Trap Code for Quantum Authentication

Notation

• $\rho$: 1-qubit input state

Protocol Description

• Encoding:

1. Input: $\rho$, pair of keys $k=(k_{1},k_{2})$
2. Apply an $[[n,1,d]]$ 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 (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 3n} -qubit register by $\pi _{k_{1}}$ according to the key 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_1}
6. Apply a Pauli encryption 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 P_{k_2}} according to key 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_2}

• Decoding:

1. Input: $\rho ^{\prime }$ (state after encoding), 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 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 P_{k_2}} according to key 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_2}
3. Apply inverse permutation 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 \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 $|+\rangle \langle +|$ 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 $\Omega$

References

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

contributed by Shraddha Singh and Isabel Nha Minh Le