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Quantum Private Queries Protocol Based on Quantum Random Access Memory
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==Protocol Description== <!-- Mathematical step-wise protocol algorithm helpful to write a subroutine. --> '''Inputs:''' * User (Alice): index <math>j</math> of the desired database element; Alice knows the <math>0^\text{th}</math> database element, <math>A_0</math>. * Server (Bob): classical database with <math>N</math> elements <math>A_0,...,A_{N-1}</math>. '''Output:''' * User (Alice): desired database entry <math>A_j</math> (in the case of honest parties). '''Protocol:''' * '''Choice of scenario:''' ** Alice chooses randomly between two scenarios ❶ and ❷. * '''Query:''' ** Alice prepares two n qubit registers: <math>|j\rangle_Q</math> and <math>(|j\rangle_Q+|0\rangle_Q)/\sqrt{2}</math>. ** Scenario ❶: Alice sends <math>|j\rangle_Q</math> to Bob. Scenario ❷: Alice sends <math>(|j\rangle_Q+|0\rangle_Q)/\sqrt{2}</math> to Bob. * '''Answer:''' ** Bob interrogates the database with the received quantum register ('query register') using a qRAM algorithm <math>\sum_j \alpha_j |j\rangle_Q \rightarrow \sum_j \alpha_j |j\rangle_Q |A_j\rangle_R</math>. The output of the qRAM algorithm is another quantum register: <math>|j\rangle_Q |A_j\rangle_R</math> in scenario ❶ or <math>(|j\rangle_Q |A_j \rangle_R+|0\rangle_Q |A_0 \rangle_R)/\sqrt{2}</math> in scenario ❷. ** Bob sends to Alice the output of the qRAM algorithm: <math>|j\rangle_Q |A_j\rangle_R</math> in scenario ❶ or <math>(|j\rangle_Q |A_j \rangle_R+|0\rangle_Q |A_0 \rangle_R)/\sqrt{2}</math> in scenario ❷ * '''Query:''' ** Scenario ❶: Alice sends <math>(|j\rangle_Q+|0\rangle_Q)/\sqrt{2}</math> to Bob. Scenario ❷: Alice sends <math>|j\rangle_Q</math> to Bob. * '''Answer:''' ** Bob interrogates the database with the received quantum register ('query register') using the qRAM algorithm <math>\sum_j \alpha_j |j\rangle_Q \rightarrow \sum_j \alpha_j |j\rangle_Q |A_j\rangle_R</math>. The output of the qRAM algorithm is another quantum register: <math>(|j\rangle_Q |A_j \rangle_R+|0\rangle_Q |A_0 \rangle_R)/\sqrt{2}</math> in scenario ❶ or <math>|j\rangle_Q |A_j\rangle_R</math> in scenario ❷. ** Bob sends to Alice the output of the qRAM algorithm: <math>(|j\rangle_Q |A_j \rangle_R+|0\rangle_Q |A_0 \rangle_R)/\sqrt{2}</math> in scenario ❶ or <math>|j\rangle_Q |A_j\rangle_R</math> in scenario ❷. * '''Retrieval & Honesty test:''' ** Alice performs a test on the received quantum registers to check if the server cheated. First, she measures <math>|j\rangle_Q |A_j\rangle_R</math> to obtain the desired database element <math>A_j</math>. Based on this first measurement outcome, she tailors the measurement that she performs on <math>(|j\rangle_Q |A_j \rangle_R+|0\rangle_Q |A_0 \rangle_R)/\sqrt{2}</math> in such a way that if the measurement outcome is not a certain outcome, Alice knows for sure that Bob cheated. Otherwise, Alice assumes that Bob is honest (which is not guaranteed).
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