Table 2 Characteristic of a two-cascaded CZ gate on metasurface

From: Quantum CZ gates on a single gradient metasurface

(a) Encoding rules of signal polarization qubit

 

Signal

Auxiliary

Polarization state

\({| R\left.\right\rangle }_{-1}\)

\({| L\left.\right\rangle }_{-1}\)

\({| R\left.\right\rangle }_{+2}\)

\({| L\left.\right\rangle }_{-2}\)

Encoded qubit state

\({| 0\left.\right\rangle }_{S}\)

\({| 1\left.\right\rangle }_{S}\)

None

None

Mode label

\({\hat{S}}_{0}^{R}\)

\({\hat{S}}_{1}^{L}\)

\({\hat{V}}^{R}\)

\({\hat{V}}^{L}\)

(b) Truth table of 3-qubit input states

S&T in standard basis

S&T in Hadamard basis

 

Input

Output

Input

Output

Post-selection probability

\({| 0\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

1/27

\({| 0\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

1/27

\({| 0\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\(-{| 0\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

1/27

\({| 0\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\(-{| 0\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

\({| 0\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

1/27

\({| 1\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

1/27

\({| 1\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\(-{| 1\left.\right\rangle }_{C}{| 0\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| +\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

1/27

\({| 1\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 0\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

1/27

\({| 1\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\(-{| 1\left.\right\rangle }_{C}{| 1\left.\right\rangle }_{S}{| 1\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| -\left.\right\rangle }_{T}\)

\({| 1\left.\right\rangle }_{C}{| -\left.\right\rangle }_{S}{| +\left.\right\rangle }_{T}\)

1/27

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