Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit

No-cloning theorem forbids perfect cloning of an unknown quantum state. A universal quantum cloning machine (UQCM), capable of producing two copies of any input qubit with the optimal fidelity, is of fundamental interest and has applications in quantum information processing. This is enabled by delicately tailored nonclassical correlations between the input qubit and the copying qubits, which distinguish the UQCM from a classical counterpart, but whose experimental demonstrations are still lacking. We here implement the UQCM in a superconducting circuit, and investigate these correlations. The measured entanglements well agree with our theoretical prediction that they are independent of the input state and thus constitute a universal quantum behavior of the UQCM that was not previously revealed. Another feature of our experiment is the realization of deterministic and individual cloning, in contrast to previously demonstrated UQCMs, which either were probabilistic or did not constitute true cloning of individual qubits.

reported in nuclear magnetic resonance systems [10, 11], but where the true cloning of individual quantum systems cannot be achieved due to the ensemble aspect. Huang et al. presented a proof-of-principle demonstration in an optical system [12], where only a single photon was involved; its polarization state was copied onto one path freedom degree.
Several optical experiments have been reported, where the state of a photon was copied onto another photon [13][14][15][16][17], but the cloning processes are probabilistic for lack of a deterministic two-qubit controlled gate between different photons in these experiments.
Besides the limitation of ensemble aspect or probabilistic nature, previous experiments did not reveal the nonclassical correlations between the original input qubit and the copying qubits. These correlations enable the information carried by the input state to be equally imprinted on the clones with the optimal fidelity, and represent the most fundamental difference between the UQCM and a classical cloning machine. Quantitative characterization of these correlations is important for revealing the genuine quantum behavior of the UQCM, which is closely related to the universality and optimality of the copying operation.
We here adapt a scheme proposed in the context of cavity quantum electrodynamics [18] to a superconducting circuit involving Xmon qubits controllably coupled to a bus resonator. The high degree of control over the qubit-qubit interactions enables realizations of all gate operations required for approximately cloning the state of each qubit in a deterministic way. We indicate the universality of the implemented UQCM, and quantitatively characterize the entanglement between the input qubit and each of the copy qubits. The results confirm our theoretical prediction that this entanglement is also inputstate-independent, and represents a universal quantum feature of the UQCM. The entanglement between the two copy qubits is also measured.

II. RESULTS
A. Implementation of UQCM.
The sample used to perform the experiment involves five Xmon qubits [19], three of which are employed in our experiment and labeled from Q 1 to Q 3 ; these qubits are almost symmetrically coupled to a central bus resonator, as sketched in Fig. 1a. The resonator has a fixed frequency of ω r /2π = 5.588 GHz, while the frequencies of the qubits are individually adjustable, which enables us to tailor the system dynamics to accomplish the copying task. The Hamiltonian for the total system is where a † and a are the photonic creation and annihilation operators for the resonator, respectively, S + j = |1 j 0 j | and S − j = |0 j 1 j | are the flip operators for Q j , with |0 j and |1 j being its ground and first excited states separated by an energy gap ω q,j , g j are the corresponding qubit-resonator coupling strengths, and is the reduced Planck constant. In our sample these coupling strengths are almost identical, e.g., g j g 2π×20 MHz. The system parameters are detailed in Supplementary Note 3. The qubit frequency tunability makes the system dynamics programable.
When two or more qubits are detuned from the resonator by the same amount much larger than g, they are coupled by virtual photon exchange [20][21][22][23][24][25][26][27][28][29][30]. In our experiment, Q 1 acts as the original qubit whose state is to be cloned, and Q 2 and Q 3 are used as the copying qubits.
The experimental sequence for realizing the UQCM the qubits' manipulation and their coupling to the resonator and the measurement are controlled by microwave pulses injected onto the circuit sample. b Before the copying operation, all qubits are initialized to their ground state at the corresponding idle frequencies. The whole procedure can be divided into four steps: Preparation of the input state through a unitary rotation at the idle frequency, denoted as U ; entanglement of Q2 and Q3 with Bell type, achieved by a π rotation Xπ on Q3, the Q2-Q3 √ iSWAP gate, and a small Z pulse on Q3 realizing a rotation R z θ for phase compensation; cloning of the input state onto Q2 and Q3, realized by resonator-induced couplings C1,2,3 and C2,3; output state tomography. C1,2,3 is implemented by tuning Q2 and Q3 on resonance with Q1 at the working frequency, while C2,3 realized by tuning Q1 back to its idle frequency, leaving Q2 and Q3 coupled to each other. Note that in our experiment, steps 1 and 2 are completed simultaneously for the sake of reducing qubits' decoherence, see Supplementary Note 4 for the details.
with our setup is shown in Fig. 1b. The experiment starts with initializing the resonator to the vacuum state |0 r and the qubits to their ground state |0 1 0 2 0 3 at their idle frequencies. These idle frequencies are highly detuned from the resonator frequency and off-resonant with each other, ensuring each qubit to be effectively decoupled from the resonator and other qubits when staying at its idle frequency. After the initialization, a suitable rotation is applied to Q 1 to prepare it in the state to be cloned where α and β are complex numbers, satisfying |α| 2 + GHz, where the rotation R z θ = e iθ|13 13| is realized after a duration of 30 ns.
After the production of ψ + 2,3 , Q 2 and Q 3 are tuned on resonance with Q 1 at the working frequency, where these qubits are red-detuned from the resonator by the same amount ∆ = 2π × 148 MHz. With this setting, the resonator does not exchange photons with the qubits due to the large detuning, but can mediate a coupling of strength λ = g 2 /∆ between any two of these qubits.
The resonator will remain in the ground state during this Under this Hamiltonian, Q 2 and Q 3 symmetrically interact with Q 1 through excitation exchange, with the number of the total excitations being conserved. After an interaction time τ = 2π/9λ, the three-qubit coupling C 1,2,3 evolves Q 1 , Q 2 and Q 2 to the entangled state Then Q 1 is tuned back to its idle frequency of 5.367 GHz and decoupled from Q 2 and Q 3 , which remain at the working frequency and continue to interact with each other. The state components |0 2 |0 3 and |1 2 |1 3 are eigenstates of the two-qubit interaction Hamiltonian with the zero eigenvalue, while ψ + 2,3 is an eigenstate of H e with the eigenvalue of −λ. As a result, this swapping interaction does not affect |0 2 |0 3 and |1 2 |1 3 , but produces a phase shift λτ to ψ + 2,3 , with τ being the interaction time. With the choice τ = π/3λ, the two-qubit coupling C 2,3 cancels the phase factor e −iπ/3 associated with ψ + 2,3 , evolving the three qubits to [18] where the phase φ is due to the frequency shift of Q 1 during the Q 2 -Q 3 interaction, which does not affect the reduced density matrices for both Q 2 and Q 3 , each of which in the basis {|0 , |1 } is given by  For the perfect UQCM, the fidelity of these two output copiers with respect to the input state |ψ in is 5/6, irrespective of the probability amplitudes α and β associated with the components |0 and |1 . Due to the nonuniform qubit-resonator couplings and the existence of the direct but also nonuniform qubit-qubit couplings in our We characterize the performance of the UQCM by copy qubits is mainly due to direct qubit-qubit couplings.
These nonuniform couplings also make the qualities of the output states slightly depend on the input state. We note that for each of the six input states, the output state of Q 3 has a fidelity very close to the theoretical upper bound. This is partly due to the asymmetry between the two clones. The other reason is that the qubitqubit couplings during the copy process partly protect the qubits from dephasing, so that the real T 2 times of the qubits coupled at the working frequency are longer than the corresponding results listed in Supplementary To further examine the performance of the UQCM, we perform the quantum process tomography (See Supplementary Note 7), achieved by preparing the above mentioned six distinct input states, and measuring them and the corresponding output states of Q 2 and Q 3 through quantum state tomography. The measured process matrices associated with the output states of Q 2 and Q 3 , χ meas,2 and χ meas,3 , are respectively presented in Fig. 3a and 3b, respectively. The fidelities of χ meas,2 and χ meas,3 with respect to the ideal cloning process χ id , defined as For clarity, a single-qubit z-axis rotation is numerically applied to cancel the phase of the extra phase accumulated due to the qubits' frequency shift. The measured Q1-Q2 and Q1-Q3 output density matrices are displayed in the upper and lower panels, respectively. The black wire frames denote the corresponding density matrices produced by the perfect UQCM.
respectively. These process fidelities are close to the result of the perfect UQCM, 0.75, demonstrating a good quantum control over the multiqubit-resonator system. plementary Note 1). To detect these nonclassical correlations, we respectively measure the joint Q 1 -Q 2 and Q 1 -Q 3 output density matrices. The results for the six  2). With the present system parameters, the value of the modulation function of the Q 2 − Q 3 concurrence is much smaller than that of the fidelity of the output state of each copy qubit, which approximates to the maximum. Consequently, the output Q 2 −Q 3 entanglement for the input superposition state is much smaller than that for the input |0 -or |1 -state; while the output state fidelity of each qubit is almost input-state-independent. The existence of concurrence between any two of the three qubits confirm they are in a genuine three-particle entangled state, revealing the fundamental difference between a quantum cloning process and a classical one.

III. DISCUSSION
We have demonstrated universal cloning of an arbitrary state of an individual qubit with a circuit QED setup, where all the quantum operations necessary for constructing a UQCM network are deterministically realized. We characterize the performance of the UQCM by quantum state tomography, confirming the universality of the copying process. We measure the entangle-ment between each copy qubit and the original qubit, with the results being in well agreement with the theoretical prediction that this entanglement is input-stateindependent and represents a universal quantum behavior of the UQCM. We further measure the entanglement between the two clones, verifying the existence of true three-particle entanglement at the output. These results underline the fact that the universal entanglement behavior underlies the performance of the UQCM. ( The four eigenvalues of ∼ ρ in the decreasing order are λ 1 = 4/9 and λ 2 = λ 3 = λ 4 = 0, respectively. The corresponding concurrence [1], defined as C = max{ The matrix ∼ ρ is arXiv:1909.03170v3 [quant-ph] 25 May 2021 the four eigenvalues of which in the decreasing order are λ 1 = 1/9 and λ 2 = λ 3 = λ 4 = 0, respectively, corresponding to a concurrence of 1/3, which is also inputstate-independent.

SUPPLEMENTARY NOTE 2: Effects of asymmetric direct couplings
With direct asymmetric couplings between the qubits being considered, the system finally evolves to the state The reduced density operator for Q 2 and Q 3 after tracing out Q 1 is given by where The concurrence between Q 2 and Q 3 is given by For k j ≈ 1 with j = 1 to 5, we have When β = 0 or 1, we have E 2,3 ≈ 1 3 . For |β| = 1/ √ 2, This implies that the output Q 2 -Q 3 entanglement for the input superposition state may be much smaller than that for the input |0 -or |1 -state.
For k j ≈ 1, we have The corresponding fidelity is When β = 0 or 1, we have F ≈ 5 6 . For |β| = 1/ √ 2, When φ 6 ≈ 2mπ and φ 2 −φ 4 ≈ 2nπ, with m and n being integers, the fidelities for the input superposition states are almost the same as those for the input |0 -or |1 -state.

SUPPLEMENTARY NOTE 3: Device sketch, system parameters and experimental setup
The UQCM is demonstrated in a superconducting circuit consisting of five frequency-tunable Xmon qubits, labeled from Q 1 to Q 5 , coupled to a resonator with a fixed frequency of ω r /2π = 5.588 GHz [3,4]  Supplementary Figure 1: Sketch of the device and experimental setup. At the right-bottom side: the device (sample) consists of five frequency-tunable superconducting Xmon qubits, denoted by Q j with j = 1 to 5, coupled to a bus resonator, labeled as R. Q 1 acts as the input qubit, whose state is to be copied onto Q 2 and Q 3 . Q 4 and Q 5 are unused and not marked here. All the electronics and wiring is shown, from right to left, for the qubit XY control, the qubit Z control, the qubit readout and JPA control, respectively. Each qubit XY control is produced by the mixing of the signals of two independent DAC channels I/Q and a microwave source, and used for the fast qubit flipping. Each qubit Z control has two channels of signals: the Z pulse control directly produced by a DAC channel for the fast frequency tuning of the qubit and the DC control provided by a DC source for resetting its operating point. The qubit readout signal is provided also by the sideband mixing of the signals of two independent DAC channels I/Q and a microwave source, outputting a three-tone microwave pulse that targets the readout resonators of all the qubits. The output signal is amplified sequentially by JPA, HEMT and room temperature amplifiers, before being captured and demodulated by the ADC. The JPA control is provided by the pumping of an independent microwave signal with its amplification band being tunable with a DC bias applied to it. Each control line is fed with well-designed attenuators and filters to prevent the unwanted noise from disturbing the operation of the device.
Supplementary Table 1. The qubit-resonator coupling strength g j is measured and estimated through the vacuum Rabi oscillations. The qubits' energy decaying time T 1 , the Ramsey dephasing time T * 2,j , and the dephasing time with spin-echo T SE 2,j are respectively measured at qubits' idle frequencies ω j ; while the values in parentheses are those measured at the qubits' working frequency ω w /2π = 5.44 GHz where the resonator-induced qubitqubit couplings are realized and used for the Bell-state generation and the cloning operation. The leakage rate κ r j of Q j 's readout resonator R j is obtained through measuring the frequency shift of Q j induced by photons accumulated in its readout resonator [4,5]. GHz band-pass filter; the amplification band of the JPA is tunable with a DC bias applied to it. To reduce kinds of unwanted noises feeding into the signal lines, each signal from (to) DAC and Analog-to-digital convertor (ADC) is filtered with a 7.5 GHz low-pass filter, each qubit Z pulse from DAC is filtered with a 500 MHz low-pass filter, and each DC signal is filtered with a 80 MHz low-pass filter at Mixing-Chamber (MC) stage and with a RC filter at 4K state. Moreover, some attenuators are also added to the signal lines at different temperature stages to further reduce the noises influencing the operation of the device.

SUPPLEMENTARY NOTE 4: Experimental pulse sequences
The pulse sequence for UQCM is shown in Supplementary Figure 2, which is divided into 3 stages in time series. The first stage involves 2 steps for the implementation: the preparation of the state to be cloned for Q 1 and the preparation of the entangled state for the two copy qubits Q 2 and Q 3 . The experiment begins with a π rotation X π applied on Q 3 , realized by a microwave pulse with a duration of 40 ns at its idle frequency. Then Q 2 and Q 3 are biased with rectangular pulses from their respective idle frequencies (see Supplementary Each matrix element is characterized by two color bars, one for the real part and the other for the imaginary part. The black wire frames denote the corresponding matrix elements of the ideal output states.
The Q 2 -Q 3 joint output density matrices for the six input states |0 1 , (|0 1 Supplementary Figure 3a- The system's state density matrix (ρ c N ) is reconstructed by correcting the measured probabilities (ρ N ) by multiplying the matrix inversion aŝ withF defined as the tensor product of each qubit'sF c j : with X = tr(ρX) = ρ 0,1 + ρ 1,0 , Each combination of rotations needs a pulse sequence, in which we append a tomographic operation for each qubit after producing the state, and then measure the binary outcomes of the N qubits. We repeat the same sequence thousands of times to count all the probabilities.
Taking into account the condition of normalization, these measurements produce 3 N (2 N −1) numbers to determine the N qubits' density matrixρ N , that possesses 4 N − 1 degrees of freedom. Owing to the condition of 3 N (2 N − 1) ≥ 4 N − 1, the measurement outcomes give more than enough information to determineρ N . By introducing the vector formP O = P j k andρ N =ρ N , we simplify Eq. (19) asP withŨ being the tensor object formed by the combination of U j and (U j ) + . As the problem in the form of Eq. (20) is overconstrained, for our implementation we solved it by turning to a least-squares optimization as to find the optimal solution. Due to the inevitable noises involved in the measured quantities, conversely it is helpful to use the overdetermined equations to guarantee the availability of the outcomes. Nevertheless, the solvedρ N through such a method does not necessarily ensure that it is positive, Hermitian, and simi-definite with unit trace. It is still needed another optimization, that is, min( ρ N −ρ N ), to get the requiredρ N that satisfies the three conditions imposed. Then the fidelity of the qubit state can be obtained by the calculation F = tr(ρ N * ρ id ), whereρ id is the ideal density matrix. Note that the reconstructed density matrices for the fidelity calculation are calibrated according to the qubits' |0 -and |1 -state measurement fidelities, as shown in Supplementary note 5.
The quantum process tomography (QPT) characterizes certain evolution processes for quantum operations.
These quantum operations can be described by the operator-sum representation: Such a linear map M completely describes the dynamics of the system, whose states could change due to not only any unitary operation performed by kinds of quantum gates, but also projection or decoherence effect that are not unitary. The operatorsB i act on the system alone, and satisfy the completeness relation iB where χ hv = i b ih b * iv . Obviously, the χ hv are determined by theB h , and completely describe the whole mapping process provided theB h are fixed.
We now show how to determine χ hv experimentally.
Consider a N-qubit system, whose initial state can be expressed as a 2 N ×2 N matrix. The matrix can be written as the unique linear combination ofρ j , withρ j being the linearly independent basis element within the matrix. It's convenient to choose the basis operatorρ hv ≡ |h v|, and by preparing a set of preset input states, commonly, such as |j ∈ {|h , |v , (|h +|v )/ √ 2, (|h +i|v )/ √ 2}, after the process transformation due to the system's changing, it is then natural to produce the output M(ρ j ), which is formed by linear combinations of M(|j j|), as can be determined by quantum state tomography. Therefore, it is reasonable to express M(ρ j ) as a linear combination of the basis states: which determines η jk based on the known M(ρ j ).
Going back to Eq. (24), we write within it the relation: which determines ξ hv jk given theB h operators and theρ j operators. Combining Eqs. (24) and (25), we have k hv Eq. (27) immediately gives hv χ hv ξ hv jk = η jk (28) due to thatρ k is linearly independent.The generalized inverse ς of ξ hv jk can be solved with the help of specific software packages for matrix calculation. Thus χ hv jk can then be obtained to be