
A. Kruckenhauser, R. van Bijnen, T. Zache, M. Di Liberto, P. Zoller Highdimensional SO(4)symmetric Rydberg manifolds for quantum simulation,
Quantum Sci. Technol. 8 (20221219),
http://dx.doi.org/10.1088/20589565/aca996 doi:10.1088/20589565/aca996 (ID: 720885)
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We develop a toolbox for manipulating arrays of Rydberg atoms prepared in highdimensional hydrogenlike manifolds in the regime of linear Stark and Zeeman effect. We exploit the SO(4) symmetry to characterize the action of static electric and magnetic fields as well as microwave and optical fields on the wellstructured manifolds of states with principal quantum number n. This enables us to construct generalized largespin Heisenberg models for which we develop statepreparation and readout schemes. Due to the available large internal Hilbert space, these models provide a natural framework for the quantum simulation of Quantum Field Theories, which we illustrate for the case of the sineGordon and massive Schwinger models. Moreover, these highdimensional manifolds also offer the opportunity to perform quantum information processing operations for quditbased quantum computing, which we exemplify with an entangling gate and a statetransfer protocol for the states in the neighborhood of the circular Rydberg level.

M. Meth, V. Kuzmin, Rick van Bijnen, L. Postler, R. Stricker, Rainer Blatt, M. Ringbauer, Thomas Monz, Pietro Silvi, Philipp Schindler Probing phases of quantum matter with an iontrap tensornetwork quantum eigensolver,
Phys. Rev. X 12 41035 (20220328),
http://dx.doi.org/10.1103/PhysRevX.12.041035 doi:10.1103/PhysRevX.12.041035 (ID: 720826)
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TensorNetwork (TN) states are efficient parametric representations of ground states of local quantum Hamiltonians extensively used in numerical simulations. Here we encode a TN ansatz state directly into a quantum simulator, which can potentially offer an exponential advantage over purely numerical simulation. In particular, we demonstrate the optimization of a quantumencoded TN ansatz state using a variational quantum eigensolver on an iontrap quantum computer by preparing the ground states of the extended SuSchriefferHeeger model. The generated states are characterized by estimating the topological invariants, verifying their topological order. Our TN encoding as a trapped ion circuit employs only singlesite addressing optical pulses  the native operations naturally available on the platform. We reduce nearestneighbor crosstalk by selecting different magnetic sublevels with wellseparated transition frequencies to encode even and odd qubits.

C. Dlaska, K. Ender, G. B. Mbeng, A. Kruckenhauser, W. Lechner, R. van Bijnen Quantum optimization via fourbody Rydberg gates,
Phys. Rev. Lett. 128 (20220324),
http://dx.doi.org/10.1103/PhysRevLett.128.120503 doi:10.1103/PhysRevLett.128.120503 (ID: 720749)
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There is a large ongoing research effort towards obtaining a quantum advantage in the solution of combinatorial optimization problems on nearterm quantum devices. A particularly promising platform for testing and developing quantum optimization algorithms are arrays of trapped neutral atoms, lasercoupled to highly excited Rydberg states. However, encoding combinatorial optimization problems in atomic arrays is challenging due to the limited interqubit connectivity given by their native finiterange interactions. Here we propose and analyze a fast, high fidelity fourbody Rydberg parity gate, enabling a direct and straightforward implementation of the LechnerHaukeZoller (LHZ) scheme and its recent generalization, the parity architecture, a scalable architecture for encoding arbitrarily connected interaction graphs. Our gate relies on onetimeoptimized adiabatic laser pulses and is fully programmable by adjusting two holdtimes during operation. We numerically demonstrate an implementation of the quantum approximate optimization algorithm (QAOA) for a small scale test problem. Our approach allows for efficient execution of variational optimization steps with a constant number of system manipulations, independent of the system size, thus paving the way for experimental investigations of QAOA beyond the reach of numerical simulations.

C. Marciniak, T. Feldker, I. Pogorelov, C. R. Kaubrügger, D. Vasilyev, R. van Bijnen, P. Schindler, P. Zoller, R. Blatt, T. Monz Optimal metrology with variational quantum circuits on trapped ions,
Nature 603 604 (20220323),
http://dx.doi.org/10.1038/s41586022044354 doi:10.1038/s41586022044354 (ID: 720667)
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Cold atoms and ions exhibit unparalleled performance in frequency metrology epitomized in the atomic clock. More recently, such atomic systems have been used to implement programmable quantum computers and simulators with highest reported operational fidelities across platforms. Their strength in metrology and quantum information processing offers the opportunity to utilize tailored, programmable entanglement generation to approach the `optimal quantum sensor' compatible with quantum mechanics. Here we report quantum enhancement in metrology beyond squeezing through lowdepth, variational quantum circuits searching for optimal input states and measurement operators in a trapped ion platform. We perform entanglementenhanced Ramsey interferometry finding optimal parameters for variational quantum circuits using a Bayesian approach to stochastic phase estimation tailored to the sensor platform capabilities and finite dynamic range of the interferometer. We verify the performance by both directly using theory predictions of optimal parameters, and performing online quantumclassical feedback optimization to `selfcalibrate' the variational parameters. In both cases we find that variational circuits outperform classical and direct spin squeezing strategies under realistic noise and imperfections. With 26 ions we achieve 2.02(8) dB of metrological gain over classical interferometers.

S. Hollerith, S. Srakaew, D. Wei, A. Rubio López, D. Adler, P. Weckesser, A. Kruckenhauser, V. Walther, R. van Bijnen, J. Rui, C. Gross, I. Bloch, J. Zeiher Realizing distanceselective interactions in a Rydbergdressed atom array,
Phys. Rev. Lett. 128 113602 (20220314),
http://dx.doi.org/10.1103/PhysRevLett.128.113602 doi:10.1103/PhysRevLett.128.113602 (ID: 720748)
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Measurementbased quantum computing relies on the rapid creation of largescale entanglement in a register of stable qubits. Atomic arrays are well suited to store quantum information, and entanglement can be created using highlyexcited Rydberg states. Typically, isolating pairs during gate operation is difficult because Rydberg interactions feature long tails at large distances. Here, we engineer distanceselective interactions that are strongly peaked in distance through offresonant laser coupling of molecular potentials between Rydberg atom pairs. Employing quantum gas microscopy, we verify the dressed interactions by observing correlated phase evolution using manybody Ramsey interferometry. We identify atom loss and coupling to continuum modes as a limitation of our present scheme and outline paths to mitigate these effects, paving the way towards the creation of largescale entanglement.

A. Rath, R. van Bijnen, A. Elben, P. Zoller, B. Vermersch Importance sampling of randomized measurements for probing entanglement,
Phys. Rev. Lett. 127 (20211111),
http://dx.doi.org/10.1103/PhysRevLett.127.200503 doi:10.1103/PhysRevLett.127.200503 (ID: 720632)
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We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of statistical errors is obtained using classical techniques of machinelearning and tensor networks using partial information on the quantum state. In present experimental settings of engineered manybody quantum systems this effectively doubles the (sub)system sizes for which entanglement can be measured. In particular, we show an exponential reduction of the required number of measurements to estimate the purity of product states and GHZ states.

C. Kokail, B. Sundar, T. Zache, A. Elben, B. Vermersch, M. Dalmonte, R. van Bijnen, P. Zoller Quantum Variational Learning of the Entanglement Hamiltonian,
Phys. Rev. Lett. 127 170501 (20211022),
http://dx.doi.org/10.1103/PhysRevLett.127.170501 doi:10.1103/PhysRevLett.127.170501 (ID: 720649)
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Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum manybody states in analog quantum simulation. We describe a protocol where spatial deformations of the manybody Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of FermiHubbard models in quasi1D geometries, finding excellent agreement of the EH with BisognanoWichmann predictions. Subsequent ondevice spectroscopy enables a direct measurement of the entanglement spectrum, which we illustrate for a Fermi Hubbard model in a topological phase.

D. Paulson, L. Dellantonio, J. Haase, A. Celi, A. Kan, A. Jena, C. Kokail, R. van Bijnen, K. Jansen, P. Zoller, C. A. Muschik Simulating 2D effects in lattice gauge theories on a quantum computer,
PRX Quantum 2 30334 (20210825),
http://dx.doi.org/10.1103/PRXQuantum.2.030334 doi:10.1103/PRXQuantum.2.030334 (ID: 720526)
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Quantum computing is in its greatest upswing, with socalled noisyintermediatescalequantum devices heralding the computational power to be expected in the near future. While the field is progressing toward quantum advantage, quantum computers already have the potential to tackle classically intractable problems. Here, we consider gauge theories describing fundamentalparticle interactions. On the way to their fullfledged quantum simulations, the challenge of limited resources on nearterm quantum devices has to be overcome. We propose an experimental quantum simulation scheme to study groundstate properties in twodimensional quantum electrodynamics (2D QED) using existing quantum technology. Our protocols can be adapted to larger lattices and offer the perspective to connect the lattice simulation to lowenergy observable quantities, e.g., the hadron spectrum, in the continuum theory. By including both dynamical matter and a nonminimal gaugefield truncation, we provide the novel opportunity to observe 2D effects on presentday quantum hardware. More specifically, we present two variationalquantumeigensolver (VQE) based protocols for the study of magnetic field effects and for taking an important first step toward computing the running coupling of QED. For both instances, we include variational quantum circuits for qubitbased hardware. We simulate the proposed VQE experiments classically to calculate the required measurement budget under realistic conditions. While this feasibility analysis is done for trapped ions, our approach can be directly adapted to other platforms. The techniques presented here, combined with advancements in quantum hardware, pave the way for reaching beyond the capabilities of classical simulations.

D. Petter, A. Patscheider, G. Natale, M. J. Mark, M. Baranov, R. van Bijnen, S. M. Roccuzzo, A. Recati, B. Blakie, D. Baillie, L. Chomaz, F. Ferlaino Bragg scattering of an ultracold dipolar gas across the phase transition from BoseEinstein condensate to supersolid in the freeparticle regime,
Phys. Rev. A 104 L011302 (20210722),
http://dx.doi.org/10.1103/PhysRevA.104.L011302 doi:10.1103/PhysRevA.104.L011302 (ID: 720484)
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We present an experimental and theoretical study of the highenergy excitation spectra of a dipolar supersolid. Using Bragg spectroscopy, we study the scattering response of the system to a highenergy probe, enabling measurements of the dynamic structure factor. We experimentally observe a continuous reduction of the response when tuning the contact interaction from an ordinary BoseEinstein condensate to a supersolid state. Yet the observed reduction is faster than the one theoretically predicted by the BogoliubovdeGennes theory. Based on an intuitive semianalytic model and realtime simulations, we primarily attribute such a discrepancy to the outofequilibrium phase dynamics, which although not affecting the system global coherence, reduces its response.

C. Kokail, R. van Bijnen, A. Elben, B. Vermersch, P. Zoller Entanglement Hamiltonian Tomography in Quantum Simulation,
Nature Phys. (20210624),
http://dx.doi.org/10.1038/s4156702101260w doi:10.1038/s4156702101260w (ID: 720530)
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Entanglement is the crucial ingredient of quantum manybody physics, and characterizing and quantifying entanglement in closed system dynamics of quantum simulators is an outstanding challenge in today's era of intermediate scale quantum devices. Here we discuss an efficient tomographic protocol for reconstructing reduced density matrices and entanglement spectra for spin systems. The key step is a parametrization of the reduced density matrix in terms of an entanglement Hamiltonian involving only quasi local fewbody terms. This ansatz is fitted to, and can be independently verified from, a small number of randomised measurements. The ansatz is suggested by Conformal Field Theory in quench dynamics, and via the BisognanoWichmann theorem for ground states. Not only does the protocol provide a testbed for these theories in quantum simulators, it is also applicable outside these regimes. We show the validity and efficiency of the protocol for a longrange Ising model in 1D using numerical simulations. Furthermore, by analyzing data from 10 and 20 ion quantum simulators [Brydges \textit{et al.}, Science, 2019], we demonstrate measurement of the evolution of the entanglement spectrum in quench dynamics.

L. R. Picard, M. J. Mark, F. Ferlaino, R. van Bijnen Deep LearningAssisted Classification of SiteResolved Quantum Gas Microscope Images,
Measurement Science and Technology 31 25201 (20191105),
http://dx.doi.org/10.1088/13616501/ab44d8 doi:10.1088/13616501/ab44d8 (ID: 720262)

C. R. Kaubrügger, P. Silvi, C. Kokail, R. van Bijnen, A. M. Rey, J. Ye, A. Kaufman, P. Zoller Variational spinsqueezing algorithms on programmable quantum sensors,
Phys. Rev. Lett. 123 260505 (20190822),
http://dx.doi.org/10.1103/PhysRevLett.123.260505 doi:10.1103/PhysRevLett.123.260505 (ID: 720356)
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Arrays of atoms trapped in optical tweezers combine features of programmable analog quantum simulators with atomic quantum sensors. Here we propose variational quantum algorithms, tailored for tweezer arrays as programmable quantum sensors, capable of generating entangled states ondemand for precision metrology. The scheme is designed to generate metrological enhancement by optimizing it in a feedback loop on the quantum device itself, thus preparing the best entangled states given the available quantum resources. We apply our ideas to generate spinsqueezed states on Sr atom tweezer arrays, where finiterange interactions are generated through Rydberg dressing. The complexity of experimental variational optimization of our quantum circuits is expected to scale favorably with system size. We numerically show our approach to be robust to noise, and surpassing known protocols.

G. Natale, R. van Bijnen, A. Patscheider, D. Petter, M. J. Mark, L. Chomaz, F. Ferlaino Excitation spectrum of a trapped dipolar supersolid and its experimental evidence,
Phys. Rev. Lett. 123 50402 (20190801),
http://dx.doi.org/10.1103/PhysRevLett.123.050402 doi:10.1103/PhysRevLett.123.050402 (ID: 720313)
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We study the spectrum of elementary excitations of a trapped dipolar Bose gas across the superfluidsupersolid phase transition. Our calculations, accounting for the experimentally relevant case of confined systems, show that, when entering the supersolid phase, two distinct excitation branches appear, respectively connected to crystal or superfluid orders. These results confirm infinitesystem predictions, showing that finitesize effects play only a small qualitative role. Experimentally, we probe compressional excitations in an Er quantum gas across the phase diagram. While in the BEC regime the system exhibits an ordinary quadrupole oscillation, in the supersolid regime, we observe a striking twofrequency response of the system, related to the two spontaneously broken symmetries.

C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, P. Zoller Selfverifying variational quantum simulation of lattice models,
Nature 569 360 (20190515),
http://dx.doi.org/10.1038/s4158601911774 doi:10.1038/s4158601911774 (ID: 720076)
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Hybrid classicalquantum algorithms aim at variationally solving optimisation problems, using a feedback loop between a classical computer and a quantum coprocessor, while benefitting from quantum resources. Here we present experiments demonstrating selfverifying, hybrid, variational quantum simulation of lattice models in condensed matter and highenergy physics. Contrary to analog quantum simulation, this approach forgoes the requirement of realising the targeted Hamiltonian directly in the laboratory, thus allowing the study of a wide variety of previously intractable target models. Here, we focus on the Lattice Schwinger model, a gauge theory of 1D quantum electrodynamics. Our quantum coprocessor is a programmable, trappedion analog quantum simulator with up to 20 qubits, capable of generating families of entangled trial states respecting symmetries of the target Hamiltonian. We determine ground states, energy gaps and, by measuring variances of the Schwinger Hamiltonian, we provide algorithmic error bars for energies, thus addressing the longstanding challenge of verifying quantum simulation.

D. Petter, G. Natale, R. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz, F. Ferlaino Probing the roton excitation spectrum of a stable dipolar Bose gas,
Phys. Rev. Lett. 122 183401 (20190508),
http://dx.doi.org/10.1103/PhysRevLett.122.183401 doi:10.1103/PhysRevLett.122.183401 (ID: 720098)
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We measure the excitation spectrum of a stable dipolar BoseEinstein condensate over a wide momentumrange via Bragg spectroscopy. We precisely control the relative strength, εdd, of the dipolar to the contact interactions and observe that the spectrum increasingly deviates from the linear phononic behavior for increasing εdd. Reaching the dipolar dominated regime εdd>1, we observe the emergence of a roton minimum in the spectrum and its softening towards instability. We characterize how the excitation energy and the strength of the densitydensity correlations at the roton momentum vary with εdd. Our findings are in excellent agreement with numerical calculations based on meanfield Bogoliubov theory. When including beyondmeanfield corrections, in the form of a LeeHuangYang potential, we observe a quantitative deviation from the experiment, questioning the validity of such a description in the roton regime.

L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, F. Ferlaino LongLived and Transient Supersolid Behaviors in Dipolar Quantum Gases,
Phys. Rev. X 9 21012 (20190419),
http://dx.doi.org/10.1103/PhysRevX.9.021012 doi:10.1103/PhysRevX.9.021012 (ID: 720203)
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By combining theory and experiments, we demonstrate that dipolar quantum gases of both 166Er and 164Dy support a state with supersolid properties, where a spontaneous density modulation and a global phase coherence coexist. This paradoxical state occurs in a well defined parameter range, separating the phases of a regular BoseEinstein condensate and of an insulating droplet array, and is rooted in the roton mode softening, on the one side, and in the stabilization driven by quantum fluctuations, on the other side. Here, we identify the parameter regime for each of the three phases. In the experiment, we rely on a detailed analysis of the interference patterns resulting from the free expansion of the gas, quantifying both its density modulation and its global phase coherence. Reaching the phases via a slow interaction tuning, starting from a stable condensate, we observe that 166Er and 164Dy exhibit a striking difference in the lifetime of the supersolid properties, due to the different atom loss rates in the two systems. Indeed, while in 166Er the supersolid behavior only survives a few tens of milliseconds, we observe coherent density modulations for more than 150ms in 164Dy. Building on this long lifetime, we demonstrate an alternative path to reach the supersolid regime, relying solely on evaporative cooling starting from a thermal gas.

J. Zeiher, J. Choi, A. Rubio Abadal, T. Pohl, R. van Bijnen, I. Bloch, C. Gross Coherent manybody spin dynamics in a longrange interacting Ising chain,
Phys. Rev. X 7 41063 (20171214),
http://dx.doi.org/10.1103/PhysRevX.7.041063 doi:10.1103/PhysRevX.7.041063 (ID: 719831)
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Coherent manybody quantum dynamics lies at the heart of quantum simulation and quantum computation. Both require coherent evolution in the exponentially large Hilbert space of an interacting manybody system. To date, trapped ions have defined the state of the art in terms of achievable coherence times in interacting spin chains. Here, we establish an alternative platform by reporting on the observation of coherent, fully interactiondriven quantum revivals of the magnetization in Rydbergdressed Ising spin chains of atoms trapped in an optical lattice. We identify partial manybody revivals at up to about ten times the characteristic time scale set by the interactions. At the same time, singlesiteresolved correlation measurements link the magnetization dynamics with interspin correlations appearing at different distances during the evolution. These results mark an enabling step towards the implementation of Rydberg atom based quantum annealers, quantum simulations of higher dimensional complex magnetic Hamiltonians, and itinerant longrange interacting quantum matter.
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J. Cui, R. van Bijnen, T. Pohl, S. Montangero, T. Calarco Optimal control of Rydberg lattice gases,
Quantum Sci. Technol. 2 35006 (20170802),
http://dx.doi.org/10.1088/20589565/aa7daf doi:10.1088/20589565/aa7daf (ID: 719830)
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We present optimal control protocols to prepare different manybody quantum states of Rydberg atoms in optical lattices. Specifically, we show how to prepare highly ordered manybody ground states, GHZ states as well as some superposition of symmetric excitation number Fock states, that inherit the translational symmetry from the Hamiltonian, within sufficiently short excitation times minimizing detrimental decoherence effects. For the GHZ states, we propose a twostep detection protocol to experimentally verify the optimal preparation of the target state based only on standard measurement techniques. Realistic experimental constraints and imperfections are taken into account by our optimization procedure making it applicable to ongoing experiments.
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A. Glätzle, R. van Bijnen, P. Zoller, W. Lechner A Coherent Quantum Annealer with Rydberg Atoms,
Nat. Commun. 8 15813 (20170622),
http://dx.doi.org/10.1038/ncomms15813 doi:10.1038/ncomms15813 (ID: 719686)
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There is a significant ongoing effort in realizing quantum annealing with different physical platforms. The challenge is to achieve a fully programmable quantum device featuring coherent adiabatic quantum dynamics. Here we show that combining the welldeveloped quantum simulation toolbox for Rydberg atoms with the recently proposed LechnerHaukeZoller~(LHZ) architecture allows one to build a prototype for a coherent adiabatic quantum computer with alltoall Ising interactions and, therefore, a novel platform for quantum annealing. In LHZ a infiniterange spinglass is mapped onto the low energy subspace of a spin1/2 lattice gauge model with quasilocal 4body parity constraints. This spin model can be emulated in a natural way with Rubidium and Cesium atoms in a bipartite optical lattice involving laserdressed RydbergRydberg interactions, which are several orders of magnitude larger than the relevant decoherence rates. This makes the exploration of coherent quantum enhanced optimization protocols accessible with stateoftheart atomic physics experiments.
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