Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions

::::#Server generates superposition on distribution D: <math>\sum_{\mu\in\{0,1\},r}\sqrt{D(\mu,r)}|\mu,r\rangle</math>

::::#Server entangles above superposition and <math>|\psi\rangle</math> with a third register:<math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu,r)}|a,b\rangle|\mu,r\rangle|f_a(r)\rangle</math>, such that </br><math>f_0=\mathrm{AltHE.Enc}_{pk}()</math>;</br><math>f_1(\mu_1,r_1)=f_0 (\mu_0,r_0)\oplus_H \hat{c}=\mathrm{AltHE.Enc}_{pk}(\mu_0,r_0)\oplus_H \mathrm{AltHE.Enc}_{pk}(s)</math> </br><math>\therefore \mu_0\oplus\mu_1=s</math>

***'''Encrypted CNOT operation:'''<div style="background-color: gray; border: solid thin black;title=Functionality Description;"><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}CNOT_{a,b}^s|a,b\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a,b\oplus a\cdot s\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a,b\oplus a\cdot(\mu_0\oplus\mu_1)\rangle</math></br><math>=\sum_{b\in\{0,1\}}\alpha_{0b}|0,b\oplus \mu_0\oplus\mu_0\rangle+\alpha_{1b}|1,b\oplus \mu_0\oplus\mu_1\rangle</math>,  <math>\because q\oplus q=0</math></br><math>=\sum_{b\in\{0,1\}}\alpha_{0b}|0\rangle\otimes X^{\mu_0}|b\oplus \mu_0\rangle+\alpha_{1b}|1\rangle \otimes X^{\mu_0}|b\oplus \mu_1\rangle</math>, <math>\because |q\oplus y\rangle=X^y|q\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}|a\rangle\otimes X^{\mu_0}|b\oplus \mu_a\rangle</math></br><math>=\sum_{a,b\in\{0,1\}}\alpha_{ab}(I\otimes X^{\mu_0})|a,b\oplus \mu_a\rangle</math></br> </div>

::::#Server XORs the second qubit of first register with <math>\mu_a</math> to get:</br><math>\sum_{a,b\in\{0,1\}}\alpha_{ab}\sqrt{D(\mu_0,r_0)}(I\otimes X^{\mu_0})CNOT_{a,b}^s|a,b\rangle\otimes|\mu_a,r_a\rangle|y\rangle</math>