Editing Secure Multiparty Delegated Quantum Computation (section)

====Multiparty Quantum Computing Protocol====
* A quantum input <math>\rho_{in}</math> and measurement angles <math>\{\phi_j\}_{j=1}^q</math> for qubits <math>j\in O^c</math> .
    
'''Preparation phase'''

''Quantum input'': For <math>j\in I</math> 

# Client <math>C_j</math> applies a one-time pad <math>X^{a_j}Z(\theta_j^j)</math> to his qubit, where <math>a_j\in_R\{0,1\}</math> and <math>\theta_j^j\in_R\{l\pi/4\}_{l=0}^7</math> and sends it to the Server. He secret-shares the values <math>a_j</math> and <math>\theta_j^j</math> with the other clients.
# Each client <math>C_k (k\neq j)</math>, runs Protocol 1 with the Server. If all clients pass the test, the Server at the end has <math>n-1</math> states <math>|+_{\theta_j^k}\rangle=\frac{1}{\sqrt{2}}\big(|0\rangle+e^{i\theta_j^k}|1\rangle \big)</math> for <math>k\neq j</math>.
# The Server runs Protocol2 and announces outcome vector <math>\mathbf{t}_j</math>. 
At this point the Server has the state  <math>\rho'_{in}=\big(X^{a_1}Z(\theta_1)\otimes \dots \otimes X^{a_n} Z(\theta_n)\otimes \mathbf{1}_{\mathcal{R}}\big)\cdot \rho_{in}</math>, where 
<math>\theta_j=\theta_j^j+\sum_{k=1, k\neq j}^n (-1)^{\bigoplus_{i=k}^n t_j^i+a_j}\theta_j^k </math>
# [non-output / non-input qubits:] For <math>j\in O^c\setminus I</math> 
        \begin{enumerate}
            \item[4.] All clients </math>C_k</math>, </math>k\in[n]</math>  run Protocol \ref{Algo1} with the Server. If all clients pass the test, the Server at the end has </math>n</math> states </math>\ket{+_{\theta_j^k}}=\frac{1}{\sqrt{2}}\big(\ket{0}+e^{i\theta_j^k}\ket{1}  \big)</math> for </math>k=1,\dots,n</math>.
            \item[5.] The Server runs Protocol \ref{Algo3} getting outcome vector </math>\mathbf{t}_j</math>. He ends up with the state </math>\ket{+_{\theta_j}}</math>, where:
            \begin{equation}\label{eq:entangle2}
                \theta_j=\theta_j^n+\sum_{k=1}^{n-1} (-1)^{\bigoplus_{i=k}^{n-1} t_j^i}\theta_j^k
            \end{equation}
        \end{enumerate}
        \item[output qubits:] For </math>j\in O</math>, the Server prepares </math>\ket{+}</math> states.
        \item[graph state:] The Server entangles the </math>n+q</math> qubits to a brickwork state by applying ctrl-</math>Z</math> gates.
    
    \end{description}
        
    \begin{flushleft}
        \underline{\emph{Computation phase}}
        \vspace{-7pt}
    \end{flushleft}
    
    \begin{description}
        \item[non-output qubits:] For </math>j\in O^c</math>
        \begin{enumerate}
            \item All clients </math>C_k</math>, </math>k=1,\dots,n</math> choose random </math>r_j^k\in\{0,1\}</math>, which they secret-share with the other clients. Then using a computation oracle, they compute the measurement angle of qubit </math>j</math>:
            \begin{equation}\label{angle}
                \delta_j:=\phi'_j+\pi r_j+\theta_j
            \end{equation}
            where undefined values are equal to zero, or otherwise:
            \begin{itemize}
                \item </math>\phi'_j=(-1)^{a_j+s_j^X}\phi_j+s^Z_j\pi+a_{f^{-1}(j)}\pi</math>.
                \item </math>r_j=\bigoplus\limits_{k=1}^n r_j^k</math>.
                \item </math>s_i=b_i\oplus r_i</math>, for </math>i\leq j</math>.
            \end{itemize}
            \item The Server receives </math>\delta_j</math> and measures qubit </math>j</math> in basis </math>\{\ket{+_{\delta_j}},\ket{-_{\delta_j}}\}</math>, getting result </math>b_j</math>. He announces </math>b_j</math> to the clients.
        \end{enumerate}
        \item[output qubits:] For </math>j\in O</math>, the Server sends the ``encrypted'' quantum state to client </math>C_{j-q}</math>. All participants jointly compute </math>s_j^X</math> and </math>s_j^Z</math> and send it to client </math>C_{j-q}</math>, who applies operation </math>Z^{s_j^Z}X^{s_j^X}</math> to retrieve the actual quantum output. 
    \end{description}

\end{algorithm}


<div style='text-align: right;'>''*contributed by Natansh Mathur''</div>