Clifford Code for Quantum Authentication

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

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.

Notations

• ${\mathcal {S}}$: 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.

Protocol Description

Input: $\rho$, $d$, $k$

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

• Encoding:

1. ${\mathcal {S}}$ appends an auxiliary register of $d$ qubits in state $|0\rangle \langle 0|$ to the quantum message $\rho$, which results in $\rho \otimes |0\rangle \langle 0|^{\otimes d}$.
2. ${\mathcal {S}}$ then applies $C_{k}$ for a uniformly random $k\in {\mathcal {K}}$ on the total state.
3. ${\mathcal {S}}$ 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:

1. ${\mathcal {A}}$ applies the inverse Clifford $C_{k}^{\dagger }$ to the received state, which is denoted by $\rho ^{\prime }$.
2. ${\mathcal {A}}$ 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
  $${\mathcal {D}}_{k}:\rho ^{\prime }\mapsto \mathrm {tr} _{0}\left({\mathcal {P}}_{\mathrm {acc} }C_{k}^{\dagger }(\rho ^{\prime })C_{k}{\mathcal {P}}_{\mathrm {acc} }^{\dagger }\right)\otimes |\mathrm {ACC} \rangle \langle \mathrm {ACC} |+\mathrm {tr} \left({\mathcal {P}}_{\mathrm {rej} }C_{k}^{\dagger }(\rho ^{\prime })C_{k}{\mathcal {P}}_{\mathrm {rej} }^{\dagger }\right)\Omega \otimes |\mathrm {REJ} \rangle \langle \mathrm {REJ} |.$$
  In the above, $\mathrm {tr} _{0}$ 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
  $${\mathcal {P}}_{\mathrm {acc} }=\mathbb {1} ^{\otimes n}\otimes |0\rangle \langle 0|^{\otimes d}$$
  and
  $${\mathcal {P}}_{\mathrm {rej} }=\mathbb {1} ^{\otimes (n+d)}-{\mathcal {P}}_{\mathrm {acc} }.$$

References

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