Clifford Code for Quantum Authentication: Difference between revisions
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The Clifford Authentication Scheme was introduced in the paper [https://arxiv.org/pdf/0810.5375.pdf| Interactive Proofs For Quantum Computations by Aharanov et al.]. | The Clifford Authentication Scheme was introduced in the paper [https://arxiv.org/pdf/0810.5375.pdf| Interactive Proofs For Quantum Computations by Aharanov et al.]. It applies a random Clifford operator to the quantum message and an auxiliary register and then measures the auxiliary register to decide whether to accept or abort for [[Authentication of Quantum Messages|quantum authentication]]. | ||
'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocol]][[Category:Two Party Protocols]] | |||
==Outline== | ==Outline== | ||
The Clifford code encodes a | The Clifford code encodes a quantum message by appending an auxiliary register with each qubit in state <math>|0\rangle</math> and then applying a random Clifford operator on all qubits. The authenticator then measures only the auxiliary register. If all qubits in the auxiliary register are still in state <math>|0\rangle</math>, the authenticator accepts and decodes the quantum message. Otherwise, the authenticator aborts the process. | ||
==Notations== | ==Notations== | ||
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*<math>\rho</math>: <math>m</math>-qubit state to be transmitted | *<math>\rho</math>: <math>m</math>-qubit state to be transmitted | ||
*<math>d\in\mathbb{N}</math>: security parameter defining the number of qubits in the auxiliary register | *<math>d\in\mathbb{N}</math>: security parameter defining the number of qubits in the auxiliary register | ||
*<math>\{C_k\}</math>: set of Clifford operations on <math>n</math> qubits labelled by a classical key <math>k\in\mathcal{K}</math> | |||
*<math>\{C_k\}</math>: set of Clifford operations on <math>n</math> qubits labelled by key <math>k\in\mathcal{K}</math> | |||
==Properties== | ==Properties== | ||
*The Clifford code is quantum authentication scheme with security <math>2^{-d}</math> | *The Clifford code makes use of <math>n=m+d+1</math> qubits | ||
*The Clifford code is [[Authentication of Quantum Messages|quantum authentication]] scheme with security <math>2^{-d}</math> | |||
*The qubit registers used can be divided into a message register with <math>m</math> qubits, an auxiliary register with <math>d</math> qubits, and a flag register with <math>1</math> qubit. | |||
==Protocol Description== | ==Protocol Description== | ||
*'''''Encoding:''''' <math>\mathcal{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger</math> | *'''Input:''' <math>\rho</math>, <math>d</math>, <math>k</math> | ||
*'''Output:''' Receiver accepts or rejects | |||
**'''''Encoding:''''' <math>\mathcal{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger</math> | |||
#<math>\mathcal{S}</math> appends an auxiliary register of <math>d</math> qubits in state <math>|0\rangle\langle 0|</math> to the quantum message <math>\rho</math>, which results in <math>\rho\otimes|0\rangle\langle0|^{\otimes d}</math>. | #<math>\mathcal{S}</math> appends an auxiliary register of <math>d</math> qubits in state <math>|0\rangle\langle 0|</math> to the quantum message <math>\rho</math>, which results in <math>\rho\otimes|0\rangle\langle0|^{\otimes d}</math>. | ||
#<math>\mathcal{S}</math> then applies <math>C_k</math> for a uniformly random <math>k\in\mathcal{K}</math> on the total state. | #<math>\mathcal{S}</math> then applies <math>C_k</math> for a uniformly random <math>k\in\mathcal{K}</math> on the total state. | ||
#<math>\mathcal{S}</math> sends the result to <math>\mathcal{A}</math>. | #<math>\mathcal{S}</math> sends the result to <math>\mathcal{A}</math>. | ||
*'''''Decoding:''''' Mathematically, the decoding process is described by <math display=block>\mathcal{D}_k: \rho^\prime \mapsto \mathrm{tr}_0\left( \mathcal{P}_\mathrm{acc} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{acc}^\dagger \right) \otimes |\mathrm{ACC}\rangle\langle \mathrm{ACC}| + \mathrm{tr}\left( \mathcal{P}_\mathrm{rej} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{rej}^\dagger \right) \Omega \otimes |\mathrm{REJ}\rangle\langle\mathrm{REJ}|</math> In the above, <math>\mathrm{tr}_0</math> is the trace over the auxiliary register only, and <math>\mathrm{tr}</math> is the trace over the quantum message system and the auxiliary system. Furthermore, <math>\mathcal{P}_\mathrm{acc}=\mathbb{1}^{\otimes n} \otimes |0\rangle\langle 0|^{\otimes d}</math> and <math>\mathcal{P}_\mathrm{rej}=\mathbb{1}^{\otimes (n+d)} - \mathcal{P}_\mathrm{acc}</math> are projective measurement operators. | **'''''Decoding:''''' Mathematically, the decoding process is described by <math display=block>\mathcal{D}_k: \rho^\prime \mapsto \mathrm{tr}_0\left( \mathcal{P}_\mathrm{acc} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{acc}^\dagger \right) \otimes |\mathrm{ACC}\rangle\langle \mathrm{ACC}| + \mathrm{tr}\left( \mathcal{P}_\mathrm{rej} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{rej}^\dagger \right) \Omega \otimes |\mathrm{REJ}\rangle\langle\mathrm{REJ}|</math> In the above, <math>\mathrm{tr}_0</math> is the trace over the auxiliary register only, and <math>\mathrm{tr}</math> is the trace over the quantum message system and the auxiliary system. Furthermore, <math>\mathcal{P}_\mathrm{acc}=\mathbb{1}^{\otimes n} \otimes |0\rangle\langle 0|^{\otimes d}</math> and <math>\mathcal{P}_\mathrm{rej}=\mathbb{1}^{\otimes (n+d)} - \mathcal{P}_\mathrm{acc}</math> are projective measurement operators. | ||
#<math>\mathcal{A}</math> applies the inverse Clifford <math>C_k^\dagger</math> to the received state, which is denoted by <math>\rho^\prime</math>. | #<math>\mathcal{A}</math> applies the inverse Clifford <math>C_k^\dagger</math> to the received state, which is denoted by <math>\rho^\prime</math>. | ||
#<math>\mathcal{A}</math> measures the auxiliary register in the computational basis.</br>a. If all <math>d</math> auxiliary qubits are 0, the state is accepted and an additional flag qubit in state <math>|\mathrm{ACC}\rangle\langle\mathrm{ACC}|</math> is appended.</br>b. Otherwise, the remaining system is traced out and replaced with a fixed <math>m</math>-qubit state <math>\Omega</math> and an additional flag qubit in state <math>|\mathrm{REJ}\rangle\langle \mathrm{REJ}|</math> is appended. | #<math>\mathcal{A}</math> measures the auxiliary register in the computational basis.</br>a. If all <math>d</math> auxiliary qubits are 0, the state is accepted and an additional flag qubit in state <math>|\mathrm{ACC}\rangle\langle\mathrm{ACC}|</math> is appended.</br>b. Otherwise, the remaining system is traced out and replaced with a fixed <math>m</math>-qubit state <math>\Omega</math> and an additional flag qubit in state <math>|\mathrm{REJ}\rangle\langle \mathrm{REJ}|</math> is appended. | ||
Revision as of 13:41, 22 December 2021
The Clifford Authentication Scheme was introduced in the paper Interactive Proofs For Quantum Computations by Aharanov et al.. It applies a random Clifford operator to the quantum message and an auxiliary register and then measures the auxiliary register to decide whether to accept or abort for quantum authentication.
Tags: Two Party Protocol
Outline
The Clifford code encodes a quantum message by appending an auxiliary register with each 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 |0\rangle} and then applying a random Clifford operator on all qubits. The authenticator then measures only the auxiliary register. If all qubits in the auxiliary register are still 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} , the authenticator accepts and decodes the quantum message. Otherwise, the authenticator aborts the process.
Notations
- : suppliant (sender)
- 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 \mathcal{A}} : authenticator (prover)
- 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} : 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 m} -qubit state to be transmitted
- 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\in\mathbb{N}} : security parameter defining the number of qubits in the auxiliary register
- : set of Clifford operations on 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 labelled by a classical 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\in\mathcal{K}}
Properties
- The Clifford code makes use 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=m+d+1} qubits
- The Clifford code is quantum authentication scheme with security
- The qubit registers used can be divided into a message register with 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 m} qubits, an auxiliary register with qubits, and a flag register with 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 1} qubit.
Protocol Description
- Input: , ,
- Output: Receiver accepts or rejects
- Encoding: 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 \mathcal{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger}
- 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 \mathcal{S}} appends an auxiliary 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 d} qubits in state to the quantum message 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} , which results 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 \rho\otimes|0\rangle\langle0|^{\otimes d}} .
- 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 \mathcal{S}} then applies 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 C_k} for a uniformly random 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\in\mathcal{K}} on the total 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 \mathcal{S}} sends the result to 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 \mathcal{A}} .
- Decoding: Mathematically, the decoding process is described 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 \mathcal{D}_k: \rho^\prime \mapsto \mathrm{tr}_0\left( \mathcal{P}_\mathrm{acc} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{acc}^\dagger \right) \otimes |\mathrm{ACC}\rangle\langle \mathrm{ACC}| + \mathrm{tr}\left( \mathcal{P}_\mathrm{rej} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{rej}^\dagger \right) \Omega \otimes |\mathrm{REJ}\rangle\langle\mathrm{REJ}|} In the above, is the trace over the auxiliary register only, 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 \mathrm{tr}} is the trace over the quantum message system and the auxiliary system. Furthermore, 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 \mathcal{P}_\mathrm{acc}=\mathbb{1}^{\otimes n} \otimes |0\rangle\langle 0|^{\otimes d}} 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 \mathcal{P}_\mathrm{rej}=\mathbb{1}^{\otimes (n+d)} - \mathcal{P}_\mathrm{acc}} are projective measurement operators.
- 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 \mathcal{A}} applies the inverse Clifford 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 C_k^\dagger} to the received state, which is denoted 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 \rho^\prime} .
- 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 \mathcal{A}}
measures the auxiliary register in the computational basis.
a. If all 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} auxiliary qubits are 0, the state is accepted and 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.
b. Otherwise, the remaining system is traced out and replaced with a fixed -qubit 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} and 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.
References
contributed by Shraddha Singh and Isabel Nha Minh Le