Gate teleportation using Parallel and Series entanglement distributions on arbitrary remote states
Mohanpur Campus
Nadia-741252
West Bengal
India
Dr. Homi Bhabha Road
Pashan
Pune-411008
Maharashtra
We consider two different entanglement distributions; Series and Parallel, between remote parties for implementing non-local controlled-Unitary gates. In parallel network, each of the control parties shares one entangled state with the target party and none of the control parties shares entanglement between them. In series network, the entanglement distribution forms a series, such that the control parties share entanglement with the adjacent control parties and only one of them shares entanglement with the target party, which is located at the end of the network. We demonstrate the protocol of simultaneous teleportation of controlled operation through both the distributions of Bell state entangled channels using local operations and classical communications (LOCC). For the parallel network, the communication cost of implementing controlled-Unitary gate on `n' qubit arbitrary state is $(n-1)$ ebits and $2(n-1)$ cbits. Whereas, in general the series network allows controlled-Hermitian gate implementation on arbitrary state using $(n-1)$ ebits and $(n^2+n-2)/2$ cbits. We then explicate the protocol for complete teleportation of controlled-controlled-Unitary gate, a generalized form of Toffoli gate and subsequently generalize the protocol for `n'-qubit gate.
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\begin{document}
\title{\bf Gate teleportation using Parallel and Series entanglement distributions on arbitrary remote states}
\author{\it Debashis Saha$^{\footnote{debashis169@gmail.com}}$, Sanket Nandan$^{\footnote{sanketnandan@gmail.com}}$  and Prasanta K. Panigrahi$^{\footnote{pprasanta@iiserkol.ac.in}}$\\
$^{*}$$^{,}$$^{\ddagger}$\it Indian Institute of Science Education and Research Kolkata,\\
 \it Mohanpur Campus, Nadia-741252, West Bengal, India\\
$^{\dagger}$\it Indian Institute of Science Education and Research Pune,\\
 \it Dr. Homi Bhabha Road, Pashan, Pune-411008, Maharashtra, India}


\maketitle
 


\begin{abstract}

We consider two different entanglement distributions; Series and Parallel, between remote parties for implementing non-local controlled-Unitary gates. In parallel network, each of the control parties shares one entangled state with the target party and none of the control parties shares entanglement between them. In series network, the entanglement distribution forms a series, such that the control parties share entanglement with the adjacent control parties and only one of them shares entanglement with the target party, which is located at the end of the network. We demonstrate the protocol of simultaneous teleportation of controlled operation through both the distributions of Bell state entangled channels using local operations and classical communications (LOCC). For the parallel network, the communication cost of implementing controlled-Unitary gate on `n' qubit arbitrary state is $(n-1)$ ebits and $2(n-1)$ cbits. Whereas, in general the series network allows controlled-Hermitian gate implementation on arbitrary state using $(n-1)$ ebits and $(n^2+n-2)/2$ cbits. We then explicate the protocol for complete teleportation of controlled-controlled-Unitary gate, a generalized form of Toffoli gate and subsequently generalize the protocol for `n'-qubit gate.

\end{abstract}

\subsubsection*{Keywords:}  Bell state, controlled-Unitary gate, controlled-Hermitian gate, Toffoli gate, Gate teleportation.

\section{INTRODUCTION}

The counter-intuitive and one of the most striking features of the quantum world is entanglement, which has found practical use in the field of communication and cryptography [1]. Various quantum communication protocols, like qubit teleportation [2,3], superdense coding [4], quantum information splitting [5], secret sharing [6], remote state preparation [7,8] have been theoretically and experimentally demonstrated through entangled channels. Gate teleportation is another important protocol implementing a multi-partite quantum gates between qubits, which are spatially distributed. The protocol requires application of a unitary gate, that can not be decomposed into individual local operations i.e., $|\psi\rangle_{\mathcal{AB}}\longrightarrow\mathcal{U}|\psi\rangle_{\mathcal{AB}}$,
where $\mathcal{U}\mathcal{\neq U_{A}\otimes U_{B}}$. This can be achieved using entangled channels shared by the remote parties.  In the context of distributed quantum computing, it is necessary to produce controlled operations between remote parties, which may be more than two in number. Together with the local single particle operations, these can produce the necessary entanglement between remote parties for implementation of quantum tasks.

As is well known, controlled-NOT, together with the single qubit Hadamard gate, form the universal gates to which other gates can be decomposed [9]. Controlled-Unitary gates play significant role in quantum computation. In principle, these gates can be teleported, using only CNOT gate teleportation protocol. 
Involving less entanglement and communication costs, several protocols have been carried out implementing non-local multi-partite operations locally by LOCC using entangled channels[10-20] and qubit communication [21,22]. Probabilistic and deterministic gate teleportation using non-maximally entangled state has been exploited [23-26].  Assisted with linear optical manipulations, photon entanglement produced from parametric down-conversion, and post-selection from the coincidence measurements, Huang {\it et al.} [27] teleported the CNOT gate experimentally. 
In contrast with the other protocols, we consider arbitrary multipartite state, either product or entangled, where all the qubits are remote placed and explicate the protocol of simultaneous and `n' qubit controlled operation in two different entanglement distribution.


The paper is organized as follows. We start with a three party scenario, where Alice and Bob simultaneously teleport controlled-Unitary gate to Charlie in parallel entangled network, which is then generalized for arbitrary multi-partite state. Sect. II deals the series entanglement distribution of Bell states among three distant parties to simultaneously teleport controlled-Hermitian gates on `$n$' remote parties. We then proceed to teleport controlled-controlled-Unitary gate and generalize it to $n$-controlled ($n$ parties having the controls) Unitary gate teleportation. Sect. IV is devoted to conclusion and directions for future work. 

\section{SIMULTANEOUS TELEPORTATION OF CONTROLLED-UNITARY GATE USING PARALLEL NETWORK}

In this configuration, we start with a system of three parties (fig. 1), where Alice and Bob are two control parties and Charlie is the target party.  The three qubit arbitrary state is possessed by the parties having one qubit each. Alice and Charlie share one EPR pair between A and $C_1$:
\begin{equation}
| \Phi \rangle_{AC_1} = \frac{1}{\sqrt{2}}( | 00 \rangle + | 11 \rangle ).
\end{equation}
Bob and Charlie share another EPR pair between B and $C_2$:
\begin{equation}
| \Phi \rangle_{BC_2} = \frac{1}{\sqrt{2}}( | 00 \rangle + | 11 \rangle ).
\end{equation}
Thus there is no connection between the control parties and each of the control parties shares entanglement with the target party forming a parallel entanglement connection.
The unknown input state comprising of the qubits a, b and c can be expressed as,
\begin{equation}
| \psi \rangle_{abc} = ( d_0 | 000 \rangle + d_1 | 001 \rangle + d_2 | 010 \rangle + d_3 | 011 \rangle + d_4 | 100 \rangle + d_5 | 101 \rangle + d_6 | 110 \rangle + d_7 | 111 \rangle ),
\end{equation}
with $ \Sigma_{i=0}^{7}| d_i |^2 =1$.
\begin{figure}
\centering
\includegraphics[height=9cm,width=14cm]{fig1}
\caption{Simultaneous $C^{\mathcal{U}}$ teleportation through parallel network.}
\end{figure}
The combined state of all the qubits possessed by Alice, Bob and Charlie is given by,
\begin{equation}
\begin{split}
| \zeta \rangle_{aAbBC_1C_2c}& = (| \psi \rangle_{abc} \otimes | \Phi \rangle_{AC_1} \otimes | \Phi \rangle_{BC_2})\\
& = \frac{1}{2}[( d_0 | 0000000 \rangle + d_1 | 0000001 \rangle + d_2 | 0010000 \rangle + d_3 | 0010001 \rangle\\
& \quad \quad + d_4 | 1000000 \rangle+ d_5 | 1000001 \rangle + d_6 | 1010000 \rangle + d_7 | 1010001 \rangle )\\
& \quad + ( d_0 | 0001010 \rangle + d_1 | 0001011 \rangle + d_2 | 0011010 \rangle + d_3 | 0011011 \rangle\\
& \quad \quad + d_4 | 1001010 \rangle+ d_5 | 1001011 \rangle + d_6 | 1011010 \rangle + d_7 | 1011011 \rangle )\\
& \quad + ( d_0 | 0100100 \rangle + d_1 | 0100101 \rangle + d_2 | 0110100 \rangle + d_3 | 0110101 \rangle\\
& \quad \quad + d_4 | 1100100 \rangle+ d_5 | 1100101 \rangle + d_6 | 1110100 \rangle + d_7 | 1110101 \rangle )\\
& \quad + ( d_0 | 0101110 \rangle + d_1 | 0101111 \rangle + d_2 | 0111110 \rangle + d_3 | 0111111 \rangle\\
& \quad \quad + d_4 | 1101110 \rangle+ d_5 | 1101111 \rangle + d_6 | 1111110 \rangle + d_7 | 1111111 \rangle )]\\
\end{split}
\end{equation}
{\bf Step 1:} In order to achieve remote teleportation on this three qubit gate, Alice first applies controlled-NOT gate $C^N_{aA}$ ($`a$' as the control bit and $`A$' as target bit) and Bob applies controlled-NOT gate $C^N_{bB}$. Step 2 and 3 are given in table 1 and 2 respectively.
%\begin{equation}
%\begin{split}
%C^N_{aA} C^N_{bB} | \zeta \rangle_{aAbBC_1C_2c}& = \frac{1}{2}[( d_0 | 0000000 \rangle + d_1 | 0000001 \rangle + d_2 | 0011000 \rangle + d_3 | 0011001 \rangle\\
%& \quad \quad + d_4 | 1100000 \rangle+ d_5 | 1100001 \rangle + d_6 | 1111000 \rangle + d_7 | 1111001 \rangle )\\
%& \quad + ( d_0 | 0001010 \rangle + d_1 | 0001011 \rangle + d_2 | 0010010 \rangle + d_3 | 0010011 \rangle\\
%& \quad \quad + d_4 | 1101010 \rangle+ d_5 | 1101011 \rangle + d_6 | 1110010 \rangle + d_7 | 1110011 \rangle )\\
%& \quad + ( d_0 | 0100100 \rangle + d_1 | 0100101 \rangle + d_2 | 0111100 \rangle + d_3 | 0111101 \rangle\\
%& \quad \quad + d_4 | 1000100 \rangle+ d_5 | 1000101 \rangle + d_6 | 1011100 \rangle + d_7 | 1011101 \rangle )\\
%& \quad + ( d_0 | 0101110 \rangle + d_1 | 0101111 \rangle + d_2 | 0110110 \rangle + d_3 | 0110111 \rangle\\
%& \quad \quad + d_4 | 1001110 \rangle+ d_5 | 1001111 \rangle + d_6 | 1010110 \rangle + d_7 | 1010111 \rangle )]\\
%\end{split}
%\end{equation}


\begin{table}[tp]
\caption{\label{tab1} The outcomes of the measurements of qubits $A$ and $B$ performed by Alice and Bob with the corresponding local operation applied by Charlie and the combined state obtained by them (here $\mathcal{U} | \psi \rangle = \p{| \psi \rangle}$ and $\mathcal{U}^2 | \psi \rangle = \widetilde{| \psi \rangle}$)}
{\bf Step 2}
\begin{center}

\begin{tabular}{|c|c|c|}
\hline
{}&{}&{}\\
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{}\\
{\bf measurements}&{\bf operations}&{\bf Combined state of qubits a,b,c,$\bf C_1$,$\bf C_2$}\\
{\bf on $\bf A$ and $\bf B$}&{\bf after}&{}\\
{}&{\bf measurements}&{}\\
{}&{}&{}\\
\hline
{}&{}&{}\\
$| 00 \rangle_{AB}$&$C^{\mathcal{U}}_{C_1c}C^{\mathcal{U}}_{C_2c}$&$$\\
{}&{}&{}\\
$| 01 \rangle_{AB}$&$C^{\mathcal{U}}_{C_1c}C^{\mathcal{U}}_{C_2c}  ~ \sigma^{C_{2}}_x$&$d_0 | 00000 \rangle + d_1 | 00001 \rangle + d_2 | 0101\p0 \rangle + d_3 | 0101\p1 \rangle$\\
{}&{}&{}\\
$| 10 \rangle_{AB}$&$C^{\mathcal{U}}_{C_1c}C^{\mathcal{U}}_{C_2c}   ~   \sigma^{C_{1}}_x$&$ + d_4 | 1010\p0 \rangle + d_5 | 1010\p1 \rangle + d_6 | 1111\widetilde{0} \rangle + d_7 | 1111\widetilde{1} \rangle$\\
{}&{}&{}\\
$| 11 \rangle_{AB}$&$C^{\mathcal{U}}_{C_1c}C^{\mathcal{U}}_{C_2c}~   \sigma^{C_{1}}_x   \sigma^{C_{2}}_x$&$$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[tp]
\caption{\label{tab1} The outcomes of the measurements of qubits $C_1$ and $C_2$ performed by Charlie and the local operation applied by Alice and Bob to obtain the gate teleportation ($\mathcal{U} | \psi \rangle = \p{| \psi \rangle}$ and $\mathcal{U}^2 | \psi \rangle = \widetilde{| \psi \rangle}$)}
{\bf Step 3}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{}&{}&{}\\
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{}\\
{\bf measurements}&{\bf operations}&{\bf Combined state of qubits a,b,c}\\
{\bf on $\bf C_1$ and $\bf C_2$}&{\bf after}&{}\\
{}&{\bf measurements}&{}\\
{}&{}&{}\\
\hline
{}&{}&{}\\
$| ++ \rangle_{C_1 C_2}$&$\mathcal{I}$&$$\\
{}&{}&{}\\
$| +- \rangle_{C_1 C_2}$&$\sigma^{b}_Z$&$d_0 | 000 \rangle + d_1 | 001 \rangle + d_2 | 01\p0 \rangle + d_3 | 01\p1 \rangle $\\
{}&{}&{}\\
$| -+ \rangle_{C_1 C_2}$&$\sigma^{a}_Z$&$+ d_4 | 10\p0 \rangle + d_5 | 10\p1 \rangle + d_6 | 11\widetilde{0} \rangle + d_7 | 11\widetilde{1} \rangle$\\
{}&{}&{}\\
$| -- \rangle_{C_1 C_2}$&$\sigma^{b}_Z\sigma^{a}_Z$&$$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}


The pictorial representation of for simultaneous $ C^{\mathcal{H}}$ teleportation through parallel Bell state network has been depicted in figure 1.

The simultaneous remote teleportation of generalized controlled-Unitary gate, from two parties to one, consumes 2 ebits represented by the states $| \Phi \rangle_{AC_1}$ and $| \Phi \rangle_{BC_2}$, and total 4 cbits to communicate the measurement outcomes.

\subsubsection*{\bf $n$-party generalized gate teleportation :}


Here, we generalize the simultaneous teleportation protocol of $(n-1)$ control parties and one target party, using the above parallel network.
Each of the control parties share a Bell state ($| \Phi\rangle$) with the target party, the unknown state being,
\begin{equation}
| \chi \rangle = \Sigma_{m=0}^{2^n-1}a_m | i \rangle,
\end{equation}  where ` i ' is the binary representation of decimal number ` m '. In this state, each of the first $(n-1)$ qubits are possessed by the $(n-1)$ control parties, the target party possesses the last qubit.

Each of the control parties first apply CNOT gates, taking the unknown qubit as control, then measure their Bell shared qubit in computational basis and communicate the outcomes to the target party. The target party applies $\sigma_x$ on the Bell shared qubits if the measurement outcomes of the corresponding shared qubit is $|1\rangle$. After that the party applies simultaneous $C^{\mathcal{U}}$ gate (n-1) times, taking the shared Bell qubit as control, and the unknown qubit as target. Then the target party does measurement of all shared qubits in Hadamard basis. After communicating the outcomes of the measurement, control parties apply $\sigma_z$ gate if the outcome is $|-\rangle$ of the corresponding Bell shared qubit. The teleportation consumes $(n-1)$ ebit and $2(n-1)$ cbits.


\section{SIMULTANEOUS TELEPORTATION OF CONTROLLED-HERMITIAN GATE USING SERIES ENTANGLEMENT NETWORK}

Here, we illustrate the scheme for the simultaneous teleportation of controlled-Hermitian (as well as unitary) gates from two parties to one, where the entangled network is in series connection. Suppose, Alice, Bob and Charlie possess qubits 1,4 and 7 respectively of the arbitrary state,
\begin{equation}
| \psi \rangle_{147} = ( d_0 | 000 \rangle + d_1 | 001 \rangle + d_2 | 010 \rangle + d_3 | 011 \rangle + d_4 | 100 \rangle + d_5 | 101 \rangle + d_6 | 110 \rangle + d_7 | 111 \rangle )
\end{equation}
with $ \Sigma_{i=0}^{7}| d_i |^2 =1$.
Alice and Bob share a Bell state between their respective qubits 2 and 3; Bob and Charlie share a Bell state between their respective qubits 5 and 6 :
\begin{equation}
| \Phi \rangle_{23} = \frac{1}{\sqrt{2}}( | 00 \rangle + | 11 \rangle ),
\end{equation}
and
\begin{equation}
| \Phi \rangle_{56} = \frac{1}{\sqrt{2}}( | 00 \rangle + | 11 \rangle ).
\end{equation}
Here, Alice shares entanglement with another control party, Bob. And Bob shares entanglement with the target party, Charlie. Thus there is no direct entanglement between Alice and Charlie, which makes a series entanglement connection.

\begin{figure}
\centering
\includegraphics[height=9cm,width=14cm]{fig2}
\caption{Simultaneous $C^{\mathcal{H}}$ teleportation through series network.}
\end{figure}


Now the combined state of all the qubits possessed by Alice, Bob and Charlie is given by,
\begin{equation}
\begin{split}
| \zeta \rangle_{1234567}& = (| \psi \rangle_{147} \otimes | \Phi \rangle_{23} \otimes | \Phi \rangle_{56})\\
& = \frac{1}{2}[( d_0 | 0000000 \rangle + d_1 | 0000001 \rangle + d_2 | 0001000 \rangle + d_3 | 0001001 \rangle\\
& \quad \quad + d_4 | 1000000 \rangle+ d_5 | 1000001 \rangle + d_6 | 1001000 \rangle + d_7 | 1001001 \rangle )\\
& \quad + ( d_0 | 0000110 \rangle + d_1 | 0000111 \rangle + d_2 | 0001110 \rangle + d_3 | 0001111 \rangle\\
& \quad \quad + d_4 | 1000110 \rangle + d_5 | 1000111 \rangle + d_6 | 1001110 \rangle + d_7 | 1001111 \rangle )\\
& \quad + ( d_0 | 0110000 \rangle + d_1 | 0110001 \rangle + d_2 | 0111000 \rangle + d_3 | 0111001 \rangle\\
& \quad \quad + d_4 | 1110000 \rangle + d_5 | 1110001 \rangle + d_6 | 1111000 \rangle + d_7 | 1111001 \rangle )\\
& \quad + ( d_0 | 0110110 \rangle + d_1 | 0110111 \rangle + d_2 | 0111110 \rangle + d_3 | 0111111 \rangle\\
& \quad \quad + d_4 | 1110110 \rangle + d_5 | 1110111 \rangle + d_6 | 1111110 \rangle + d_7 | 1111111 \rangle )].
\end{split}
\end{equation}

{\bf Step 1:} To achieve the goal of complete remote teleportation of this three qubit gate, Alice applies controlled-NOT gate {$C^N_{12}$} on her qubits 1 and 2 and then measures qubit 2 in computational basis. Step 2, 3 and 4 are shown in table 3, 4 and 5 respectively.


\begin{table}[tp]
\caption{\label{tab1} The outcomes of the measurements of qubit 2, local operations by Bob and the combined state obtained by them}
{\bf Step 2}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{}&{}&{}\\
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{}\\
{\bf measurements}&{\bf operations}&{\bf Combined state after}\\
{\bf on $2$}&{\bf after}&{\bf measurement and operations}\\
{}&{\bf measurements}&{}\\
{}&{}&{}\\
\hline
{}&{}&{}\\
$| 0 \rangle_{2}$&$ C^N_{35} C^N_{45}$&$ d_0 | 000000 \rangle + d_1 | 000001 \rangle + d_2 | 001100 \rangle + d_3 | 001101 \rangle$\\
$$&$$&$ + d_0 | 000110 \rangle + d_1 | 000111 \rangle + d_2 | 001010 \rangle + d_3 | 001011 \rangle$\\
$| 1 \rangle_{2}$&$ C^N_{35} C^N_{45} ~ \sigma^{3}_x$&$ + d_4 | 110100 \rangle + d_5 | 110101 \rangle + d_6 | 111000 \rangle + d_7 | 111001 \rangle$\\
$$&$$&$ + d_4 | 110010 \rangle + d_5 | 110011 \rangle + d_6 | 111110 \rangle + d_7 | 111111 \rangle$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[tp]
\caption{\label{tab1} The outcomes of the measurements of qubit 5 in computational basis, local unitary gates applied by Charlie and the combined state obtained by them}
{\bf Step 3}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{}\\
{\bf measurements}&{\bf operations}&{\bf Combined state after}\\
{\bf on $5$}&{\bf after}&{\bf measurement and operations}\\
{}&{\bf measurements}&{}\\
{}&{}&{}\\
\hline
{}&{}&{}\\
$| 0 \rangle_{AB}$&$C^{\mathcal{H}}_{67}$&$ d_0 | 00000 \rangle + d_1 | 00001 \rangle + d_2 | 0011 \rangle \mathcal{H} | 0 \rangle + d_3 | 0011 \rangle \mathcal{H} | 1 \rangle$\\
$$&$$&$$\\
$| 1 \rangle_{AB}$&$ C^{\mathcal{H}}_{67} ~ \sigma^{6}_x$&$ + d_4 | 1101 \rangle \mathcal{H} | 0 \rangle + d_5 | 1101 \rangle \mathcal{H} | 1 \rangle + d_6 | 11100 \rangle + d_7 | 11101 \rangle$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[tp]
\caption{\label{tab1} The outcomes of the measurements of qubits 3 and 6 in Hadamard basis performed by Bob and Charlie and the corresponding unitary operations to obtain the desired state}
{\bf Step 4}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{}&{}&{}\\
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{}\\
{\bf measurements}&{\bf operations}&{\bf Combined state of qubits a,b,c}\\
{\bf on $3$ and $6$}&{\bf after}&{\bf after measurement and operations}\\
{}&{\bf measurements}&{}\\
{}&{}&{}\\
\hline
{}&{}&{}\\
$| ++ \rangle_{36}$&$\mathcal{I}$&$ $\\
$| +- \rangle_{36}$&$\sigma^{1}_z \sigma^{4}_z$&$ d_0 | 000 \rangle + d_1 | 001 \rangle + d_2 | 01 \rangle \mathcal{H} | 0 \rangle + d_3 | 01  \rangle \mathcal{H} | 1 \rangle $\\
$| -+ \rangle_{36}$&$\sigma^{1}_z$&$ + d_4 | 10 \rangle \mathcal{H} | 0 \rangle + d_5 | 10 \rangle \mathcal{H} | 1 \rangle + d_6 | 110 \rangle + d_7 | 111 \rangle$\\
$| -- \rangle_{36}$&$ \sigma^{4}_z$&$ $\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}
\centering
\includegraphics[width=14cm]{fig3}
\caption{Generalized simultaneous $C^{\mathcal{H}}$ teleportation using the series network}
\end{figure}

The pictorial representation of local operations and measurements for simultaneous $ C^{\mathcal{H}}$ teleportation through series network of Bell states has been depicted in figure 2.
The simultaneous remote teleportation of controlled-Hermitian gate from two parties to one consumes 2 ebits and total 5 cbits to communicate the measurement outcomes.

\subsubsection*{\bf $n$-party generalized gate teleportation :}




The generalized protocol of simultaneous $ C^{\mathcal{H}} $ gate teleportation using the above series network is given in figure 3, where the unknown state given in eq. 5.
For n parties the communication cost is $(n-1)$ ebits and $(n^2+n-2)/2$ cbits.
It is worth emphasizing the interesting feature of the series network, which enables a party, not directly connected to the target, to control an operation on the same.

We could not find any protocol where controlled-Unitary gate can be teleported using this entanglement cost. So, from the above protocol, we can say that the Unitary as well as Hermitian operators has significance in series entangled network. This operator has the additional property of involution (i.e., the operator is same as its inverse), which is responsible for making this protocol deterministic. Most of the important gates like controlled-Pauli gates, controlled-Hadamard gate etc., belong to this category, making this implementation powerful.

~~~~

~~~~

~~~~

~~~~


\section{TOFFOLI GATE TELEPORTATION USING SERIES ENTANGLEMENT NETWORK}

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

\begin{figure}
\centering
\includegraphics[height=9cm,width=14cm]{fig4}
\caption{Controlled-controlled-Unitary gate teleportation through series network.}
\end{figure}
It has been shown that a more general form of Toffoli gate, i.e., controlled-controlled-Unitary
gate can be deterministically teleported in parallel network, using two Bell pairs (2 ebits of entanglement) and 4 cbits to communicate the measurement outcomes [10]. In this present protocol we implement that same non-local gate using series entanglement distribution with the same communication cost, given in figure 4.

For illustration, we consider the qubit distribution described in eq. 9 :
\begin{equation}
| \zeta \rangle_{1234567} = (| \psi \rangle_{147} \otimes | \Phi \rangle_{23} \otimes | \Phi \rangle_{56})
\end{equation}

{\bf Step 1}: Alice first applies controlled-NOT gates $C^N_{12}$ and measure her qubit 2 in computational basis. Other steps are given in table 6 to 9.


\begin{table}[tp]
\caption{\label{tab1}The outcomes of the measurements of qubit 2 with unitary gate performed
by Bob and the combined state after operations.}
{\bf Step 2}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{\bf Combined state}\\
{\bf measurements}&{\bf operations}&{\bf obtained}\\
{\bf on qubit 2}&{\bf by Bob}&{\bf after}\\
{}&{}&{\bf measurements and operations}\\
\hline
{}&{}&{}\\
$$&$$&$$\\
$| 0 \rangle_2$&$ C^{N}_{345}$&$$\\
$$&$$&$ d_0 | 000000 \rangle_ + d_1 | 000001 \rangle + d_2 | 001000 \rangle + d_3 | 001001 \rangle$\\
$$&$$&$+ d_0 | 000110 \rangle + d_1 | 000111 \rangle + d_2 | 001110 \rangle + d_3 | 001111 \rangle$\\
$$&$$&$+ d_4 | 110000 \rangle+ d_5 | 110001 \rangle + d_6 | 111100 \rangle + d_7 | 111101 \rangle$\\
$| 1 \rangle_2$&$ C^{N}_{345} \sigma^3_{x}$&$+ d_4 | 110110 \rangle + d_5 | 110111 \rangle + d_6 | 111010 \rangle + d_7 | 111011 \rangle$\\
$$&$$&$$\\
$$&$$&$$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[tp]
\caption{\label{tab1}The outcomes of the measurements of qubit 5 with unitary gate performed
by Charlie and the combined state after operations.}
{\bf Step 3}
\begin{center}
\begin{tabular}{|c|c|c|}

\hline
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{\bf Combined state }\\
{\bf measurements}&{\bf operations}&{\bf obtained}\\
{\bf on qubit 5}&{\bf by Bob}&{\bf after}\\
{}&{}&{\bf measurements and operations}\\
\hline
{}&{}&{}\\
$| 0 \rangle_5$&$C^\mathcal{U}_{67}$&$$\\
$$&$$&$ d_0 | 00000 \rangle + d_1 | 00001 \rangle + d_2 | 00100 \rangle + d_3 | 00101 \rangle$\\
$| 1 \rangle_5$&$C^\mathcal{U}_{67} \sigma_x^6$&$+ d_4 | 11000 \rangle+ d_5 | 11001 \rangle + d_6 | 1111\p0 \rangle + d_7 | 1111\p1 \rangle$\\
$$&$$&$$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}

\end{table}


\begin{table}[tp]
\caption{\label{tab1}The outcomes of the measurements of qubit 6 in Hadamard basis with unitary gate performed
by Bob and the combined state after operations.}
{\bf Step 4}
\begin{center}
\begin{tabular}{|c|c|c|}

\hline
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{\bf Combined state }\\
{\bf measurements}&{\bf operations}&{\bf obtained}\\
{\bf on qubit 6}&{\bf by Bob}&{\bf after}\\
{}&{}&{\bf measurements and operations}\\
\hline
{}&{}&{}\\
$| + \rangle_6$&$\mathcal{I}$&$$\\
$$&$$&$ d_0 | 0000 \rangle + d_1 | 0001 \rangle + d_2 | 0010 \rangle + d_3 | 0011 \rangle$\\
$| - \rangle_6$&$C^{\sigma_{z}}_{34}$&$+ d_4 | 1100 \rangle+ d_5 | 1101 \rangle + d_6 | 111\p0 \rangle + d_7 | 111\p1 \rangle$\\
$$&$$&$$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[tp]
\caption{\label{tab1}The outcomes of the measurements of qubit 3 in Hadamard basis with unitary gate performed
by Alice and the combined desired state obtained by the three parties}
{\bf Step 5}
\begin{center}
\begin{tabular}{|c|c|c|}

\hline
{}&{}&{}\\
{\bf Outcomes of}&{\bf Local}&{\bf Combined state }\\
{\bf measurements}&{\bf operations}&{\bf obtained}\\
{\bf on qubit 3}&{\bf by Bob}&{\bf after}\\
{}&{}&{\bf measurements and operations}\\
\hline
{}&{}&{}\\
$| + \rangle_3$&$\mathcal{I}$&$$\\
$$&$$&$ d_0 | 000 \rangle + d_1 | 001 \rangle + d_2 | 010 \rangle + d_3 | 011 \rangle$\\
$| - \rangle_3$&$\sigma_{z}^{1}$&$+ d_4 | 100 \rangle+ d_5 | 101 \rangle + d_6 | 11\p0 \rangle + d_7 | 11\p1 \rangle$\\
$$&$$&$$\\
{}&{}&{}\\
\hline
\end{tabular}
\end{center}
\end{table}

\subsubsection*{\bf $n$-party generalized gate teleportation :}

We now generalize the above procedure to teleport a `n'-qubit
gate, where $(n-1)$ qubits are controls and the unitary operator
acts on the target qubit, only if, all the control qubits are $|1\rangle$s.
The protocol is shown in figure 5 and for n-qubit gate the communication cost is $(n-1)$ ebits and $2(n-1)$ cbits which is same as for the parallel network, given in Ref.[10].

\begin{figure}
\centering
\includegraphics[height=9cm,width=14cm]{fig5}
\caption{Generalized controlled-Unitary gate teleportation using series entangled network.}
\end{figure}

~
\section{CONCLUSION}

In conclusion, we have demonstrated gate teleportation protocols of simultaneous controlled gate operations in two different configurations, which are efficient, as evident from the associated communication and entanglement costs. All the protocols have been generalized to `n'-qubit gates. Interestingly, in the series configuration, the implemented generalized gate and communication cost is different from the parallel distribution for the  simultaneous gate teleportation. Optimal protocol can be investigated for the simulstaneous controlled-Unitary gate teleportation in series entangled channels. In parallel to the qubit case of remote state preparation, the gate teleportation with partial information needs to be investigated [7,8]. The fact that, Bell states are realized in laboratory conditions makes our protocol experimentally achievable [3]. In future, we would like to study the teleportation protocols with minimum communication costs of other important gates, which do not belong to this present category. The question of teleportation witness [22,23,24] for the present protocol needs careful study.


\section{ACKNOWLEDGMENT}

The authors acknowledge Professor Vijay A. Singh of Homi Bhabha Centre
for Science Education (HBCSE\textemdash{}TIFR), Mumbai, India for continuous encouragement. This work is supported by
the `National Initiative on Undergraduate Science' (NIUS) program,
undertaken by HBCSE. 


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~~~
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