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 $[[n,1,d]]$ 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 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 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 |+\rangle\langle +|} and 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 |0\rangle\langle 0|} , 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{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 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 \Omega}

References

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

contributed by Shraddha Singh and Isabel Nha Minh Le