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Distributed Ballot Based Protocol: Difference between revisions
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*'''Setup phase''': | *'''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 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>, 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. | # 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''': | *'''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 | #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 | ||