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 $3n$-qubit register by $\pi _{k_{1}}$ according to the key $k_{1}$
6. Apply a Pauli encryption $P_{k_{2}}$ according to key $k_{2}$

• Decoding:

1. Input: $\rho ^{\prime }$ (state after encoding), pair of keys $k=(k_{1},k_{2})$
2. Apply $P_{k_{2}}$ according to key $k_{2}$
3. Apply inverse permutation $\pi _{k_{1}}^{\dagger }$ 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).