Revision History for: Clifford Code for Quantum Authentication
- 2025-06-24 at 21:38 by JuliaM
implements Authentication of Quantum Messages
Introduction
The Clifford Authentication Scheme is a non-interactive protocol for quantum authentication. 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.
Outline
The Clifford code encodes a quantum message by appending an auxiliary register with each qubit in state $|0\\\rangle$ 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 $|0\\\rangle$, 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.
Assumptions
acheck
Requirements
No content has been added to this section, yet!
Notation
- $\\\mathcal{S}$: suppliant (sender)
- $\\\mathcal{A}$: authenticator (prover)
- $\\\rho$: $m$-qubit state to be transmitted
- $d \\\in \\\mathbb{N}$: security parameter defining the number of qubits in the auxiliary register
- $\\\{C_{k}\\\}$: set of Clifford operations on $n$ qubits labelled by a classical key $k \\\in \\\mathcal{K}$
Properties
- The Clifford code makes use of $n = m + d + 1$ qubits.
- The Clifford code is quantum authentication scheme with security $2^{-d}$.
- The qubit registers used can be divided into a message register with $m$ qubits, an auxiliary register with $d$ qubits, and a flag register with $1$ qubit.
Technical Description
Input: $\\\rho$, $d$, $k$
Output: Quantum state $\\\rho’$ if the protocol accepts; fixed quantum state $\\\Omega$ if the protocol aborts
Encoding:
- $\\\mathcal{S}$ appends an auxiliary register of $d$ qubits in state $|0\\\rangle \\\langle 0|$ to the quantum message $\\\rho$, which results in $\\\rho \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d}$.
- $\\\mathcal{S}$ then applies $C_k$ for a uniformly random $k \\\in \\\mathcal{K}$ on the total state.
- $\\\mathcal{S}$ sends the result to $\\\mathcal{A}$.
Mathematical Encoding Description:
Mathematically, the encoding process can be described by $$\\\mathcal{E}{k} : \\\rho \\\mapsto C{k}\\\left(\\\rho \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d}\\\right)C_{k}^{\\\dagger}.$$Decoding:
- $\\\mathcal{A}$ applies the inverse Clifford $C_k^{\\\dagger}$ to the received state, which is denoted by $\\\rho’$.
- $\\\mathcal{A}$ measures the auxiliary register in the computational basis.
a. If all $d$ auxiliary qubits are 0, the state is accepted and an additional flag qubit in state $|\\\mathrm{ACC}\\\rangle \\\langle \\\mathrm{ACC}|$ is appended.
b. Otherwise, the remaining system is traced out and replaced with a fixed $m$-qubit state $\\\Omega$ and an additional flag qubit in state $|\\\mathrm{REJ}\\\rangle \\\langle \\\mathrm{REJ}|$ is appended.
Mathematical Decoding Description:
Mathematically, the decoding process is described by
$$\\\mathcal{D}k : \\\rho’ \\\mapsto \\\mathrm{tr}0\\\left(\\\mathcal{P}{\\\mathrm{acc}} C_k^{\\\dagger}(\\\rho’) C_k \\\mathcal{P}{\\\mathrm{acc}}^{\\\dagger}\\\right) \\\otimes |\\\mathrm{ACC}\\\rangle \\\langle \\\mathrm{ACC}| + \\\mathrm{tr}\\\left(\\\mathcal{P}{\\\mathrm{rej}} C_k^{\\\dagger}(\\\rho’) C_k \\\mathcal{P}{\\\mathrm{rej}}^{\\\dagger}\\\right) \\\Omega \\\otimes |\\\mathrm{REJ}\\\rangle \\\langle \\\mathrm{REJ}|.$$In the above, $\\\mathrm{tr}0$ is the trace over the auxiliary register only, and $\\\mathrm{tr}$ is the trace over the quantum message system and the auxiliary system. Furthermore, $\\\mathcal{P}{\\\mathrm{acc}}$ and $\\\mathcal{P}{\\\mathrm{rej}}$ refer to the measurement projectors that determine whether the protocol accepts or aborts the received quantum message. It is $$\\\mathcal{P}{\\\mathrm{acc}} = \\\mathbb{1}^{\\\otimes n} \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d} \\\text{ and } \\\mathcal{P}{\\\mathrm{rej}} = \\\mathbb{1}^{\\\otimes (n+d)} – \\\mathcal{P}{\\\mathrm{acc}}.$$
Experimental Implementations
No content has been added to this section, yet!
Further Information
No content has been added to this section, yet!
References
- Aharonov, Dorit, Michael Ben-Or, Elad Eban, and Urmila Mahadev. “Interactive proofs for quantum computations.” arXiv preprint arXiv:1704.04487 (2017).
- 2025-05-16 at 14:08 by JuliaM
implements Authentication of Quantum Messages
Introduction
The Clifford Authentication Scheme is a non-interactive protocol for quantum authentication. 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.
Outline
The Clifford code encodes a quantum message by appending an auxiliary register with each qubit in state $|0\\\rangle$ 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 $|0\\\rangle$, 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.
Assumptions
acheck
Requirements
No content has been added to this section, yet!
Notation
- $\\\mathcal{S}$: suppliant (sender)
- $\\\mathcal{A}$: authenticator (prover)
- $\\\rho$: $m$-qubit state to be transmitted
- $d \\\in \\\mathbb{N}$: security parameter defining the number of qubits in the auxiliary register
- $\\\{C_{k}\\\}$: set of Clifford operations on $n$ qubits labelled by a classical key $k \\\in \\\mathcal{K}$
Properties
- The Clifford code makes use of $n = m + d + 1$ qubits.
- The Clifford code is quantum authentication scheme with security $2^{-d}$.
- The qubit registers used can be divided into a message register with $m$ qubits, an auxiliary register with $d$ qubits, and a flag register with $1$ qubit.
Technical Description
Input: $\\\rho$, $d$, $k$
Output: Quantum state $\\\rho\’$ if the protocol accepts; fixed quantum state $\\\Omega$ if the protocol aborts
Encoding:
- $\\\mathcal{S}$ appends an auxiliary register of $d$ qubits in state $|0\\\rangle \\\langle 0|$ to the quantum message $\\\rho$, which results in $\\\rho \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d}$.
- $\\\mathcal{S}$ then applies $C_k$ for a uniformly random $k \\\in \\\mathcal{K}$ on the total state.
- $\\\mathcal{S}$ sends the result to $\\\mathcal{A}$.
Mathematical Encoding Description:
Mathematically, the encoding process can be described by $$\\\mathcal{E}{k} : \\\rho \\\mapsto C{k}\\\left(\\\rho \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d}\\\right)C_{k}^{\\\dagger}.$$Decoding:
- $\\\mathcal{A}$ applies the inverse Clifford $C_k^{\\\dagger}$ to the received state, which is denoted by $\\\rho\’$.
- $\\\mathcal{A}$ measures the auxiliary register in the computational basis.
a. If all $d$ auxiliary qubits are 0, the state is accepted and an additional flag qubit in state $|\\\mathrm{ACC}\\\rangle \\\langle \\\mathrm{ACC}|$ is appended.
b. Otherwise, the remaining system is traced out and replaced with a fixed $m$-qubit state $\\\Omega$ and an additional flag qubit in state $|\\\mathrm{REJ}\\\rangle \\\langle \\\mathrm{REJ}|$ is appended.
Mathematical Decoding Description:
Mathematically, the decoding process is described by
$$\\\mathcal{D}k : \\\rho\’ \\\mapsto \\\mathrm{tr}0\\\left(\\\mathcal{P}{\\\mathrm{acc}} C_k^{\\\dagger}(\\\rho\’) C_k \\\mathcal{P}{\\\mathrm{acc}}^{\\\dagger}\\\right) \\\otimes |\\\mathrm{ACC}\\\rangle \\\langle \\\mathrm{ACC}| + \\\mathrm{tr}\\\left(\\\mathcal{P}{\\\mathrm{rej}} C_k^{\\\dagger}(\\\rho\’) C_k \\\mathcal{P}{\\\mathrm{rej}}^{\\\dagger}\\\right) \\\Omega \\\otimes |\\\mathrm{REJ}\\\rangle \\\langle \\\mathrm{REJ}|.$$In the above, $\\\mathrm{tr}0$ is the trace over the auxiliary register only, and $\\\mathrm{tr}$ is the trace over the quantum message system and the auxiliary system. Furthermore, $\\\mathcal{P}{\\\mathrm{acc}}$ and $\\\mathcal{P}{\\\mathrm{rej}}$ refer to the measurement projectors that determine whether the protocol accepts or aborts the received quantum message. It is $$\\\mathcal{P}{\\\mathrm{acc}} = \\\mathbb{1}^{\\\otimes n} \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d} \\\text{ and } \\\mathcal{P}{\\\mathrm{rej}} = \\\mathbb{1}^{\\\otimes (n+d)} – \\\mathcal{P}{\\\mathrm{acc}}.$$
Experimental Implementations
No content has been added to this section, yet!
Further Information
No content has been added to this section, yet!
References
- Aharonov, Dorit, Michael Ben-Or, Elad Eban, and Urmila Mahadev. “Interactive proofs for quantum computations.” arXiv preprint arXiv:1704.04487 (2017).
- 2025-05-16 at 14:08 by JuliaM
implements Authentication of Quantum Messages
Introduction
The Clifford Authentication Scheme is a non-interactive protocol for quantum authentication. 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.
Outline
The Clifford code encodes a quantum message by appending an auxiliary register with each qubit in state $|0\\\rangle$ 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 $|0\\\rangle$, 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.
Assumptions
acheck
Requirements
No content has been added to this section, yet!
Notation
- $\\\mathcal{S}$: suppliant (sender)
- $\\\mathcal{A}$: authenticator (prover)
- $\\\rho$: $m$-qubit state to be transmitted
- $d \\\in \\\mathbb{N}$: security parameter defining the number of qubits in the auxiliary register
- $\\\{C_{k}\\\}$: set of Clifford operations on $n$ qubits labelled by a classical key $k \\\in \\\mathcal{K}$
Properties
- The Clifford code makes use of $n = m + d + 1$ qubits.
- The Clifford code is quantum authentication scheme with security $2^{-d}$.
- The qubit registers used can be divided into a message register with $m$ qubits, an auxiliary register with $d$ qubits, and a flag register with $1$ qubit.
Technical Description
Input: $\\\rho$, $d$, $k$
Output: Quantum state $\\\rho\’$ if the protocol accepts; fixed quantum state $\\\Omega$ if the protocol aborts
Encoding:
- $\\\mathcal{S}$ appends an auxiliary register of $d$ qubits in state $|0\\\rangle \\\langle 0|$ to the quantum message $\\\rho$, which results in $\\\rho \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d}$.
- $\\\mathcal{S}$ then applies $C_k$ for a uniformly random $k \\\in \\\mathcal{K}$ on the total state.
- $\\\mathcal{S}$ sends the result to $\\\mathcal{A}$.
Mathematical Encoding Description:
Mathematically, the encoding process can be described by $$\\\mathcal{E}{k} : \\\rho \\\mapsto C{k}\\\left(\\\rho \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d}\\\right)C_{k}^{\\\dagger}.$$Decoding:
- $\\\mathcal{A}$ applies the inverse Clifford $C_k^{\\\dagger}$ to the received state, which is denoted by $\\\rho\’$.
- $\\\mathcal{A}$ measures the auxiliary register in the computational basis.
a. If all $d$ auxiliary qubits are 0, the state is accepted and an additional flag qubit in state $|\\\mathrm{ACC}\\\rangle \\\langle \\\mathrm{ACC}|$ is appended.
b. Otherwise, the remaining system is traced out and replaced with a fixed $m$-qubit state $\\\Omega$ and an additional flag qubit in state $|\\\mathrm{REJ}\\\rangle \\\langle \\\mathrm{REJ}|$ is appended.
Mathematical Decoding Description:
Mathematically, the decoding process is described by
$$\\\mathcal{D}k : \\\rho\’ \\\mapsto \\\mathrm{tr}0\\\left(\\\mathcal{P}{\\\mathrm{acc}} C_k^{\\\dagger}(\\\rho\’) C_k \\\mathcal{P}{\\\mathrm{acc}}^{\\\dagger}\\\right) \\\otimes |\\\mathrm{ACC}\\\rangle \\\langle \\\mathrm{ACC}| + \\\mathrm{tr}\\\left(\\\mathcal{P}{\\\mathrm{rej}} C_k^{\\\dagger}(\\\rho\’) C_k \\\mathcal{P}{\\\mathrm{rej}}^{\\\dagger}\\\right) \\\Omega \\\otimes |\\\mathrm{REJ}\\\rangle \\\langle \\\mathrm{REJ}|.$$In the above, $\\\mathrm{tr}0$ is the trace over the auxiliary register only, and $\\\mathrm{tr}$ is the trace over the quantum message system and the auxiliary system. Furthermore, $\\\mathcal{P}{\\\mathrm{acc}}$ and $\\\mathcal{P}{\\\mathrm{rej}}$ refer to the measurement projectors that determine whether the protocol accepts or aborts the received quantum message. It is $$\\\mathcal{P}{\\\mathrm{acc}} = \\\mathbb{1}^{\\\otimes n} \\\otimes |0\\\rangle \\\langle 0|^{\\\otimes d} \\\text{ and } \\\mathcal{P}{\\\mathrm{rej}} = \\\mathbb{1}^{\\\otimes (n+d)} – \\\mathcal{P}{\\\mathrm{acc}}.$$
Experimental Implementations
No content has been added to this section, yet!
Further Information
No content has been added to this section, yet!
References
- Aharonov, Dorit, Michael Ben-Or, Elad Eban, and Urmila Mahadev. “Interactive proofs for quantum computations.” arXiv preprint arXiv:1704.04487 (2017).