Classical Fully Homomorphic Encryption for Quantum Circuits: Difference between revisions

### if <math>c_i=</math>CNOT with m as target bit and n as control bit then <div class="floatright">//CNOT</div></br>(<math>\tilde {x}^{[l]},\tilde{z}^{[l]};\tilde {x}^{[n]},\tilde{z}^{[n]})\rightarrow (\tilde {x}^{[l]},\tilde{z}^{[l]}\oplus \tilde {z}^{[n]};\tilde{x}^{[l]}\oplus \tilde {x}^{[n]},\tilde{z}^{[n]})</math>

## If <math>c_i=T</math> gate then <div class="floatright">//Toffoli Gate on <math>l_{th}, n_{th}, o_{th}</math> key bits</div></br> The Toffoli gate application can be deduced as follows:</br><math>TZ^{z^{[l]}}X^{x^{[l]}}Z^{z^{[n]}}X^{x^{[n]}}Z^{z^{[o]}}X^{x^{[o]}}|\psi\rangle</math></br><math>=TZ^{z^{[l]}}X^{x^{[l]}}Z^{z^{[n]}}X^{x^{[n]}}Z^{z^{[o]}}X^{x^{[o]}}T\dagger T|\psi\rangle</math></br><math>=CNOT_{l,o}^{x^{[n]}}CNOT_{n,o}^{x^{[l]}}CZ_{l,n}^{z^{[o]}}Z^{z^{[l]}}X^{x^{[l]}}T|\psi\rangle</math></br><math>=CNOT_{l,o}^{x^{[n]}}CNOT_{n,o}^{x^{[l]}}H_nCNOT_{l,n}^{z^{[o]}}H_{n}Z^{z^{[l]}}X^{x^{[l]}}T|\psi\rangle</math></br><math>=C_{zx}P_{zx}T|\psi\rangle</math>, where <math>C\epsilon \{\text{CNOT,H}\}</math> and <math>P\epsilon\{X,Z\}</math>

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

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