\right\rangle -r_{0}\left|R^{\downarrow},\,\uparrow_{s}\right\rangle \\
\left|L^{\uparrow},\,\downarrow_{s}\right\rangle \rightarrow r_{1}\left|R^{\downarrow},\,\downarrow_{s}\right\rangle +t_{1}\left|L^{\uparrow},\,\downarrow_{s}\right\rangle 
\end{array}\label{eq:45}$$ ]{}

Photon 1 first passes through HWP1, then it travels through the optical micro-cavity and then it passes through HWP2. The two switches used in the CNOT are denoted SW1 and SW2, the transmittance and reflectance coefficients of SW1 are denoted $T_{1,2}^{1}$, $T_{2,1}^{1}$, $R_{1,1}^{1}$ and $R_{2,2}^{1}$, while they are $T_{1,2}^{2}$, $T_{2,1}^{2}$, $R_{1,1}^{2}$ and $R_{2,2}^{2}$ for SW2. No details were given about SW1 and SW2 in [@Wang_2013], and they were supposed to switch between photons 1 and 2 perfectly. After a certain time defined by DL1, photon 2 is switched by SW1 and injected to the spin cavity system, but before entering and after leaving the system, two Hadamard transforms are performed on the electron spin state, through $\nicefrac{\pi}{2}$ microwave pulses [@Wang_2013; @Bonato_2010], which transforms the state [$\left|\uparrow_{s}\right\rangle \rightarrow\left(\left|\uparrow_{s}\right\rangle +\left|\downarrow_{s}\right\rangle \right)/\sqrt{2}$]{} and[ $\left|\downarrow_{s}\right\rangle \rightarrow\left(\left|\uparrow_{s}\right\rangle -\left|\downarrow_{s}\right\rangle \right)/\sqrt{2}$]{}. After being switched by SW2, photon 2 is delayed by DL2 to wait for the interaction between photon 1 and it’s clone.

For the inputs of equation \[eq:2\], the state at the output of the CNOT is then transformed as follows:

[ $$\begin{array}{c}
\left|\Psi_{ph}^{1}\right\rangle \otimes\left|\Psi_{ph}^{2}\right\rangle \otimes\left|\Psi_{s}\right\rangle \rightarrow\\
\left(\alpha\delta\left|R_{1}\right\rangle \left|R_{2}\right\rangle +\alpha\gamma\left|R_{1}\right\rangle \left|L_{2}\right\rangle -\beta\delta\left|L_{1}\right\rangle \left|L_{2}\right\rangle -\beta\gamma\left|L_{1}\right\rangle \left|R_{2}\right\rangle \right)\left|\uparrow_{s}\right\rangle \\
+\left(\alpha\delta\left|R_{1}\right\rangle \left|R\right\rangle _{2}+\alpha\gamma\left|R_{1}\right\rangle \left|L_{2}\right\rangle +\beta\delta\left|L_{1}\right\rangle \left|L_{2}\right\rangle +\beta\gamma\left|L_{1}\right\rangle \left|R_{2}\right\rangle \right)\left|\downarrow_{s}\right\rangle 
\end{array}\label{eq:4}$$ ]{}

It is clear from equation \[eq:4\] that the CNOT function is correctly entangled to the spin state $\left|\downarrow_{s}\right\rangle $, but a $\left(-\right)$ sign is introduced to the CNOT when photon 1 is in the state $\left|L_{1}\right\rangle $ and both photons are entangled with the spin state $\left|\uparrow_{s}\right\rangle $. A measurement of the spin is required to determine the spin state, and then decide whether to apply an identity $\left(I\right)$ or a negation gate ($\sigma_{z}$) on photon 1, to get correct CNOT entangled with both $\left|\uparrow_{s}\right\rangle $ and $\left|\downarrow_{s}\right\rangle $ states. This heralded function has a fidelity of 94 %, which means that the correct CNOT performed only with 47 % of success probability. However, without spin measurement, this CNOT model cannot be used in a serial or parallel combination, since the success probability of the entire circuit will decrease exponentially.

Our main idea is to eliminate this $\left(-\right)$ sign at the output, in order to be independent of the spin state for further circuit realization. The idea is to apply a $\sigma_{z}$ transform on photon 1, being at the state $\left|L_{1}\right\rangle $, only when the spin state at the output is $\left|\uparrow_{s}\right\rangle $. A measurement of the spin state inside a QD has been addressed in [@Hu_2009]: if we have an horizontally-polarized $\left(\left|H\right\rangle \right)$ or vertically-polarized $\left(\left|V\right\rangle \right)$ single photon at the input of QD spin system, being initially at the state $\mu\left|\uparrow_{s}\right\rangle +\nu\left|\downarrow_{s}\right\rangle $, it is possible using a QWP after the QD system to transmit the state of the electron to the photon as $\mu\left|R\right\rangle +\nu\left|L\right\rangle $. Based on this idea, we use in our architecture a UC to clone photon 1 and produce another photon 1’ in the same state $\left|\Psi_{ph}^{1'}\right\rangle =\left|\Psi_{ph}^{1}\right\rangle =\alpha\left|R_{1'}\right\rangle +\beta\left|L_{1'}\right\rangle $. $\sigma_{z}$ gate. After traversing QWP1, the state of photon 1’ becomes $\left|\Psi_{ph}^{1'}\right\rangle =\alpha\left|H_{1'}\right\rangle +\beta\left|V_{1'}\right\rangle $. Photon 1’ is then delayed by DL3 to wait for photons 1 and 2 to pass the QD system and alter the spin state (the spin state is initially given by equation \[eq:2\], and after two imperfect Hadamard gates, it becomes $\mu\left|\uparrow_{s}\right\rangle +\nu\left|\downarrow_{s}\right\rangle $, for $\mu\approx\nu\approx\nicefrac{1}{2}$). Photon 1’ passes through the QD spin system and after QWP2, it’s state becomes $\left|\Psi_{ph}^{1'}\right\rangle =\mu\left|R_{1}\right\rangle +\nu\left|L_{1}\right\rangle $. CPBS4 transmits $\mu\left|R_{1}\right\rangle $ while $\nu\left|L_{1}\right\rangle $ is discarded. The transmitted $\mu\left|R_{1}\right\rangle $ is flipped to $\mu\left|L_{1}\right\rangle $ by HWP3. At this level, photon 1’ is present with same probability amplitude $\mu$ of the electron spin being at the state $\left|\uparrow_{s}\right\rangle $, moreover, it is exactly in the same mode of photon 1, this allows it to serve as control for the $\sigma_{z}$ gate. This gate should perform a $\left(-\right)$ sign only to photon 1 being at the state $\left|L_{1}\right\rangle $, this is the role of CPBS2 and CPBS3. Finally, the time interval between all photons, the paths lengths traveled by photons and the time delay of DL1, DL2 and DL3, should take into consideration cavity photon lifetime and single charged electron spin coherence time [@Wang_2013].

We consider $\xi_{1}$ and $\xi_{2}$ to be the errors related to HWP1 and HWP2. We suppose that $\left(\tau_{R}^{1},\,\tau_{L}^{1}\right)$,$\left(\tau_{R}^{2},\,\tau_{L}^{2}\right)$, $\left(\tau_{R}^{3},\,\tau_{L}^{3}\right)$ and $\left(\tau_{R}^{4},\,\tau_{L}^{4}\right)$ are the errors related to CPBS1, CPBS2, CPBS3 and CPBS4, respectively. For simplicity, we neglect errors due to QWP1, QWP2 and HWP3. For the same inputs of equation \[eq:2\], we compute the output of the optimized CNOT and we obtain:

[ $$\begin{array}{c}
\left|\Psi_{ph}^{1}\right\rangle \otimes\left|\Psi_{ph}^{2}\right\rangle \otimes\left|\Psi_{s}\right\rangle \rightarrow\sqrt{T_{1,2}^{1}\,R_{2,2}^{1}\,T_{1,2}^{2}\,R_{1,1}^{2}\,F_{cloner}}\\
\times\left(\left(\eta_{1}\left|R_{1}\right\rangle \left|R_{2}\right\rangle +\eta_{2}\left|R_{1}\right\rangle \left|L_{2}\right\rangle +\eta_{3}\left|L_{1}\right\rangle \left|L_{2}\right\rangle +\eta_{4}\left|L_{1}\right\rangle \left|R_{2}\right\rangle \right)\left|\uparrow_{s}\right\rangle \right.\\
\left.+\left(\eta_{5}\left|R_{1}\right\rangle \left|R_{2}\right\rangle +\eta_{6}\left|R_{1}\right\rangle \left|L_{2}\right\rangle +\eta_{7}\left|L_{1}\right\rangle \left|L_{2}\right\rangle +\eta_{8}\left|L_{1}\right\rangle \left|R_{2}\right\rangle \right)\left|\downarrow_{s}\right\rangle \right)
\end{array}\label{eq:5}$$ ]{}

[where:]{}

[ $$\begin{array}{c}
\eta_{1}=\frac{\sqrt{1-\xi_{2}}}{2\sqrt{2}}\left(\left(a_{2}a_{2}^{'}+a_{4}a_{4}^{'}-a_{1}a_{1}^{'}-a_{3}a_{3}^{'}\right)\left(\delta\delta^{'}+\gamma\gamma^{'}\right)+\right.\\
\left.\left(a_{2}a_{2}^{"}+a_{4}a_{4}^{"}-a_{1}a_{1}^{"}-a_{3}a_{3}^{"}\right)\left(\delta\delta^{"}+\gamma\gamma^{"}\right)\right)\\
a_{1}=\left(\alpha+\beta\right)\sqrt{\left(1-\tau_{R}^{1}\right)\left(1-\xi_{1}\right)/2}\\
a_{2}=\left(\alpha+\beta\right)\sqrt{\tau_{R}^{1}\left(1-\xi_{1}\right)/2}\\
a_{3}=\left(\alpha-\beta\right)\sqrt{\left(1-\tau_{L}^{1}\right)\left(1+\xi_{1}\right)/2}\\
a_{4}=\left(\alpha-\beta\right)\sqrt{\tau_{L}^{1}\left(1+\xi_{1}\right)/2}\\
a_{1}^{'}=\sqrt{\left(1-\tau_{R}^{1}\right)}\left(t_{0}+t_{1}\right)+\sqrt{\left(1-\tau_{L}^{1}\right)}\left(r_{0}+r_{1}\right)\\
a_{2}^{'}=\sqrt{\tau_{R}^{1}}\left(t_{0}+t_{1}\right)+\tau_{L}^{1}\left(r_{0}+r_{1}\right)\\
a_{3}^{'}=\sqrt{\left(1-\tau_{R}^{1}\right)}\left(r_{0}+r_{1}\right)+\sqrt{\left(1-\tau_{L}^{1}\right)}\left(t_{0}+t_{1}\right)\\
a_{4}^{'}=\sqrt{\tau_{R}^{1}}\left(r_{0}+r_{1}\right)+\sqrt{\tau_{L}^{1}}\left(t_{0}+t_{1}\right)\\
a_{1}^{"}=\sqrt{\left(1-\tau_{R}^{1}\right)}\left(t_{0}-t_{1}\right)+\sqrt{\left(1-\tau_{L}^{1}\right)}\left(r_{0}-r_{1}\right)\\
a_{2}^{"}=\sqrt{\tau_{R}^{1}}\left(t_{1}-t_{0}\right)+\tau_{L}^{1}\left(r_{1}+r_{0}\right)\\
a_{3}^{"}=\sqrt{\left(1-\tau_{R}^{1}\right)}\left(r_{0}-r_{1}\right)+\sqrt{\left(1-\tau_{L}^{1}\right)}\left(t_{0}-t_{1}\right)\\
a_{4}^{"}=\sqrt{\tau_{R}^{1}}\left(r_{1}+r_{0}\right)+\sqrt{\tau_{L}^{1}}\left(t_{1}-t_{0}\right)\\
\delta^{'}=t_{1}\tau_{R}^{1}-t_{0}\left(1-\tau_{R}^{1}\right)\\
\gamma^{'}=r_{1}\sqrt{\tau_{R}^{1}\tau_{L}^{1}}-r_{0}\sqrt{\left(1-\tau_{R}^{1}\right)\left(1-\tau_{L}^{1}\right)}\\
\delta^{"}=t_{1}\left(1-\tau_{R}^{1}\right)-t_{0}\tau_{R}^{1}\\
\gamma^{"}=r_{1}\sqrt{\left(1-\tau_{R}^{1}\right)\left(1-\tau_{L}^{1}\right)}-r_{0}\sqrt{\tau_{R}^{1}\tau_{L}^{1}}
\end{array}\label{eq:6}$$ ]{}

and $\left\{ \eta_{i}\right\} _{2\leq i\leq8}$ have all the same form of equation \[eq:6\].

It is worth highlighting the fact that [$\Theta=\left(-1\right)\times\sqrt{\left(1-\tau_{L}^{2}\right)\left(1-\tau_{L}^{3}\right)\left(1-\tau_{R}^{4}\right)}$ i]{}s the operator that will eliminate the (-) sign for the CNOT being entangled with $\left|\uparrow_{s}\right\rangle $, this operator appears only in $\eta_{3}$ and $\eta_{4}$ of equation \[eq:5\]. This is the main contribution in this work since the CNOT function is correctly entangled with both $\left|\uparrow_{s}\right\rangle $ and $\left|\downarrow_{s}\right\rangle $ spin states. To measure the performance of our optimized CNOT, we refer to the fidelity denoted $F_{CNOT}$ and given by [@Wang_2013]:

[ $$F_{CNOT}=\left\langle \overline{\Psi_{in}|U_{CNOT}^{\dagger}\rho_{t}U_{CNOT}|\Psi_{in}}\right\rangle \label{eq:9}$$ ]{}

where the upper line indicates that the fidelity is obtained according to the average over all possible four input states $\left|\Psi_{in}\right\rangle $, $U_{CNOT}$ is the ideal CNOT transform, $\rho_{t}=\left|\Psi_{out}\right\rangle \left\langle \Psi_{out}\right|$, with $\left|\Psi_{out}\right\rangle $ is the state at the output of the CNOT for the specific $\left|\Psi_{in}\right\rangle $ input.

Simulation results and experimental challenges
==============================================

A first simulation concerns only the original proposed CNOT, where we study the impact of the errors of HWP1, HWP2 and CPBS1. To this end, we consider perfect SW1 and SW2[ $\left(T_{1,2}^{1}=T_{1,2}^{2}=R_{2,2}^{1}=R_{1,1}^{2}=1\right)$]{}, and we vary all errors around a realistic range of $10^{-2}$.

We illustrate in figures \[fig:3a\] and \[fig:3b\], the average fidelities for spin $\left|\uparrow_{s}\right\rangle $ and $\left|\downarrow_{s}\right\rangle $ states, denoted $\overline{F}_{CNOT}^{\uparrow}$ and $\overline{F}_{CNOT}^{\downarrow}$, versus the normalized coupling strength. Here we have set $\gamma=0.1\kappa$.

In [@Wang_2013], errors due to HWP1, HWP2 and CPBS1 have been neglected and it has been shown that the CNOT provides best $\overline{F}_{CNOT}^{\uparrow}$ or $\overline{F}_{CNOT}^{\downarrow}$ value around 93.74% for the strong coupling regime (obtained for $g>\left(\kappa_{s}+\kappa\right)/4$), and 32.34 % for the weak coupling regime (obtained when $g<\left(\kappa_{s}+\kappa\right)/4$). If we consider the same parameters used for the strong coupling ($\kappa_{s}=0.05\kappa$ and $g=2.5\kappa$) and weak coupling ($\kappa_{s}=1.0\kappa$ and $g=0.45\kappa$), our simulation shows that the errors affect the fidelities $\overline{F}_{CNOT}^{\uparrow}$ and $\overline{F}_{CNOT}^{\downarrow}$ , and we obtain best values around 87.89% and 30.02%, respectively.

Another simulation concerns our optimized model while taking into consideration realistic features of all devices. In this case, we consider SW1 and SW2 realized according to [@Shomroni_2014]. For SW1 being initially in the state $m_{F}=+1$, the experimental results obtained are $T_{1,2}^{1}=89.9\%$, $R_{2,2}^{1}=65\%$. For SW2 being initially in the ground state $m_{F}=-1$, the obtained coefficients are $T_{1,2}^{2}=95.6\%$ and $R_{1,1}^{2}=64.8\%$. We consider the universal cloning of polarization state experimentally realized and providing $F_{cloner}=0.82$ [@Fasel_2002]. We consider arbitrary errors around $10^{-2}$ affecting separately all devices of the CNOT of figure \[fig:2\] (except QWP1, QWP2 and HWP3). With these assumptions, we show in Figure \[fig:4a\] the average fidelity of the CNOT function being correctly entangled with both $\left|\uparrow_{s}\right\rangle $ and $\left|\downarrow_{s}\right\rangle $ , denoted $\overline{F}_{CNOT}^{\uparrow\downarrow}$. Best values according to figure \[fig:4a\] in the strong coupling regime is $\overline{F}_{CNOT}^{\uparrow\downarrow}=26.27\%$.

It is clear that $\overline{F}_{CNOT}^{\uparrow\downarrow}$ value is highly sensitive to the fidelity of the cloner and SW1 and SW2 imperfections, we have considered only the strong coupling regime and the optimal cloner with $F_{UC}=5/6$. We denote $E_{rr}$ the set of all errors affecting the CNOTs components and we vary them in $\left[10^{-4}..10^{-1}\right]$. We consider also the same range of errors separately altering the coefficients $T_{1,2}^{1}$, $T_{1,2}^{2}$, $R_{2,2}^{1}$ and $R_{1,1}^{2}$, therefore, we denote $P_{SW}\simeq T_{1,2}^{1}$$\simeq T_{1,2}^{2}$$\simeq R_{2,2}^{1}$$\simeq R_{1,1}^{2}$, and we illustrate $\overline{F}_{CNOT}^{\uparrow\downarrow}$ depending on $E_{rr}$ and $P_{SW}$ in figure \[fig:4b\]. The best fidelity permitted by our CNOT for lowest error range and $P_{SW}$ approaching unity is $\overline{F}_{CNOT}^{\uparrow\downarrow}$=78%. This fidelity is very close to $F_{UC}$ and our optimized CNOT is very advantageous since neither a measurement of the electron spin state nor an extra treatment are required to allow using the CNOT outputs as inputs for another circuits.

Conclusion
==========

We propose a quantum CNOT gate that overcomes the inefficiencies of a previously published CNOT design based on Quantum-Dot system. The previous proposal provides fidelity of 94 % but with a heralded functioning, which means that the correct CNOT performed only with 47 % probability success. Our CNOT functioning is not heralded by any spin states and it provides a success of 78%. This is a highly innovative result since it uses the quantum Cloner. This design will lead to another CNOT that uses the cloner with better fidelity approaching the cloner optimal limit of 5/6 and will allow possible generalization of the CNOT to all $C^{n}NOT$ photonic gates.
