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, 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 d=2t+1} )
3. Append an additional trap register of 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 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 |0\rangle\langle 0|^{\otimes n}}
4. Append a second additional trap register of 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 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 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 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 $k_{2}$

• Decoding:

1. Input: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\mathrm {REJ} \rangle \langle \mathrm {REJ} |} is appended and the (disturbed) encoded quantum message is replaced by a fixed state Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega }

References

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

contributed by Shraddha Singh and Isabel Nha Minh Le