Experimental demonstration that qubits can be cloned at will, if encrypted with a single-use decryption key

TL;DR

Encrypted cloning is stable under hardware noise, even when used as a module, namely in parallel, series or interleaved, while preserving pre-existing entanglement, establishes it as a versatile quantum primitive for practical use.

Abstract

The no-cloning theorem forbids the creation of identical copies of qubits, thereby imposing strong limitations on quantum technologies. A recently-proposed protocol, encrypted cloning, showed, however, that the creation of perfect clones is theoretically possible - if the clones are simultaneously encrypted with a single-use decryption key. It has remained an open question, however, whether encrypted cloning is stable under hardware noise and thus practical as a quantum primitive. This is nontrivial because spreading quantum information widely could dilute it until barely exceeding the noise level, leading to catastrophic fidelity decay. Given the complexity of hardware noise, theory and classical simulation are insufficient to settle this. Here, we settle this question experimentally, on IBM Heron-R2 superconducting processors using up to 154 qubits. We find that encrypted cloning is stable under hardware noise, even when used as a module, namely in parallel, series or interleaved, while preserving pre-existing entanglement. This establishes it as a versatile quantum primitive for practical use, and it necessitates a refinement to our understanding of the no-cloning theorem: quantum information can be spread at will, in theory and in practice, without dilution or degradation, if encrypted or obscured. The actual constraint is that the decryption mechanism must be single-use.

Figures (11)

• Figure 1: Experiment 1: Proof on hardware that encrypted cloning can serve as a quantum primitive in the sense that it respects its input's pre-existing entanglement: pre-existing entanglement of qubit $A$ is measurably recovered after decrypting one of up to seven encrypted clones of $A$. The drop in entanglement fidelity is dominated by the number of 2-qubit gate layers. Results depend on when and which chip is used, but on a fixed chip, for a given time, the sampling errors (for 1000s of shots) are very small (see Extended Data: Table \ref{['tab:EC_fidelity']}, Extended Data: Fig. \ref{['fig:comp_EM']}, and Supplementary Information). Results confirmed with two independent measurement methods: Bell state measurement (BSM) and parity oscillations method (POM) (see Methods). They possess similar circuit depths and correspondingly similar entanglement fidelities.
• Figure 2: Experiment 2: (a) Setup to test if the two parts of encrypted cloning, namely the creation of encrypted clones on one hand and the choosing and decrypting of one of the encrypted clones on the other can be treated, as a quantum primitive on hardware, as independent modules that can be interleaved, for example, with a measurement operation. To this end, the measurement on the ancilla $\tilde{A}$ (top rail) is performed either before, at the same time or after choosing and decrypting an encrypted clone, i.e., at time 1, 2 or 3 for experiment Scenarios 2-1, 2-2 or 2-3 respectively. Notice that this is a nontrivial test because, for example, in Scenario 2-1, the measurement on $\tilde{A}$ to test for the preservation of its pre-existing entanglement with $A$ is performed even before the choice is made which encrypted clone to decrypt and then measure. (b) The results show in each scenario that encrypted cloning respects pre-existing entanglement of its input, in the sense that the CHSH inequality is violated, proving quantumness of the correlations, on the given hardware, for up to three encrypted clones. The deterioration of the performance is dominated by the duration of the circuit. For example, the lower performance in Scenario 2-1 arises from the extra idling time during the measurement of $\tilde{A}$. The statistical error bars from $10,000$ shots are very small (see also Extended Data: Table \ref{['tab:S_vs_n']}).
• Figure 3: Experiment 4: We prepare GHZ states for $r$ up to $15$. We then independently create $3$ encrypted clones of each of the $r$ qubits, i.e., we produce $r$ independent groups of three encrypted clones. After the encrypted cloning, in each group, one of the clones is decrypted. Finally, the fidelity $F_r$ between the recovered GHZ state and the initial GHZ state is measured using the POM with 10 000 shots per measurement setting. The signal always remains well above the noise floor and reaches fidelities above the entanglement witness threshold for GHZ states with up to 4 qubits. These results support that encrypted cloning can serve as a candidate quantum primitive in the sense that it respects not only bipartite but also genuine multipartite entanglement of input qubits embedded in larger circuits, and encrypted cloning is a quantum primitive in the sense that it can also be applied in true modular fashion in parallel to multiple qubits.
• Figure 4: Entanglement fidelities $F_e$ of Experiment 1 under the influence of hardware variability and error mitigation methods (a) Comparison between multiple transpilation runs on ibm_kingston and ibm_aachen by averaging over 3 unique transpilation runs. Error bars are the standard deviation over the 3 runs. (b) Comparison between Experiment 1 using no error mitigation, dynamical decoupling and Pauli twirling in combination with dynamical decoupling.
• Figure 5: Gate decomposition of first exponential factor of Eq. \ref{['eq:encoding_operation']}, i.e., $e^{-\mathrm{i} \frac{\pi}{4}\sigma_{3}^{(A)}\otimes \left(\bigotimes_{i=1}^n\sigma_{3}^{(S_i)}\right)}$ when $n=2m$ with $m\in\mathbb{Z}_{>0}$. When $n=2m+1$, the encrypted cloning operation requires two additional CNOT gates applied to $S_{2m+1}$ and $S_{2m}$. The second exponential factor, $e^{-\mathrm{i} \frac{\pi}{4} \sigma_{1}^{(A)}\otimes \left(\bigotimes_{i=1}^n\sigma_{1}^{(S_i)}\right)}$, can be implemented in the similar way, since $e^{-\mathrm{i} \frac{\pi}{4} \sigma_{1}^{(A)}\otimes \left(\bigotimes_{i=1}^n\sigma_{1}^{(S_i)}\right)}=H^{\otimes (n+1)}e^{-\mathrm{i} \frac{\pi}{4} \sigma_{3}^{(A)}\otimes \left(\bigotimes_{i=1}^n\sigma_{3}^{(S_i)}\right)}H^{\otimes (n+1)}$.