
D. S. Wild, D. Sels, H. Pichler, C. Zanoci, M. Lukin Quantum Sampling Algorithms for NearTerm Devices,
Phys. Rev. Lett. 127 100504 (20210902),
http://dx.doi.org/10.1103/PhysRevLett.127.100504 doi:10.1103/PhysRevLett.127.100504 (ID: 720685)
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Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples, including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on nearterm quantum devices, as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.

D. S. Wild, D. Sels, H. Pichler, C. Zanoci, M. Lukin Quantum sampling algorithms, phase transitions, and computational complexity,
Phys. Rev. A 104 32602 (20210902),
http://dx.doi.org/10.1103/PhysRevA.104.032602 doi:10.1103/PhysRevA.104.032602 (ID: 720684)
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Drawing independent samples from a probability distribution is an important computational problem with applications in Monte Carlo algorithms, machine learning, and statistical physics. The problem can be solved in principle on a quantum computer by preparing a quantum state that encodes the entire probability distribution followed by a projective measurement. We investigate the complexity of adiabatically preparing such quantum states for the Gibbs distributions of various classical models including the Ising chain, hardsphere models on different graphs, and a model encoding the unstructured search problem. By constructing a parent Hamiltonian, whose ground state is the desired quantum state, we relate the asymptotic scaling of the state preparation time to the nature of transitions between distinct quantum phases. These insights enable us to identify adiabatic paths that achieve a quantum speedup over classical Markov chain algorithms. In addition, we show that parent Hamiltonians for the problem of sampling from independent sets on certain graphs can be naturally realized with neutral atoms interacting via highly excited Rydberg states.

S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletic, M. Lukin Quantum phases of matter on a 256atom programmable quantum simulator,
Nature 595 232 (20210707),
http://dx.doi.org/10.1038/s41586021035824 doi:10.1038/s41586021035824 (ID: 720676)
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Motivated by farreaching applications ranging from quantum simulations of complex processes in physics and chemistry to quantum information processing1, a broad effort is currently underway to build largescale programmable quantum systems. Such systems provide insights into strongly correlated quantum matter2,3,4,5,6, while at the same time enabling new methods for computation7,8,9,10 and metrology11. Here we demonstrate a programmable quantum simulator based on deterministically prepared twodimensional arrays of neutral atoms, featuring strong interactions controlled by coherent atomic excitation into Rydberg states12. Using this approach, we realize a quantum spin model with tunable interactions for system sizes ranging from 64 to 256 qubits. We benchmark the system by characterizing highfidelity antiferromagnetically ordered states and demonstrating quantum critical dynamics consistent with an Ising quantum phase transition in (2 + 1) dimensions13. We then create and study several new quantum phases that arise from the interplay between interactions and coherent laser excitation14, experimentally map the phase diagram and investigate the role of quantum fluctuations. Offering a new lens into the study of complex quantum matter, these observations pave the way for investigations of exotic quantum phases, nonequilibrium entanglement dynamics and hardwareefficient realization of quantum algorithms.

R. Samajdar, W. Ho, H. Pichler, M. Lukin, S. Sachdev Quantum phases of Rydberg atoms on a kagome lattice,
Proc. Natl. Acad. U.S.A. 118 e2015785118 (20210126),
http://dx.doi.org/10.1073/pnas.2015785118 doi:10.1073/pnas.2015785118 (ID: 720622)
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Programmable quantum simulators based on Rydberg atom arrays have recently emerged as versatile platforms for exploring exotic manybody phases and quantum dynamics of strongly correlated systems. In this work, we theoretically investigate the quantum phases that can be realized by arranging such Rydberg atoms on a kagome lattice. Along with an extensive analysis of the states which break lattice symmetries due to classical correlations, we identify an intriguing regime that constitutes a promising candidate for hosting a phase with longrange quantum entanglement and topological order. Our results provide a route to experimentally realizing and probing highly entangled quantum matter.

J. Perczel, J. Borregaard, D. Chang, H. Pichler, S. F. Yelin, P. Zoller, M. Lukin Photonic band structure of twodimensional atomic lattices,
Phys. Rev. A 96 63801 (20171204),
http://dx.doi.org/10.1103/PhysRevA.96.063801 doi:10.1103/PhysRevA.96.063801 (ID: 719916)
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Twodimensional atomic arrays exhibit a number of intriguing quantum optical phenomena, including subradiance, nearly perfect reflection of radiation, and longlived topological edge states. Studies of emission and scattering of photons in such lattices require complete treatment of the radiation pattern from individual atoms, including longrange interactions. We describe a systematic approach to perform the calculations of collective energy shifts and decay rates in the presence of such longrange interactions for arbitrary twodimensional atomic lattices. As applications of our method, we investigate the topological properties of atomic lattices both in free space and near plasmonic surfaces.
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H. Pichler, S. Choi, P. Zoller, M. Lukin Universal photonic quantum computation via timedelayed feedback,
PNAS 114 (20171017),
http://dx.doi.org/10.1073/pnas.1711003114 doi:10.1073/pnas.1711003114 (ID: 720008)
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We propose and analyze a deterministic protocol to generate twodimensional photonic cluster states using a single quantum emitter via timedelayed quantum feedback. As a physical implementation, we consider a single atom or atomlike system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits, while the mirror allows the implementation of feedback. We identify the class of manybody quantum states that can be produced using this approach and characterize them in terms of 2D tensor network states.

H. Pichler, S. Choi, P. Zoller, M. Lukin Photonic tensor networks produced by a single quantum emitter,
PNAS 114 11362 (20171010),
http://dx.doi.org/10.1073/pnas.1711003114 doi:10.1073/pnas.1711003114 (ID: 719751)
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We propose and analyze a protocol to generate two dimensional tensor network states using a single quantum system that sequentially interacts with a 1D string of qubits. This is accomplished by using parts of the string itself as a quantum queue memory. As a physical implementation, we consider a single atom or atom like system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits while the mirror allows the implementation of the queue memory. We identify the class of manybody quantum states that can be produced using this approach. These include universal resources for measurement based quantum computation and states associated with topologically ordered phases. We discuss an explicit protocol to deterministically create a 2D cluster state in a quantum nanophotonic experiment, that allows for a realization of a quantum computer using a single atom coupled to light.
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A. Glätzle, K. Ender, D. S. Wild, S. Choi, H. Pichler, M. Lukin, P. Zoller Quantum Spin Lenses in Atomic Arrays,
Phys. Rev. X 7 31049 (20170920),
http://dx.doi.org/10.1103/PhysRevX.7.031049 doi:10.1103/PhysRevX.7.031049 (ID: 719790)
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We propose and discuss `quantum spin lenses', where quantum states of delocalized spin excitations in an atomic medium are `focused' in space in a coherent quantum process down to (essentially) single atoms. These can be employed to create controlled interactions in a quantum lightmatter interface, where photonic qubits stored in an atomic ensemble are mapped to a quantum register represented by single atoms. We propose Hamiltonians for quantum spin lenses as inhomogeneous spin models on lattices, which can be realized with Rydberg atoms in 1D, 2D and 3D, and with strings of trapped ions. We discuss both linear and nonlinear quantum spin lenses: in a nonlinear lens, repulsive spinspin interactions lead to focusing dynamics conditional to the number of spin excitations. This allows the mapping of quantum superpositions of delocalized spin excitations to superpositions of spatial spin patterns, which can be addressed by light fields and manipulated. Finally, we propose multifocal quantum spin lenses as a way to generate and distribute entanglement between distant atoms in an atomic lattice array.
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P. Guimond, M. Pletyukhov, H. Pichler, P. Zoller Delayed Coherent Quantum Feedback from a Scattering Theory and a Matrix Product State Perspective,
Quantum Sci. Technol. 2 44012 (20170908),
http://dx.doi.org/10.1088/20589565/aa7f03 doi:10.1088/20589565/aa7f03 (ID: 719814)
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We study the scattering of photons propagating in a semiinfinite waveguide terminated by a mirror and interacting with a quantum emitter. This paradigm constitutes an example of coherent quantum feedback, where light emitted towards the mirror gets redirected back to the emitter. We derive an analytical solution for the scattering of twophoton states, which is based on an exact resummation of the perturbative expansion of the scattering matrix, in a regime where the time delay of the coherent feedback is comparable to the timescale of the quantum emitter's dynamics. We compare the results with numerical simulations based on matrix product state techniques simulating the full dynamics of the system, and extend the study to the scattering of coherent states beyond the lowpower limit.
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J. Perczel, J. Borregaard, D. Chang, H. Pichler, S. F. Yelin, P. Zoller, M. Lukin Topological Quantum Optics in TwoDimensional Atomic Arrays,
Phys. Rev. Lett. 119 23603 (20170714),
http://dx.doi.org/10.1103/PhysRevLett.119.023603 doi:10.1103/PhysRevLett.119.023603 (ID: 719767)
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We demonstrate that twodimensional atomic arrays with subwavelength spacing can be used to create topologically protected quantum optical systems where the photon propagation is robust against large imperfections while losses associated with free space emission are strongly suppressed. Breaking timereversal symmetry with a magnetic field results in gapped photonic bands with nontrivial Chern numbers. Such a system displays topologically protected bound states and unidirectional emission by individual atoms into longlived edge states. Possible experimental realizations and applications are discussed.
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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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P. Lodahl, S. Mahmoodian, S. Stobbe, P. Schneeweiss, J. Volz, A. Rauschenbeutel, H. Pichler, P. Zoller Chiral Quantum Optics,
Nature 541 473 (20170126),
http://dx.doi.org/10.1038/nature21037 doi:10.1038/nature21037 (ID: 719634)

M. Łącki, M. Baranov, H. Pichler, P. Zoller NanoScale `Dark State' Optical Potentials for Cold Atoms,
Phys. Rev. Lett. 117 233001 (20161130),
http://dx.doi.org/10.1103/PhysRevLett.117.233001 doi:10.1103/PhysRevLett.117.233001 (ID: 719619)
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We discuss generation of subwavelength optical barriers on the scale of tens of nanometers, as conservative optical potentials for cold atoms. These arise from nonadiabatic corrections to BornOppenheimer potentials from dressed `dark states' in atomic Λ configurations. We illustrate the concepts with a double layer potential for atoms obtained from inserting an optical subwavelength barrier into a well generated by an offresonant optical lattice, and discuss bound states of pairs of atoms interacting via magnetic dipolar interactions. The subwavelength optical barriers represent an optical `KronigPenney' potential. We present a detailed study of the bandstructure in optical `KronigPenney' potentials, including decoherence from spontaneous emission and atom loss to open `bright' channels.

P. Guimond, H. Pichler, A. Rauschenbeutel, P. Zoller Chiral quantum optics with Vlevel atoms and coherent quantum feedback,
Phys. Rev. A 94 33829 (20160916),
http://dx.doi.org/10.1103/PhysRevA.94.033829 doi:10.1103/PhysRevA.94.033829 (ID: 719590)
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We study the dissipative dynamics of an atom in a Vlevel configuration driven by lasers and coupled to a semiinfinite waveguide. The coupling to the waveguide is chiral, in that each transition interacts only with the modes propagating in a given direction, and this direction is opposite for the two transitions. The waveguide is terminated by a mirror which coherently feeds the photon stream emitted by one transition back to the atom. First, we are interested in the dynamics of the atom in the Markovian limit where the timedelay in the feedback is negligible. Specifically, we study the conditions under which the atom evolves towards a pure "dark" stationary state, where the photons emitted by both transitions interfere destructively thanks to the coherent feedback, and the overall emission vanishes. This is a singleatom analogue of the quantum dimer, where a pair of laserdriven twolevel atoms is coupled to a unidirectional waveguide and dissipates towards a pure entangled dark state. Our setup should be feasible with current stateoftheart experiments. Second, we extend our study to nonMarkovian regimes and investigate the effect of the feedback retardation on the steadystate.
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T. Ramos, B. Vermersch, P. Hauke, H. Pichler, P. Zoller NonMarkovian Dynamics in Chiral Quantum Networks with Spins and Photons,
Phys. Rev. A 93 62104 (20160602),
http://dx.doi.org/10.1103/PhysRevA.93.062104 doi:10.1103/PhysRevA.93.062104 (ID: 719497)
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We study the dynamics of chiral quantum networks consisting of nodes coupled by unidirectional or asymmetric bidirectional quantum channels. In contrast to the familiar photonic networks consisting of driven twolevel atoms exchanging photons via 1D photonic nanostructures, we propose and study a setup where interactions between the atoms are mediated by spin excitations (magnons) in 1D XXspin chains representing a spin waveguide. While Markovian quantum network theory eliminates quantum channels as structureless reservoirs in a BornMarkov approximation to obtain a master equation for the nodes, we are interested in nonMarkovian dynamics. This arises from the nonlinear character of the dispersion with bandedge effects, and from finite spin propagation velocities leading to time delays in interactions. To account for the nonMarkovian dynamics we treat the quantum degrees of freedom of the nodes and connecting channel as a composite spin system with the surrounding of the quantum network as a Markovian bath, allowing for an efficient solution with timedependent density matrix renormalization group techniques. We illustrate our approach showing nonMarkovian effects in the drivendissipative formation of quantum dimers, and we present examples for quantum information protocols involving quantum state transfer with engineered elements as basic building blocks of quantum spintronic circuits.
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H. Pichler, G. Zhu, A. Seif, P. Zoller, M. Hafezi A Measurement Protocol for the Entanglement Spectrum of Cold Atoms,
Phys. Rev. X 6 41033 (20160517),
http://dx.doi.org/10.1103/PhysRevX.6.041033 doi:10.1103/PhysRevX.6.041033 (ID: 719569)
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Entanglement plays a major role in characterizing manybody quantum systems. In particular, the entanglement spectrum holds a great promise to characterize essential physics of quantum manybody systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to this date, due to the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of manybody states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian, and is based on the ability to produce several copies of the state under investigation together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) BoseHubbard model where such a measurement protocol reveals topological features of the Haldane phase.
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H. Pichler, P. Zoller Photonic Quantum Circuits with Time Delays,
Phys. Rev. Lett. 116 93601 (20160303),
http://dx.doi.org/10.1103/PhysRevLett.116.093601 doi:10.1103/PhysRevLett.116.093601 (ID: 719362)
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We study the dynamics of photonic quantum circuits consisting of nodes coupled by quantum channels. We are interested in the regime where the time delay in communication between the nodes is significant. This includes the problem of quantum feedback, where a quantum signal is fed back on a system with a time delay. We develop a matrix product state approach to solve the quantum stochastic Schrödinger equation with time delays, which accounts in an efficient way for the entanglement of nodes with the stream of emitted photons in the waveguide, and thus the nonMarkovian character of the dynamics. We illustrate this approach with two paradigmatic quantum optical examples: two coherently driven distant atoms coupled to a photonic waveguide with a time delay, and a driven atom coupled to its own output field with a time delay as an instance of a quantum feedback problem.
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M. Łącki, H. Pichler, A. Sterdyniak, A. Lyras, V. Lembessis, O. AlDossary, J. Budich, P. Zoller Quantum Hall Physics with Cold Atoms in Cylindrical Optical Lattices,
Phys. Rev. A 93 13604 (20160107),
http://dx.doi.org/10.1103/PhysRevA.93.013604 doi:10.1103/PhysRevA.93.013604 (ID: 719282)
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We propose and study various realizations of a HofstadterHubbard model on a cylinder geometry with fermionic cold atoms in optical lattices. The cylindrical optical lattice is created by counterpropagating LaguerreGauss beams, i.e. light beams carrying orbital angular momentum. By strong focusing of the light beams we create a real space optical lattice in the form of rings, which are offset in energy. A second set of LaguerreGauss beams then induces a Ramanhopping between these rings, imprinting phases corresponding to a synthetic magnetic field (artificial gauge field). In addition, by rotating the lattice potential, we achieve a slowly varying flux through the hole of the cylinder, which allows us to probe the Hall response of the system as a realization of Laughlin's thought experiment. We study how in the presence of interactions fractional quantum Hall physics could be observed in this setup.
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