Editing Distributed Ballot Based Protocol (section)

==Protocol Description==
*'''Setup phase''':
# T prepares an N-qudit ballot state <math>|\Phi\rangle= \dfrac{1}{\sqrt{D}}\sum_{j=0}^{D-1}|j\rangle ^{\otimes N}</math>. <p>The states <math> |j\rangle, j = 0,...,D-1,</math> form an orthonormal basis for the D-dimensional Hilbert space, and D > N. The k-th qudit of <math>\Phi</math> is <math>V_k</math>'s blank ballot.</p>
# T sends to <math>V_k</math> the corresponding blank ballot and two option qudits,for the "yes" and "no" option:<p><math> yes:|\psi(\theta_y)\rangle=\dfrac{1}{\sqrt{D}}\sum_{j=0}^{D-1}e^{ij\theta_y}|j\rangle</math></p>,<p> no:<math>|\psi(\theta_n)\rangle=\dfrac{1}{\sqrt{D}}\sum_{j=0}^{D-1}e^{ij\theta_n}|j\rangle</math>.</p> For <math> v\in \{y, n\}</math> we have <math>\theta_v = (2\pi l_v/D) + \delta</math>, where <math>l_v \in \{0,...,D- 1\}</math> and <math>\delta \in [0, 2\pi/D)</math>. Values <math>l_y</math> and <math>\delta</math> are chosen uniformly at random from their domain and <math>l_n</math> is chosen such that <math>N(l_y - l_n \text{ }mod\text{ } D)</math> < D.
*'''Casting phase''':
#Each <math>V_k</math> appends the corresponding option qudit to the blank ballot and performs a 2-qudit measurement <math> R =\sum^{D-1}_{r=0}rP_r</math> where <math> P_r=\sum_{j=0}^{D-1}|j+r\rangle\langle j+r | \otimes |j\rangle \langle j|.</math><p> According to the result <math>r_k, V_k</math> performs a unitary correction <math>U_{r_k} = I \otimes \sum_{j=0}^{D-1}|j+r_k\rangle \langle j |</math> and sends the 2-qudits ballot and <math>r_k</math> back to T
*'''Tally phase''':
#The global state of the system is: <math> \dfrac{1}{\sqrt{D}}\sum_{j=0}^{D-1}\Pi^{N}_{k=1}\alpha_{j,r_k}|j\rangle^{\otimes 2N}</math> where , <math display="block">\alpha_{j,r_k}=
\begin{cases}
e^{i(D+j-r_k)\theta^{k}_{v}},\text{ }0 \leq j \leq r_k -1,\\
e^{i(j-r_k)\theta^{k}_{v}}\text{ }r_k \leq j \leq D -1
\end{cases}
</math><p>For every k, T applies <math> W_k=\sum_{j=0}^{r_k-1}e^{-iD\delta}|j\rangle|\langle j|+\sum_{j=r_k}^{D-1}|j\rangle|\langle j| </math> on one of the qudits in the global state.</p>
# By applying the unitary operator <math> \sum_{j=0}^{D-1}e^{-ijN\theta_n}|j\rangle \langle j|</math>on one of the qudits we have <math>|\phi_q\rangle=\dfrac{1}{\sqrt{D}}\sum_{j=0}^{D-1}e^{2\pi ijq/D}|j\rangle^{\otimes 2N}</math> where <math>q=m(l_y-l_n)</math>. with the corresponding measurement, T retrieves q and uses values <math>l_y,l_n</math> to compute m.