## Entanglement goes large

The success of quantum computing relies on the ability to entangle large-scale systems. Various platforms are being pursued, with architectures based on superconducting qubits and trapped atoms being the most advanced. By entangling up to 20 qubits, Omran *et al.* and Song *et al.*—working with Rydberg atom qubits and superconducting qubits, respectively—demonstrate how far these platforms have reached. The demonstrated controllable generation and detection of entanglement on such quantum systems is promising for the development of large-scale quantum processors.

## Abstract

Quantum entanglement involving coherent superpositions of macroscopically distinct states is among the most striking features of quantum theory, but its realization is challenging because such states are extremely fragile. Using a programmable quantum simulator based on neutral atom arrays with interactions mediated by Rydberg states, we demonstrate the creation of “Schrödinger cat” states of the Greenberger-Horne-Zeilinger (GHZ) type with up to 20 qubits. Our approach is based on engineering the energy spectrum and using optimal control of the many-body system. We further demonstrate entanglement manipulation by using GHZ states to distribute entanglement to distant sites in the array, establishing important ingredients for quantum information processing and quantum metrology.

Greenberger-Horne-Zeilinger (GHZ) states constitute an important class of entangled many-body states (*1*). Such states provide an important resource for applications that range from quantum metrology (*2*) to quantum error correction (*3*). However, these are among the most fragile many-body states because a single error on any one of the *N* qubits collapses the superposition, resulting in a statistical mixture. Remarkably, despite their highly entangled nature, GHZ states can be characterized by just two diagonal and two off-diagonal terms in the *N*-particle density matrix. In contrast to quantifying the degree of entanglement in general many-body states, which is extremely challenging (*4*–*6*), the GHZ state fidelity *N*-partite entanglement (*7*). For these reasons, GHZ state creation can serve as an important benchmark for characterizing the quality of any given quantum hardware. Such states have been previously generated and characterized by using systems of nuclear spins (*8*, *9*), individually controlled optical photons (*10*–*12*), trapped ions (*7*, *13*–*15*), and superconducting quantum circuits (*16*, *17*). Large-scale superposition states have also been generated in systems of microwave photons (*18*) and atomic ensembles without individual particle addressing (*2*).

Here, we demonstrate the preparation of *N*-particle GHZ states^{87}Rb atoms, in which the qubits are encoded in an atomic ground state *19*)]. Our entangling operation relies on the strong van der Waals interaction between atoms in states *t*) and detuning Δ(*t*) (*20*, *21*). In addition, we used addressing beams to introduce local energy shifts δ* _{i}* on specific sites

*i*along the array (Fig. 1A). The resulting many-body Hamiltonian is

*(*

_{i}*t*) = Δ(

*t*) + δ

*is the local effective detuning set by the Rydberg laser and the local light shift,*

_{i}*i*, and

*V*is the interaction strength of two Rydberg atoms on neighboring sites. The separation between adjacent sites was chosen so that the nearest-neighbor interaction

*V*= 2π ⋅ 24 MHz ≫ Ω results in the Rydberg blockade (

*22*–

*24*), forbidding the simultaneous excitation of adjacent atoms into the state

To prepare GHZ states, we used arrays with an even number *N* of atoms. For large negative detuning Δ of the Rydberg laser, the many-body ground state of the Hamiltonian (Eq. 2) is * _{i}* = Δ, the ground-state manifold consists of

*N*/2 + 1 nearly degenerate classical configurations with

*N*/2 Rydberg excitations. These include in particular the two target antiferromagnetic configurations

*25*) as well as other states with nearly identical energy (up to a weak second-nearest neighbor interaction), with both edges excited, such as

*using off-resonant laser beams at 840 nm, generated with an acousto-optic deflector (AOD), which energetically penalize the excitation of edge atoms (Fig. 1A) and effectively eliminate the contribution of undesired components. In this case, the ground state for positive detuning is given by the GHZ state (*

_{e}*1*), and there exists in principle an adiabatic pathway that transforms the state

*t*) from negative to positive values (Fig. 1B).

In practice, the time necessary to adiabatically prepare such a GHZ state grows with system size and becomes prohibitively long for large *N*, owing to small energy gaps in the many-body spectrum. To address this limitation, we used optimal control methods to find laser pulses that maximize the GHZ state preparation fidelity while minimizing the amount of time necessary. Our specific implementation, the remote dressed chopped-random basis algorithm (RedCRAB) (*26*, *27*), yields optimal shapes of the laser intensity and detuning for the given experimental conditions (*19*). For *N* ≤ 8 atoms, we performed this optimization using δ* _{e}*/(2π) ≈ –4.5 MHz light shifts on the edge atoms. For larger systems of

*N*> 8, the preparation was found to be more robust by increasing the edge light shifts to δ

*/(2π) ≈ –6 MHz and adding δ*

_{e}_{4,}

_{N}_{– 3}/(2π) ≈ –1.5 MHz light shifts on the third site from both edges.

Our experiments are based on the optical tweezer platform and experimental procedure described previously (*21*). After the initialization of a defect-free *N*-atom array, the traps were switched off, while the atoms were illuminated with the Rydberg and local light shift beams. We subsequently measured the internal state of the atoms by imaging state *28*). The results of such experiments for a 20-atom array are demonstrated in Fig. 2. After applying the optimized pulse shown in Fig. 2B, we measured the probability of observing different patterns ^{20} possible patterns in a 20-atom array is shown in Fig. 2A. The states

To characterize the experimentally prepared state ρ, we evaluated the GHZ state fidelity*c _{N}* acquires a phase ϕ at a rate of

*19*,

*29*). In our experiments, the staggered field was implemented by applying off-resonant focused beams of equal intensity at 420 nm, generated by another AOD, to every other site of the array (Fig. 1C), resulting in a local energy shift δ

*(*

_{p}*27*). Subsequently, we drove the atoms resonantly, applying a unitary operation

*19*), so that a measurement of the parity

*c*. The measured parity is shown in Fig. 2C as a function of the phase accumulated on each atom, demonstrating the coherence of the created state.

_{N}To extract the entanglement fidelity for large atomic states, we carefully characterized our detection process used to identify atoms in *19*). Subsequently, we used a maximum-likelihood estimation procedure to infer the properties of created states on the basis of the raw measurement results. Using this procedure, we inferred a probability of preparing states

This protocol was applied for multiple system sizes of 4 ≤ *N* ≤ 20, using 1.1-μs control pulses optimized for each *N* individually. Consistent with expected GHZ dynamics (Fig. 1C) (*13*), the frequency of the measured parity oscillations grows linearly with *N* (Fig. 3A). Extracting the GHZ fidelity from these measurements shows that we surpass the threshold of *3*, *14*).

As an application of our entanglement-manipulation technique, we demonstrate its use for entanglement distribution between distant atoms. Specifically, we consider the preparation of Bell states between atoms at the two opposite edges of the array. Our approach was based on first creating the GHZ state by using the above procedure, followed by an operation that disentangles all but two target atoms. The latter is realized by shifting the transition frequencies of the two target edge atoms by using two strong, blue-detuned addressing beams at 420 nm. Subsequently, we performed a reverse detuning sweep of the Rydberg laser that effectively disentangles all atoms except those at the edges. The resulting state corresponds to a coherent superposition of two pinned excitations that can be converted into a Bell state

To demonstrate this protocol experimentally, we prepared a GHZ state of eight atoms and turned on the detuned 420-nm addressing beams on the edge atoms, resulting in a shift of δ_{1,8}/(2π) = 6 MHz. We then used an optimized Rydberg laser pulse to distribute the entanglement and observed the patterns

Regarding our experimental observations, the optimal control provides a substantial improvement over naïve analytic pulses (Fig. 3B) while bringing our protocol close to the speed set by a more conventional protocol of building up entanglement through a series of two-qubit operations (*19*). By contrast, a simple linear detuning sweep only allows for the creation of GHZ states for *N* ≤ 16 within a fixed 1.1-μs window (Fig. 3B), even under ideal conditions. Our analysis reveals that the reason for this improvement stems from diabatic excitations and de-excitations in the many-body spectrum, related to the recently proposed mechanisms for quantum optimization speedup (*19*, *30*, *31*).

The measured entanglement fidelity is partially limited by imperfect qubit rotations used for parity measurements. Specifically, the qubit rotation operation *19*). The resulting evolution can be understood in terms of quantum many-body scars (*21*, *32*), which gives rise to coherent qubit rotations, even in the presence of strong interactions. The deviations from an ideal parity measurement arise from the Rydberg blockade constraint and long-range interactions (*19*). These grow with the system size, resulting in finite fidelities even for a perfect initial GHZ state (Fig. 3B, gray shaded area). Our quoted fidelity values do not include the correction for this imperfection and represent the lower bound on the actual GHZ state fidelities.

Entanglement generation, manipulation, and lifetime are further limited by several sources of decoherence. The finite temperature of the atoms leads to random Doppler shifts on every site as well as position fluctuations that influence interaction energies. These thermal dephasing mechanisms lead to a Gaussian decay of the GHZ state coherence, whose time scale decreases with the system size as *19*). We can attribute this discrepancy to several additional sources of errors. Laser phase noise likely contributes to the finite fidelity of the state preparation. Drifts in the beam positions of the Rydberg lasers can lead to changing light shifts, giving rise to uncontrolled detunings, and drifts in the addressing beam positions can lead to an imbalance in the local energy shifts and thereby in the populations of the two GHZ components, limiting the maximum possible coherence. This analysis highlights the utility of GHZ states for uncovering sources of errors. We emphasize that all of these known error sources can be mitigated through technical improvements (*19*).

Our experiments demonstrate a new promising approach for the deterministic creation and manipulation of large-scale entangled states, enabling the realization of GHZ-type entanglement in system sizes of up to *N* = 20 atoms. These results show the utility of this approach for benchmarking quantum hardware, demonstrating that Rydberg atom arrays constitute a competitive platform for quantum information science and engineering. Specifically, the entanglement generation and distribution could potentially be used for applications that range from quantum metrology and quantum networking to quantum error correction and quantum computation. Our method can be extended by mapping the Rydberg qubit states used here to ground-state hyperfine sublevels, so that the entangled atoms can remain trapped and maintain their quantum coherence over very long times (*19*, *23*, *24*, *33*). This could enable the sophisticated manipulation of entanglement and realization of deep quantum circuits for applications such as quantum optimization (*30*, *31*).

During the completion of this work, we became aware of related results that demonstrate large GHZ state preparation using superconducting quantum circuits (*34*, *35*).

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank D. Sels and C. Reimer for helpful discussions.

**Funding:**The authors acknowledge financial support from the Center for Ultracold Atoms, the National Science Foundation, Vannevar Bush Faculty Fellowship, the U.S. Department of Energy, and the Office of Naval Research. H.L. acknowledges support from the National Defense Science and Engineering Graduate (NDSEG) fellowship. G.S. acknowledges support from a fellowship from the Max Planck/Harvard Research Center for Quantum Optics. J.C., S.M., and T.C. acknowledge funding from the EC H2020 grants 765267 (QuSCo), 817482 (PASQuANS), and QuantERA QTFLAG; the DFG SPP 1929 (GiRyd) and TWITTER; the IQST Alliance; and the Italian PRIN 2017.

**Author contributions:**A.O., H.L., A.K., G.S., T.T.W., S.E., H.B., A.S.Z., and M.E. built the experimental setup. A.O., H.L., A.K., and G.S. performed the measurements and analyzed the data. H.P., S.C., J.C., and S.M. performed theoretical analysis. J.C., M.R., P.R., S.M., and T.C. provided and maintained the optimal control server. All work was supervised by M.G., V.V., and M.D.L. All authors discussed the results and contributed to the manuscript.

**Competing interests:**M.G., V.V., and M.D.L. have an equity interest in and serve on the advisory board of QuEra Computing.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and the supplementary materials.