
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.

A. Neven, J. Carrasco, V. Vitale, C. Kokail, A. Elben, M. Dalmonte, P. Calabrese, P. Zoller, B. Vermersch, R. Kueng, B. Kraus Symmetryresolved entanglement detection using partial transpose moments,
npj Quantum Information 7 (20211020),
http://dx.doi.org/10.1038/s4153402100487y doi:10.1038/s4153402100487y (ID: 720635)
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We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The kth condition involves comparing moments of the partially transposed density operator up to order k. Remarkably, the union of all moment inequalities reproduces the PeresHorodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. nonidentical, but independent, state copies.

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.

Z. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, M. Hafezi Manybody Chern number from statistical correlations of randomized measurements,
Phys. Rev. Lett. 126 50501 (20210201),
http://dx.doi.org/10.1103/PhysRevLett.126.050501 doi:10.1103/PhysRevLett.126.050501 (ID: 720496)
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One of the main topological invariants that characterizes several topologicallyordered phases is the manybody Chern number (MBCN). Paradigmatic examples include several fractional quantum Hall phases, which are expected to be realized in different atomic and photonic quantum platforms in the near future. Experimental measurement and numerical computation of this invariant is conventionally based on the linearresponse techniques which require having access to a family of states, as a function of an external parameter, which is not suitable for many quantum simulators. Here, we propose an ancillafree experimental scheme for the measurement of this invariant, without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wavefunction. Remarkably, our results apply to disklike geometries that are more amenable to current quantum simulator architectures.

A. Elben, R. Kueng, H. Huang, R. van Bijnen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, B. Vermersch Mixedstate entanglement from local randomized measurements,
Phys. Rev. Lett. 125 (20201111),
http://dx.doi.org/10.1103/PhysRevLett.125.200501 doi:10.1103/PhysRevLett.125.200501 (ID: 720525)
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We propose a method for detecting bipartite entanglement in a manybody mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with singlequbit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al, Science 364, 260 (2019)].

A. Celi, B. Vermersch, O. Viyuela, H. Pichler, M. Lukin, P. Zoller Emerging 2D Gauge theories in Rydberg configurable arrays,
Phys. Rev. X 10 (20200616),
http://dx.doi.org/10.1103/PhysRevX.10.021057 doi:10.1103/PhysRevX.10.021057 (ID: 720355)
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Solving strongly coupled gauge theories in two or three spatial dimensions is of fundamental importance in several areas of physics ranging from highenergy physics to condensed matter. On a lattice, gauge invariance and gauge invariant (plaquette) interactions involve (at least) fourbody interactions that are challenging to realize. Here we show that Rydberg atoms in configurable arrays realized in current tweezer experiments are the natural platform to realize scalable simulators of the RokhsarKivelson Hamiltonian a 2D U(1) lattice gauge theory that describes quantum dimer and spinice dynamics. Using an electromagnetic duality, we implement the plaquette interactions as Rabi oscillations subject to Rydberg blockade. Remarkably, we show that by controlling the atom arrangement in the array we can engineer anisotropic interactions and generalized blockade conditions for spins built of atom pairs.
We describe how to prepare the resonating valence bond and the crystal phases of the RokhsarKivelson Hamiltonian adiabatically, and probe them and their quench dynamics by onsite measurements of their quantum correlations.
We discuss the potential applications of our Rydberg simulator to lattice gauge theory and exotic spin models.

A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, B. Vermersch Manybody topological invariants from randomized measurements,
Sci. Adv. 6 (20200410),
http://dx.doi.org/10.1126/sciadv.aaz3666 doi:10.1126/sciadv.aaz3666 (ID: 720289)
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The classification of symmetryprotected topological (SPT) phases in one dimension has been recently achieved, and had a fundamental impact in our understanding of quantum phases in condensed matter physics. In this framework, SPT phases can be identified by manybody topological invariants, which are quantized nonlocal correlators for the manybody wavefunction. While SPT phases can now be realized in interacting synthethic quantum systems, the direct measurement of quantized manybody topological invariants has remained so far elusive. Here, we propose measurement protocols for manybody topological invariants for all types of protecting symmetries of onedimensional interacting bosonic systems. Our approach relies on randomized measurements implemented with local random unitaries, and can be applied to any spin system with singlesite addressability and readout. Our scheme thus provides a versatile toolbox to experimentally classify interacting SPT phases.

P. Guimond, B. Vermersch, M. L. Juan, A. Sharafiev, G. Kirchmair, P. Zoller A Unidirectional OnChip Photonic Interface for Superconducting Circuits,
npj Quantum Information 6 (20200327),
http://dx.doi.org/10.1038/s4153402002619 doi:10.1038/s4153402002619 (ID: 720478)
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We propose and analyze a passive architecture for realizing onchip, scalable cascaded quantum devices. In contrast to standard approaches, our scheme does not rely on breaking Lorentz reciprocity. Rather, we engineer the interplay between pairs of superconducting transmon qubits and a microwave transmission line, in such a way that two delocalized orthogonal excitations emit (and absorb) photons propagating in opposite directions. We show how such cascaded quantum devices can be exploited to passively probe and measure complex manybody operators on quantum registers of stationary qubits, thus enabling the heralded transfer of quantum states between distant qubits, as well as the generation and manipulation of stabilizer codes for quantum error correction.

M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, C. F. Roos Quantum information scrambling in a trappedion quantum simulator with tunable range interactions,
Phys. Rev. Lett. 124 240505 (20200107),
http://dx.doi.org/10.1103/PhysRevLett.124.240505 doi:10.1103/PhysRevLett.124.240505 (ID: 720436)
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In ergodic manybody quantum systems, locally encoded quantum information becomes, in the course of time evolution, inaccessible to local measurements. This concept of "scrambling" is currently of intense research interest, entailing a deep understanding of manybody dynamics such as the processes of chaos and thermalization. Here, we present first experimental demonstrations of quantum information scrambling on a 10qubit trappedion quantum simulator representing a tunable longrange interacting spin system, by estimating outoftime ordered correlators (OTOCs) through randomized measurements. We also analyze the role of decoherence in our system by comparing our measurements to numerical simulations and by measuring Rényi entanglement entropies.

A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. K. Joshi, R. Blatt, C. F. Roos, P. Zoller CrossPlatform Verification of Intermediate Scale Quantum Devices,
Phys. Rev. Lett. 124 10504 (20200106),
http://dx.doi.org/10.1103/PhysRevLett.124.010504 doi:10.1103/PhysRevLett.124.010504 (ID: 720357)
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We describe a protocol for crossplatform verification of quantum simulators and quantum computers. We show how to measure directly the overlap Tr[ρ1ρ2] and the purities Tr[ρ21,2], and thus a (mixedstate) fidelity, of two quantum states ρ1 and ρ2 prepared in separate experimental platforms. We require only local measurements in randomized product bases, which are communicated classically. As a proofofprinciple, we present the measurement of experimenttheory fidelities for entangled 10qubit quantum states in a trapped ion quantum simulator.

M. Schütz, B. Vermersch, G. Kirchmair, L. Vandersypen, J. I. Cirac, M. Lukin, P. Zoller Quantum Simulation and Optimization in Hot Quantum Networks,
Phys. Rev. B 99 241302 (20190627),
http://dx.doi.org/10.1103/PhysRevB.99.241302 doi:10.1103/PhysRevB.99.241302 (ID: 720058)
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We propose and analyze a setup based on (solidstate) qubits coupled to a common multimode transmission line, which allows for coherent spinspin interactions over macroscopic onchip distances, without any groundstate cooling requirements for the data bus. Our approach allows for the realization of fast deterministic quantum gates between distant qubits, the simulation of quantum spin models with engineered (longrange) interactions, and provides a flexible architecture for the implementation of quantum approximate optimization algorithms.

B. Vermersch, A. Elben, L. Sieberer, N. Y. Yao, P. Zoller Probing scrambling using statistical correlations between randomized measurements,
Phys. Rev. X 9 21061 (20190627),
http://dx.doi.org/10.1103/PhysRevX.9.021061 doi:10.1103/PhysRevX.9.021061 (ID: 720042)
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We present a protocol to study scrambling using statistical correlations between measurements, performed after evolving a quantum system from random initial states. We show that the resulting statistical correlations are directly related to OTOCs and can be used to probe scrambling in manybody systems. Our protocol, which does not require reversing time evolution or auxiliary degrees of freedom, can be realized in stateoftheart quantum simulation experiments.

A. Elben, B. Vermersch, C. F. Roos, P. Zoller Statistical correlations between locally randomized measurements: a toolbox for probing entanglement in manybody quantum states,
Phys. Rev. A 99 52323 (20190515),
http://dx.doi.org/10.1103/PhysRevA.99.052323 doi:10.1103/PhysRevA.99.052323 (ID: 720100)
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We develop a general theoretical framework for measurement protocols employing statistical correlations of randomized measurements. We focus on locally randomized measurements implemented with local random unitaries in quantum lattice models. In particular, we discuss the theoretical details underlying the recent measurement of the second Rényi entropy of highly mixed quantum states consisting of up to 10 qubits in a trappedion quantum simulator [Brydges et al., arXiv:1806.05747]. We generalize the protocol to access the overlap of quantum states, prepared sequentially in an experiment. Furthermore, we discuss proposals for quantum state tomography based on randomized measurements within our framework and the respective scaling of statistical errors with system size.

T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos Probing Renyi entanglement entropy via randomized measurements,
Science 364 260 (20190419),
http://dx.doi.org/10.1126/science.aau4963 doi:10.1126/science.aau4963 (ID: 720034)
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Entanglement is the key feature of manybody quantum systems, and the development of new tools to probe it in the laboratory is an outstanding challenge. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a new protocol for measuring entropy, based on statistical correlations between randomized measurements. Our experiments, carried out with a trappedion quantum simulator, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts  both in the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, applicable to arbitrary quantum states of up to several tens of qubits.

P. Guimond, A. Grankin, D. Vasilyev, B. Vermersch, P. Zoller Subradiant Bell States in Distant Atomic Arrays,
Phys. Rev. Lett. 122 93601 (20190305),
http://dx.doi.org/10.1103/PhysRevLett.122.093601 doi:10.1103/PhysRevLett.122.093601 (ID: 720126)
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We study collective “freespace” radiation properties of two distant singlelayer arrays of quantum emitters as twolevel atoms. We show that this system can support a longlived Bell superposition state of atomic excitations exhibiting strong subradiance, which corresponds to a nonlocal excitation of the two arrays. We describe the preparation of these states and their application in quantum information as a resource of nonlocal entanglement, including deterministic quantum state transfer with high fidelity between the arrays representing quantum memories. We discuss experimental realizations using cold atoms in optical trap arrays with subwavelength spacing, and analyze the role of imperfections.

A. Grankin, D. Vasilyev, P. Guimond, B. Vermersch, P. Zoller Freespace photonic quantum link and chiral quantum optics,
Phys. Rev. A 98 3825 (20181012),
http://dx.doi.org/10.1103/PhysRevA.98.043825 doi:10.1103/PhysRevA.98.043825 (ID: 719980)
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We present the design of a chiral photonic quantum link, where distant atoms interact by exchanging photons propagating in a single direction in free space. This is achieved by coupling each atom in a laserassisted process to an atomic array acting as a quantum phasedarray antenna. This provides a basic building block for quantum networks in free space, i.e., without requiring cavities or nanostructures, which we illustrate with highfidelity quantum state transfer protocols. Our setup can be implemented with neutral atoms using Rydbergdressed interactions.

M. Dalmonte, B. Vermersch, P. Zoller Quantum simulation and spectroscopy of entanglement Hamiltonians,
Nature Phys. 14 151 (20180521),
http://dx.doi.org/10.1038/s4156701801517 doi:10.1038/s4156701801517 (ID: 720028)
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The properties of a strongly correlated manybody quantum system, from the presence of topological order to the onset of quantum criticality, leave a footprint in its entanglement spectrum. The entanglement spectrum is composed by the eigenvalues of the density matrix representing a subsystem of the whole original system, but its direct measurement has remained elusive due to the lack of direct experimental probes. Here we show that the entanglement spectrum of the ground state of a broad class of Hamiltonians becomes directly accessible via the quantum simulation and spectroscopy of a suitably constructed entanglement Hamiltonian, building on the Bisognano–Wichmann theorem of axiomatic quantum field theory. This theorem gives an explicit physical construction of the entanglement Hamiltonian, identified as the Hamiltonian of the manybody system of interest with spatially varying couplings. On this basis, we propose a scalable recipe for the measurement of a system’s entanglement spectrum via spectroscopy of the corresponding Bisognano–Wichmann Hamiltonian realized in synthetic quantum systems, including atoms in optical lattices and trapped ions. We illustrate and benchmark this scenario on a variety of models, spanning phenomena as diverse as conformal field theories, topological order and quantum phase transitions.

A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, P. Zoller Rényi Entropies from Random Quenches in Atomic Hubbard and Spin Models,
Phys. Rev. Lett. 120 50406 (20180202),
http://dx.doi.org/10.1103/PhysRevLett.120.050406 doi:10.1103/PhysRevLett.120.050406 (ID: 719876)
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We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimension. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered timedependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in exisiting AMO quantum simulators, and used to measure for instance area law scaling of entanglement in twodimensional spin models or the entanglement growth in manybody localized systems.
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B. Vermersch, A. Elben, M. Dalmonte, J. I. Cirac, P. Zoller Unitary ndesigns via random quenches in atomic Hubbard and Spin models: Application to the measurement of Rényi entropies,
Phys. Rev. A 97 23604 (20180202),
http://dx.doi.org/10.1103/PhysRevA.97.023604 doi:10.1103/PhysRevA.97.023604 (ID: 719928)
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We present a general framework for the generation of random unitaries based on random quenches in atomic Hubbard and spin models, forming approximate unitary ndesigns, and their application to the measurement of R\'enyi entropies. We generalize our protocol presented in [Elben2017: arXiv:1709.05060, to appear in Phys. Rev. Lett.] to a broad class of atomic and spin lattice models. We further present an indepth numerical and analytical study of experimental imperfections, including the effect of decoherence and statistical errors, and discuss connections of our approach with manybody quantum chaos.
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B. Vogell, B. Vermersch, T. E. Northup, B. P. Lanyon, C. A. Muschik Deterministic quantum state transfer between remote qubits in cavities,
Quantum Sci. Technol. 2 45003 (20170908),
http://dx.doi.org/10.1088/20589565/aa868b doi:10.1088/20589565/aa868b (ID: 719794)
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Performing a faithful transfer of an unknown quantum state is a key challenge for enabling quantum networks. The realization of networks with a small number of quantum links is now actively pursued, which calls for an assessment of different state transfer methods to guide future design decisions. Here, we theoretically investigate quantum state transfer between two distant qubits, each in a cavity, connected by a waveguide, e.g., an optical fiber. We evaluate the achievable state transfer fidelities for two different protocols: standard wave packet shaping and adiabatic passage. The main loss sources are transmission losses in the waveguide and absorption losses in the cavities. While special cases studied in the literature indicate that adiabatic passages may be beneficial in this context, it remained an open question under which conditions this is the case and whether their use will be advantageous in practice. We answer these questions by providing a full analysis, showing that state transfer by adiabatic passage  in contrast to wave packet shaping  can mitigate the effects of undesired cavity losses, far beyond the regime of coupling to a single waveguide mode and the regime of lossless waveguides, as was proposed so far. We also clarify that neither protocol can avoid losses in the waveguide and discuss how the cavity parameters should be chosen.
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A. Mazloom Shahraki, B. Vermersch, M. Baranov, M. Dalmonte Adiabatic state preparation of stripe phases with strongly magnetic atoms,
Phys. Rev. A 96 33602 (20170901),
http://dx.doi.org/10.1103/PhysRevA.96.033602 doi:10.1103/PhysRevA.96.033602 (ID: 719755)
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We propose a protocol for realizing the stripe phase in two spin models on a twodimensional square lattice, which can be implemented with strongly magnetic atoms (Cr, Dy, Er, etc.) in optical lattices by encoding spin states into Zeeman sublevels of the ground state manifold. The protocol is tested with clustermeanfield timedependent variational ans\"atze, validated by comparison with exact results for small systems, which enable us to simulate the dynamics of systems with up to 64 sites during the statepreparation protocol. This allows, in particular, to estimate the time required for preparation of the stripe phase with high fidelity under real experimental conditions.
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B. Vermersch, P. Guimond, H. Pichler, P. Zoller Quantum State Transfer via Noisy Photonic and Phononic Waveguides,
Phys. Rev. Lett. 118 133601 (20170327),
http://dx.doi.org/10.1103/PhysRevLett.118.133601 doi:10.1103/PhysRevLett.118.133601 (ID: 719696)
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We describe a quantum state transfer protocol, where a quantum state of photons stored in a first cavity can be faithfully transferred to a second distant cavity via an infinite 1D waveguide, while being immune to arbitrary noise (e.g. thermal noise) injected into the waveguide. We extend the model and protocol to a cavity QED setup, where atomic ensembles, or single atoms representing quantum memory, are coupled to a cavity mode. We present a detailed study of sensitivity to imperfections, and develop a quantum error correction protocol to account for random losses (or additions) of photons in the waveguide. Our numerical analysis is enabled by Matrix Product State techniques to simulate the complete quantum circuit, which we generalize to include thermal input fields. Our discussion applies both to photonic and phononic quantum networks.
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C. Dlaska, B. Vermersch, P. Zoller Robust quantum state transfer via topologically protected edge channels in dipolar arrays,
Quantum Sci. Technol. 2 15001 (20170105),
http://dx.doi.org/10.1088/20589565/2/1/015001 doi:10.1088/20589565/2/1/015001 (ID: 719597)
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We show how to realize quantum state transfer between distant qubits using the chiral edge states of a twodimensional topological spin system. Our implementation based on Rydberg atoms allows to realize the quantum state transfer protocol in state of the art experimental setups. In particular, we show how to adapt the standard state transfer protocol to make it robust against dispersive and disorder effects.
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