Trap Code for Quantum Authentication

Notation

• $\rho$: 1-qubit input state

Protocol Description

• Encoding:

1. Input: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} , pair of keys Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k=(k_{1},k_{2})}
2. Apply an Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [[n,1,d]]} error correction code (corrects up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} errors, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=2t+1} )
3. Append an additional trap register of $n$ qubits in state Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |0\rangle \langle 0|^{\otimes n}}
4. Append a second additional trap register of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} qubits in state $|+\rangle \langle +|^{\otimes n}$
5. Permute the total Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3n} -qubit register by $\pi _{k_{1}}$ according to the key $k_{1}$
6. Apply a Pauli encryption $P_{k_{2}}$ according to key Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k_{2}}

• Decoding:

1. Input: $\rho ^{\prime }$ (state after encoding), pair of keys Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k=(k_{1},k_{2})}
2. Apply $P_{k_{2}}$ according to key Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k_{2}}
3. Apply inverse permutation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \pi _{k_{1}}^{\dagger }} according to the key $k_{1}$
4. Measure the last Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} qubits in the Hadamard basis
5. Measure the second last $n$ qubits in the computational basis
   a. If the two measurements result in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |+\rangle \langle +|} and $|0\rangle \langle 0|$, an additional flag qubit in state $|\mathrm {ACC} \rangle \langle \mathrm {ACC} |$ is appended and the quantum message is decoded according to the error correction code
   b. Otherwise, an additional flag qubit in state $|\mathrm {REJ} \rangle \langle \mathrm {REJ} |$ is appended and the (disturbed) encoded quantum message is replaced by a fixed state Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega }

References

1. Broadbent et al. (2012)
2. Broadbent and Wainewright (2016).