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 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}} according to the key $k_{1}$
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 $k=(k_{1},k_{2})$
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 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 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 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 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).