The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1/rα. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al. [arXiv:1801.03922]. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α>3D (where D is the number of dimensions).

1 aTran, Minh, Cong1 aGuo, Andrew, Y.1 aSu, Yuan1 aGarrison, James, R.1 aEldredge, Zachary1 aFoss-Feig, Michael1 aChilds, Andrew, M.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1808.0522502311nas a2200145 4500008004100000245005700041210005600098260001500154520186200169100002202031700002502053700002302078700002702101856003702128 2019 eng d00aNon-equilibrium fixed points of coupled Ising models0 aNonequilibrium fixed points of coupled Ising models c03/06/20193 aDriven-dissipative systems can exhibit non-equilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions present in these systems generically exhibit an effectively classical equilibrium behavior in spite of their quantum non-equilibrium origin. In this paper, we show that multicritical points in driven-dissipative systems can give rise to genuinely non-equilibrium behavior. We investigate a non-equilibrium driven-dissipative model of interacting bosons that exhibits two distinct phase transitions: one from a high- to a low-density phase---reminiscent of a liquid-gas transition---and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) Z2 symmetry. They, however, coalesce at a multicritical point giving rise to a non-equilibrium model of coupled Ising-like order parameters described by a Z2×Z2 symmetry. Using a dynamical renormalization-group approach, we show that a pair of non-equilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents, spiraling phase boundaries, and a complex Liouvillian gap even close to the phase transition. As direct evidence of the non-equilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths. Finally, we argue that this non-equilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.

1 aYoung, Jeremy, T.1 aGorshkov, Alexey, V.1 aFoss-Feig, Michael1 aMaghrebi, Mohammad, F. uhttps://arxiv.org/abs/1903.0256901425nas a2200157 4500008004100000245007800041210006900119260001500188520092400203100001601127700001701143700002201160700002501182700002301207856003701230 2018 eng d00aDistributed Quantum Metrology and the Entangling Power of Linear Networks0 aDistributed Quantum Metrology and the Entangling Power of Linear c2018/07/253 aWe derive a bound on the ability of a linear optical network to estimate a linear combination of independent phase shifts by using an arbitrary non-classical but unentangled input state, thereby elucidating the quantum resources required to obtain the Heisenberg limit with a multi-port interferometer. Our bound reveals that while linear networks can generate highly entangled states, they cannot effectively combine quantum resources that are well distributed across multiple modes for the purposes of metrology: in this sense linear networks endowed with well-distributed quantum resources behave classically. Conversely, our bound shows that linear networks can achieve the Heisenberg limit for distributed metrology when the input photons are hoarded in a small number of input modes, and we present an explicit scheme for doing so. Our results also have implications for measures of non-classicality.

1 aGe, Wenchao1 aJacobs, Kurt1 aEldredge, Zachary1 aGorshkov, Alexey, V.1 aFoss-Feig, Michael uhttps://arxiv.org/abs/1707.0665501538nas a2200157 4500008004100000245007800041210006900119260001500188520103700203100001601240700001701256700002201273700002501295700002301320856003701343 2018 eng d00aDistributed Quantum Metrology and the Entangling Power of Linear Networks0 aDistributed Quantum Metrology and the Entangling Power of Linear c2018/07/253 aWe derive a bound on the ability of a linear optical network to estimate a linear combination of independent phase shifts by using an arbitrary non-classical but unentangled input state, thereby elucidating the quantum resources required to obtain the Heisenberg limit with a multi-port interferometer. Our bound reveals that while linear networks can generate highly entangled states, they cannot effectively combine quantum resources that are well distributed across multiple modes for the purposes of metrology: in this sense linear networks endowed with well-distributed quantum resources behave classically. Conversely, our bound shows that linear networks can achieve the Heisenberg limit for distributed metrology when the input photons are hoarded in a small number of input modes, and we present an explicit scheme for doing so. Our results also have implications for measures of non-classicality.

1 aGe, Wenchao1 aJacobs, Kurt1 aEldredge, Zachary1 aGorshkov, Alexey, V.1 aFoss-Feig, Michael uhttps://arxiv.org/abs/1707.0665512409nas a2200169 45000080041000002450055000412100055000963000047001514900008001985201188600206100002312092700002012115700001912135700002312154700002512177856003712202 2018 eng d00aDynamical phase transitions in sampling complexity0 aDynamical phase transitions in sampling complexity a12 pages, 4 figures. v3: published version0 v1213 aWe make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time

We make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time

Studies of quantum metrology have shown that the use of many-body entangled states can lead to an enhancement in sensitivity when compared to product states. In this paper, we quantify the metrological advantage of entanglement in a setting where the quantity to be measured is a linear function of parameters coupled to each qubit individually. We first generalize the Heisenberg limit to the measurement of non-local observables in a quantum network, deriving a bound based on the multi-parameter quantum Fisher information. We then propose a protocol that can make use of GHZ states or spin-squeezed states, and show that in the case of GHZ states the procedure is optimal, i.e., it saturates our bound.

1 aEldredge, Zachary1 aFoss-Feig, Michael1 aRolston, Steven, L.1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1607.0464605805nas a2200145 4500008004100000245004900041210004900090260001500139520537700154100002305531700002005554700002305574700002505597856003705622 2017 eng d00aComplexity of sampling as an order parameter0 aComplexity of sampling as an order parameter c2017/03/153 aWe consider the classical complexity of approximately simulating time evolution under spatially local quadratic bosonic Hamiltonians for time

We prove that the entanglement entropy of any state evolved under an arbitrary 1/rα long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α > D + 1. We also prove that for any α > 2D + 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions.

1 aGong, Zhe-Xuan1 aFoss-Feig, Michael1 aBrandão, Fernando, G. S. L.1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1702.0536802145nas a2200205 4500008004100000245005800041210005700099260001500156300001100171490000700182520152100189100002301710700002101733700002201754700002101776700002501797700002101822700002701843856006901870 2017 eng d00aEmergent equilibrium in many-body optical bistability0 aEmergent equilibrium in manybody optical bistability c2017/04/17 a0438260 v953 aMany-body systems constructed of quantum-optical building blocks can now be realized in experimental platforms ranging from exciton-polariton fluids to ultracold gases of Rydberg atoms, establishing a fascinating interface between traditional many-body physics and the driven-dissipative, non-equilibrium setting of cavity-QED. At this interface, the standard techniques and intuitions of both fields are called into question, obscuring issues as fundamental as the role of fluctuations, dimensionality, and symmetry on the nature of collective behavior and phase transitions. Here, we study the driven-dissipative Bose-Hubbard model, a minimal description of numerous atomic, optical, and solid-state systems in which particle loss is countered by coherent driving. Despite being a lattice version of optical bistability---a foundational and patently non-equilibrium model of cavity-QED---the steady state possesses an emergent equilibrium description in terms of a classical Ising model. We establish this picture by identifying a limit in which the quantum dynamics is asymptotically equivalent to non-equilibrium Langevin equations, which support a phase transition described by model A of the Hohenberg-Halperin classification. Numerical simulations of the Langevin equations corroborate this picture, producing results consistent with the behavior of a finite-temperature Ising model.

1 aFoss-Feig, Michael1 aNiroula, Pradeep1 aYoung, Jeremy, T.1 aHafezi, Mohammad1 aGorshkov, Alexey, V.1 aWilson, Ryan, M.1 aMaghrebi, Mohammad, F. uhttps://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.04382602985nas a2200157 4500008004100000245004900041210004900090260001500139300001100154490000700165520255000172100002002722700002302742700002502765856003702790 2017 eng d00aExact sampling hardness of Ising spin models0 aExact sampling hardness of Ising spin models c2017/09/14 a0323240 v963 aWe study the complexity of classically sampling from the output distribution of an Ising spin model, which can be implemented naturally in a variety of atomic, molecular, and optical systems. In particular, we construct a specific example of an Ising Hamiltonian that, after time evolution starting from a trivial initial state, produces a particular output configuration with probability very nearly proportional to the square of the permanent of a matrix with arbitrary integer entries. In a similar spirit to boson sampling, the ability to sample classically from the probability distribution induced by time evolution under this Hamiltonian would imply unlikely complexity theoretic consequences, suggesting that the dynamics of such a spin model cannot be efficiently simulated with a classical computer. Physical Ising spin systems capable of achieving problem-size instances (i.e., qubit numbers) large enough so that classical sampling of the output distribution is classically difficult in practice may be achievable in the near future. Unlike boson sampling, our current results only imply hardness of *exact* classical sampling, leaving open the important question of whether a much stronger approximate-sampling hardness result holds in this context. The latter is most likely necessary to enable a convincing experimental demonstration of quantum supremacy. As referenced in a recent paper [A. Bouland, L. Mancinska, and X. Zhang, in *Proceedings of the 31st Conference on Computational Complexity (CCC 2016)*, Leibniz International Proceedings in Informatics (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, 2016)], our result completes the sampling hardness classification of two-qubit commuting Hamiltonians.

In short-range interacting systems, the speed at which entanglement can be established between two separated points is limited by a constant Lieb-Robinson velocity. Long-range interacting systems are capable of faster entanglement generation, but the degree of the speed-up possible is an open question. In this paper, we present a protocol capable of transferring a quantum state across a distance

Exactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. And conversely, the relative scarcity of solutions for non-equilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of nonequilibrium models, some of which can be viewed as dissipative analogues of the transverse-field Ising model, in that an effectively classical Hamiltonian is frustrated by dissipative processes that drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad and experimentally relevant subset of these models can be solved efficiently in any number of spatial dimensions. We leverage these solutions to prove a no-go theorem on steady-state phase transitions in a many-body model that can be realized naturally with Rydberg atoms or trapped ions, and to compute the effects of decoherence on a canonical trapped-ion-based quantum computation architecture.

1 aFoss-Feig, Michael1 aYoung, Jeremy, T.1 aAlbert, Victor, V.1 aGorshkov, Alexey, V.1 aMaghrebi, Mohammad, F. uhttps://arxiv.org/abs/1703.0462600514nas a2200157 4500008004100000245006700041210006600108260001500174300001100189490000700200100002700207700001900234700002300253700002500276856005500301 2016 eng d00aCausality and quantum criticality in long-range lattice models0 aCausality and quantum criticality in longrange lattice models c2016/03/17 a1251280 v931 aMaghrebi, Mohammad, F.1 aGong, Zhe-Xuan1 aFoss-Feig, Michael1 aGorshkov, Alexey, V. uhttp://link.aps.org/doi/10.1103/PhysRevB.93.12512801582nas a2200169 4500008004100000245006700041210006600108260001500174300001100189490000700200520107500207100002701282700001901309700002301328700002501351856003601376 2016 eng d00aCausality and quantum criticality with long-range interactions0 aCausality and quantum criticality with longrange interactions c2016/03/17 a1251280 v923 a Quantum lattice systems with long-range interactions often exhibit drastically different behavior than their short-range counterparts. In particular, because they do not satisfy the conditions for the Lieb-Robinson theorem, they need not have an emergent relativistic structure in the form of a light cone. Adopting a field-theoretic approach, we study the one-dimensional transverse-field Ising model and a fermionic model with long-range interactions, explore their critical and near-critical behavior, and characterize their response to local perturbations. We deduce the dynamic critical exponent, up to the two-loop order within the renormalization group theory, which we then use to characterize the emergent causal behavior. We show that beyond a critical value of the power-law exponent of long-range interactions, the dynamics effectively becomes relativistic. Various other critical exponents describing correlations in the ground state, as well as deviations from a linear causal cone, are deduced for a wide range of the power-law exponent. 1 aMaghrebi, Mohammad, F.1 aGong, Zhe-Xuan1 aFoss-Feig, Michael1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1508.0090601682nas a2200193 4500008004100000245006100041210006100102260001500163300001100178490000700189520113200196100002101328700002101349700001301370700002501383700002101408700002301429856003601452 2016 eng d00aCollective phases of strongly interacting cavity photons0 aCollective phases of strongly interacting cavity photons c2016/09/01 a0338010 v943 aWe study a coupled array of coherently driven photonic cavities, which maps onto a driven-dissipative XY spin-12 model with ferromagnetic couplings in the limit of strong optical nonlinearities. Using a site-decoupled mean-field approximation, we identify steady state phases with canted antiferromagnetic order, in addition to limit cycle phases, where oscillatory dynamics persist indefinitely. We also identify collective bistable phases, where the system supports two steady states among spatially uniform, antiferromagnetic, and limit cycle phases. We compare these mean-field results to exact quantum trajectories simulations for finite one-dimensional arrays. The exact results exhibit short-range antiferromagnetic order for parameters that have significant overlap with the mean-field phase diagram. In the mean-field bistable regime, the exact quantum dynamics exhibits real-time collective switching between macroscopically distinguishable states. We present a clear physical picture for this dynamics, and establish a simple relationship between the switching times and properties of the quantum Liouvillian.

1 aWilson, Ryan, M.1 aMahmud, Khan, W.1 aHu, Anzi1 aGorshkov, Alexey, V.1 aHafezi, Mohammad1 aFoss-Feig, Michael uhttp://arxiv.org/abs/1601.0685701507nas a2200145 4500008004100000245007200041210006900113260001500182520103700197100002301234700001901257700002501276700002301301856003701324 2016 eng d00aEntanglement and spin-squeezing without infinite-range interactions0 aEntanglement and spinsqueezing without infiniterange interaction c2016/12/223 aInfinite-range interactions are known to facilitate the production of highly entangled states with applications in quantum information and metrology. However, many experimental systems have interactions that decay with distance, and the achievable benefits in this context are much less clear. Combining recent exact solutions with a controlled expansion in the system size, we analyze quench dynamics in Ising models with power-law (1/r α ) interactions in D dimensions, thereby expanding the understanding of spin squeezing into a broad and experimentally relevant context. In spatially homogeneous systems, we show that for small α the scaling of squeezing with system size is identical to the infinite-range (α = 0) case. This indifference to the interaction range persists up to a critical value α = 2D/3, above which squeezing degrades continuously. Boundaryinduced inhomogeneities present in most experimental systems modify this picture, but it nevertheless remains qualitatively correct for finite-sized systems.

1 aFoss-Feig, Michael1 aGong, Zhe-Xuan1 aGorshkov, Alexey, V.1 aClark, Charles, W. uhttps://arxiv.org/abs/1612.0780501814nas a2200205 4500008004100000245007600041210006900117260001500186300001100201490000700212520120100219100001901420700002701439700001301466700002301479700002101502700002401523700002501547856003601572 2016 eng d00aKaleidoscope of quantum phases in a long-range interacting spin-1 chain0 aKaleidoscope of quantum phases in a longrange interacting spin1 c2016/05/11 a2051150 v933 aMotivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as 1/rα, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for α≲3, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a U(1) continuous symmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains. We show that the main signatures of all five phases found could be observed experimentally in the near future. 1 aGong, Zhe-Xuan1 aMaghrebi, Mohammad, F.1 aHu, Anzi1 aFoss-Feig, Michael1 aRicherme, Philip1 aMonroe, Christopher1 aGorshkov, Alexey, V. uhttp://arxiv.org/abs/1510.0210809346nas a2200181 4500008004100000245005500041210005400096260001500150520881500165100001908980700001608999700002309015700002409038700002009062700002009082700002509102856003709127 2016 eng d00aSteady-state superradiance with Rydberg polaritons0 aSteadystate superradiance with Rydberg polaritons c2016/11/023 aA steady-state superradiant laser can be used to generate ultranarrow-linewidth light, and thus has important applications in the fields of quantum information and precision metrology. However, the light produced by such a laser is still essentially classical. Here, we show that the introduction of a Rydberg medium into a cavity containing atoms with a narrow optical transition can lead to the steady-state superradiant emission of ultranarrow-linewidth