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.]. 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]].
The ''Clifford Authentication Scheme'' is a non-interactive protocol for [[Authentication of Quantum Messages|quantum authentication]] and 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 or not a eavesdropper has tampered the original quantum message.


'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocol]][[Category:Two Party Protocols]]
'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocol]][[Category:Two Party Protocols]], [[:Category:Quantum Functionality|Quantum Functionality]][[Category:Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]], [[:Category:Building Blocks|Building Block]][[Category:Building Blocks]]


==Outline==
==Outline==
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.
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 original quantum message was tampered by a third party and the authenticator aborts the process.


==Notations==
==Notations==
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==Protocol Description==
==Protocol Description==
*'''Input:''' <math>\rho</math>, <math>d</math>, <math>k</math>
'''Input:''' <math>\rho</math>, <math>d</math>, <math>k</math></br></br>
*'''Output:''' Receiver accepts or rejects
'''Output:''' Quantum state <math>\rho^\prime</math> if the protocol accepts; fixed quantum state <math>\Omega</math> if the protocol aborts
**'''''Encoding:''''' <math>\mathcal{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger</math>
*'''''Encoding:'''''  
#<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.
*'''''Mathematical Encoding Description:'''''</br>Mathematically, the encoding process can be described by <math display=block>\mathcal{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger</math>
*'''''Decoding:'''''
#<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.
*'''''Mathematical Decoding Description:''''' </br>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}</math> and <math>\mathcal{P}_\mathrm{rej}</math> refer to the measurement projectors that determine whether the protocol accepts or aborts the received quantum message. It is <math display=block>\mathcal{P}_\mathrm{acc}=\mathbb{1}^{\otimes n} \otimes |0\rangle\langle 0|^{\otimes d}</math> and <math display=block>\mathcal{P}_\mathrm{rej}=\mathbb{1}^{\otimes (n+d)} - \mathcal{P}_\mathrm{acc}.</math>




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#[https://arxiv.org/pdf/0810.5375.pdf| Aharanov et al. (2008).]
#[https://arxiv.org/pdf/0810.5375.pdf| Aharanov et al. (2008).]
#[https://arxiv.org/pdf/1607.03075.pdf| Broadbent and Wainewright (2016).]
#[https://arxiv.org/pdf/1607.03075.pdf| Broadbent and Wainewright (2016).]
<div style='text-align: right;'>''contributed by Shraddha Singh and Isabel Nha Minh Le''</div>
 
<div style='text-align: right;'>''Contributed by Isabel Nha Minh Le and Shraddha Singh''</div>
<div style='text-align: right;'>''This page was created within the [https://www.qosf.org/qc_mentorship/| QOSF Mentorship Program Cohort 4]''</div>

Latest revision as of 18:49, 16 January 2022

The Clifford Authentication Scheme is a non-interactive protocol for quantum authentication and 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 or not a eavesdropper has tampered the original quantum message.

Tags: Two Party Protocol, Quantum Functionality, Specific Task, Building Block

Outline[edit]

The Clifford code encodes a quantum message by appending an auxiliary register with each qubit in state 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 , the authenticator accepts and decodes the quantum message. Otherwise, the original quantum message was tampered by a third party and the authenticator aborts the process.

Notations[edit]

  • : 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
  • 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\}} : 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[edit]

  • 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 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 2^{-d}}
  • 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 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, 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[edit]

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} , 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} , 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}

Output: Quantum 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 \rho^\prime} if the protocol accepts; fixed quantum 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} if the protocol aborts

  • Encoding:
  1. 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 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|} 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}} .
  2. 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.
  3. 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}} .
  • Mathematical Encoding Description:
    Mathematically, the encoding process can be 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{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger}
  • Decoding:
  1. 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} .
  2. 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 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 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.
  • Mathematical Decoding Description:
    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, 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}_0} 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}} 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}} refer to the measurement projectors that determine whether the protocol accepts or aborts the received quantum message. It is 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}.}


References[edit]

  1. Aharanov et al. (2008).
  2. Broadbent and Wainewright (2016).
Contributed by Isabel Nha Minh Le and Shraddha Singh
This page was created within the QOSF Mentorship Program Cohort 4
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