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| ==Outline== | | ==Outline== |
| In the beginning, the election authority prepares an N-qudit ballot state where the kth qudit of the state corresponds to <math> V_k</math>’s blank ballot and sends the corresponding blank ballot to <math>V_k</math> together with two option qudits, one for the “yes” and one for the “no” vote. then each voter decides on “yes” or “no” by appending the corresponding option qudit to the blank ballot and performing a 2-qudit measurement, then based on its result she performs a unitary correction and sends the 2-qudits ballot along with the measurement result back to the election authority. At the end of the election, the election authority applies a unitary operation on one of the qudits in the global state and another unitary operation on one of the qudits to find the number of yes votes. | | In the beginning, the election authority prepares an N-qudit ballot state where the kth qudit of the state corresponds to V_k’s blank ballot and sends the corresponding blank ballot to V_k together with two option qudits, one for the “yes” and one for the “no” vote. then each voter decides on “yes” or “no” by appending the corresponding option qudit to the blank ballot and performing a 2-qudit measurement, then based on its result she performs a unitary correction and sends the 2-qudits ballot along with the measurement result back to the election authority. At the end of the election, the election authority applies a unitary operation on one of the qudits in the global state and another unitary operation on one of the qudits to find the number of yes votes. |
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| ==Notations== | | ==Notations== |
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| ==Protocol Description== | | ==Protocol Description== |
| *'''Setup phase''':
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| # 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>
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| # 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.
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| *'''Casting phase''':
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| #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
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| *'''Tally phase''':
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| #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}=
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| \begin{cases}
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| e^{i(D+j-r_k)\theta^{k}_{v}},\text{ }0 \leq j \leq r_k -1,\\
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| e^{i(j-r_k)\theta^{k}_{v}}\text{ }r_k \leq j \leq D -1
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| \end{cases}
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| </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>
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| # 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.
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| ==Further Information== | | ==Further Information== |
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| <div style='text-align: right;'>''*contributed by Sara Sarfaraz''</div> | | <div style='text-align: right;'>''*contributed by Sara Sarfaraz''</div> |