Editing Prepare-and-Send Verifiable Universal Blind Quantum Computation (section)

==Protocol Description==

'''Protocol for quantum output case''': <br></br>
'''Stage 1: '''Client's preparation:</br>

'''Input''': Cylindrical brickwork state, <math>|I\rangle</math>.
</br>
'''Output''': Server: <math>m</math> qubits sequentially. 

* The client prepares <math>|e\rangle = X^{x1}_1 Z_1(\theta_1) \otimes ... \otimes X^{xn}_n Z_n(\theta_n)|I\rangle</math> using QOTP.
* Client randomly chooses <math>t</math>, where <math>t \in D</math>.
* For <math>i = n+1, n+2, ....m</math> (non-input qubits):
** if <math>i \in D</math>:
*** if <math>i  != t</math> then state <math>|0\rangle</math> or <math>|1\rangle</math> is prepared
*** if <math>i == t</math> then <math>|+\rangle_{\theta_i}</math> is prepared
** else <math>|+\rangle_{\theta_i}</math> is prepared
* <math>\forall l\epsilon\{1,..,n\}</math>, Client sends all qubits to server.


'''Stage 2: '''Server's preparation:</br>
'''Input''': <math>m</math> qubits sequentially.</br>
'''Output''': entangled graph state with a disentangled trap qubit.
* Server creates an entangled state from all received qubits using CZ operations according to their indices and creates the cylinder brickwork state.

'''Stage 3: '''Interaction and Measurement:</br>
'''Input''': <math>\delta_i</math></br>
'''Output''': <math>s_i</math>

* For <math>i = 1, 2, ... m-n</math> (received qubits): 
** Client computes <math>\phi_i</math>.
*** if <math>i == t</math>, then <math>\phi_i = 0</math>
** Client randomly selects <math>r_i</math> and generates <math>\theta'_i = \theta_i + r_i</math>.
** Client then computes the angle <math>\delta_i = (-1)^{x_i + s_{f^{-1}(i)}}\phi_i + \sum_{j:i \in N_g(f(j)}\theta'_i + s_i\pi</math> and sends <math>\delta_i</math> to server.
** Server measures <math>i</math> and sends <math>b_i</math> to client.
** Client sets the value <math>s_i = b_i \oplus r_i</math> in <math>s</math>

'''Stage 4: '''Verification:</br>
'''Input''': Output qubits <math>m-n+1</math> to <math>m</math></br>
'''Output''': Verification result
* For <math>i = m-n+1, ... m</math> (output qubits): 
** Server sends <math>i</math> to client.
* Client measures the output trap qubit <math>t</math> (which was disentangled) with angle <math>\delta_t = \phi_t + r_t\pi</math>. 
** Client obtains the result <math>b_t</math>.
*** If <math>b_t == r_t</math>, then computation is accepted.
*** else, computation is rejected.