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 (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 k=(k_1, k_2)}
2. Apply an 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,1,d]]} error correction code (corrects up to $t$ errors, $d=2t+1$)
3. Append an additional trap register of $n$ qubits in state $|0\rangle \langle 0|^{\otimes n}$
4. Append a second additional trap register of $n$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{k_{2}}} according to key $k_{2}$

• Decoding:

1. Input: $\rho ^{\prime }$ (state after encoding), pair of keys $k=(k_{1},k_{2})$
2. Apply Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{k_{2}}} according to key $k_{2}$
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 $n$ 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 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 |\mathrm{REJ}\rangle\langle\mathrm{REJ}|} is appended and the (disturbed) encoded quantum message is replaced by a fixed state $\Omega$

References

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

contributed by Shraddha Singh and Isabel Nha Minh Le