Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions

### Three encrypted CNOTs are used to correct <math>C^{zx}</math> as follows under <math>\mathrm{AltHE}</math>.

####Server converts <math>\hat{c} = \mathrm{HE.Convert(c)}</math>.

#### Servers entangles above superposition and <math>\psi</math> with a third register as follows:</br><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>  

####Server measures the last register to get <math>y =\mathrm{AltHE.Enc}(\mu_0,r_0)=\mathrm{AltHE.Enc}_{pk}(\mu_1,r_1)\oplus_H AltHE.Enc_{pk}(s)</math>. The resulting superposition state is:</br><math>\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}|a,b\rangle|\mu_a,r_a\rangle|\mathrm{AltHE.Enc}(\mu_0,r_0)\rangle=\sum_{a,b,\mu\in\{0,1\},r}\alpha_{ab}\sqrt{D(\mu_0,r_0)}|a,b\rangle|\mu_a,r_a\rangle|y\rangle</math>, where <math>\beta</math> is the normalization constant.