Pauli Cloners for Pauli Channels

TL;DR

This work develops a Pauli-cloner framework that extends the Niu–Griffiths cloner to $N$-qubit Pauli channels, linking cloning performance to Mutually Unbiased Bases and the distribution of Pauli errors. It shows that NG and QID realize Pauli cloners on single qubits and can be generalized to $N$ qubits with a circuit that programs all $4^N-1$ Pauli disturbances, enabling basis-biased fidelity while preserving per-basis cloning quality. The authors provide analytic fidelity expressions for the two-qubit case, demonstrate practical tailoring of cloners to noise models in QKD scenarios (e.g., BB84 and six-state), and derive the symmetric $N$-qubit UQCM fidelity $F=rac{d+3}{2(d+1)}$ with $d=2^N$. The results have implications for QKD security analyses and quantum-money concepts, and point to extensions to higher-dimensional systems and broader noise-model generalizations.

Abstract

We present a quantum circuit architecture for the one-to-two cloning of $N$-qubit registers. It implements the broad class of Pauli cloners by extending the Niu--Griffiths architecture to multi-qubit systems. In the single-qubit case, we provide explicit constructions for asymmetric universal, phase covariant and biased cloners. We explore the fundamental relationship between Pauli errors, mutually unbiased bases and Pauli cloning. Furthermore, we demonstrate how Pauli cloners can be tailored to specific noise models in the context of quantum communication, especially quantum key distribution.

Figures (15)

• Figure 1: Quantum circuit implementation of the single-qubit Niu--Griffiths cloner. Qubit 0 serves as both input and first output, while qubit 1 is the second output. Qubit 0 can be affected by Pauli noise, but only until Eve's controlled gates begin to act. The program state is initialized with three $R_Y$-rotations, allowing only real states parametrized by the three angles $\rho$, $\phi$ and $\theta$. The hardware part is implemented with an H gate and three CNOTs.
• Figure 2: Comparison between the PCCM and the imbalanced cloner obtained with Niu--Griffiths for a noisy channel with $p_X = 0.25$. The average fidelity is displayed alongside the specific fidelities for the $X$ and $Z$ bases. By biasing the cloner towards the $X$ basis which remains invariant under the bit-flip noise, the Niu--Griffiths cloner achieves a higher average fidelity than the PCCM.
• Figure 3: Comparison between the UQCM and the generalized imbalanced cloner obtained with Niu--Griffiths for a noisy channel with $p_X = 0.25$ and $p_Z = 0.1$. The average fidelity is displayed alongside the specific fidelities for the $X$, $Y$ and $Z$ bases. By biasing the cloner towards the $X$ basis which is only affected by the weaker $Z$ errors, the Niu--Griffiths cloner achieves a higher average fidelity than the UQCM.
• Figure 4: Comparison between the best Niu--Griffiths cloner and QML results for cloning the two states of the B92 protocol. QML achieves significantly higher fidelities than the NG cloner. The QID performance (not shown here) is equal to that of the NG cloner).
• Figure 5: Quantum circuit implementing the Niu--Griffiths cloner for $N$-qubit registers. Register 0 serves as both input and first output, while register 1 is the second output. Register 0 can be affected by Pauli noise, but only before Eve's controlled gates begin to act. The $H$ gate symbol stands for $N$ single-qubit $H$ gates, such that one H gate acts on each qubit in the register. All symbols for CNOT gates in the diagram correspond to $N$ CNOT gates, with the $i$-th CNOT having the control on the $i$-th qubit of the control register, and the target in the $i$-th qubit of the target register.