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Cloning Encrypted Quantum States in Arbitrary Dimensions

Filip-Ioan Ceară

Abstract

Recently, Yamaguchi and Kempf [Phys. Rev. Lett. 136:010801, arXiv:2501.02757] proved that encrypted qubits can be cloned. In this work, we generalize the encrypted cloning protocol and prove that it also applies to higher-order quantum systems. Given that a straightforward generalization of the protocol using the exponential of the shift and phase operators fails to satisfy the unitary requirement for a quantum gate, we propose a different approach. We introduce a new operator to be used in the encryption process and show that it is unitary. We adapt the decryption operator from the reference paper to fit in the framework of multi-level quantum systems. We analyze the circuit implementation of the proposed operators and show that the overhead imposed by larger dimensions scales linearly with qudit dimension.

Cloning Encrypted Quantum States in Arbitrary Dimensions

Abstract

Recently, Yamaguchi and Kempf [Phys. Rev. Lett. 136:010801, arXiv:2501.02757] proved that encrypted qubits can be cloned. In this work, we generalize the encrypted cloning protocol and prove that it also applies to higher-order quantum systems. Given that a straightforward generalization of the protocol using the exponential of the shift and phase operators fails to satisfy the unitary requirement for a quantum gate, we propose a different approach. We introduce a new operator to be used in the encryption process and show that it is unitary. We adapt the decryption operator from the reference paper to fit in the framework of multi-level quantum systems. We analyze the circuit implementation of the proposed operators and show that the overhead imposed by larger dimensions scales linearly with qudit dimension.

Paper Structure

This paper contains 18 sections, 9 theorems, 75 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator definitions
  4. Encryption operator
  5. Decryption operator
  6. Correctness of the unitaries
  7. Implementation analysis
  8. Implementation of encryption operator
  9. Implementation of decryption operator
  10. Conclusions
  11. Useful Lemmas for multi-dimensional quantum states
  12. Unitarity of encryption operator
  13. Unitarity of decryption operator
  14. Correctness proof
  15. Proof of the theorems
  16. ...and 3 more sections

Key Result

Theorem 3.1

The following identity holds for any $d\geq 2$ and $k_1,k_2 \in \{0\dots d-1\}$, where $Q_1, Q_2$ represent the two qudits involved. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: 2D Autocorrelation for $c_{kl}$ coefficients, for different values of the qudit dimension. All the plots show a maximum value of $1$ at $(0,0)$ and the value $0$ for all the other combinations.
  • Figure 2: Quantum circuit for implementation of $V(P)$ from \ref{['new_operator']}, with $P=P_Z$. The circuit implements the controlled gate $C(X_d)$, as defined in \ref{['def_CXd']}.
  • Figure 3: Quantum circuit for implementation of $V(P)$ from \ref{['new_operator']}, with $P=P_X$. The circuit implements the controlled gate $C(X_d)$, as defined in \ref{['def_CXd']}
  • Figure 4: Quantum circuit for implementation of $T_{\overline{kl}}$ from \ref{['Tk']}.The control values for the $C_k(X_d)$ and $C_l(Z_d)$ are marked by the values in the circles, using the definitions of the quantum gates from \ref{['k_controlled']}.
  • Figure 5: Comparison of $N_E$ and $N_D$ gate counts for varying dimensions $d$ and qudit numbers $n$. Left: Single-qudit gate scaling. Right: Two-qudit gate scaling.

Theorems & Definitions (18)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • Lemma A.4
  • ...and 8 more