The many-body localization transition in quasiperiodic systems has been extensively studied in recent ultracold atom experiments. At intermediate quasiperiodic potential strength, a surprising Griffiths-like regime with slow dynamics appears in the absence of random disorder and mobility edges. In this work, we study the interacting Aubry-Andre model, a prototype quasiperiodic system, as a function of incommensurate potential strength using a novel dynamical measure, information scrambling, in a large system of 200 lattice sites. Between the thermal phase and the many-body localized phase, we find an intermediate dynamical phase where the butterfly velocity is zero and information spreads in space as a power-law in time. This is in contrast to the ballistic spreading in the thermal phase and logarithmic spreading in the localized phase. We further investigate the entanglement structure of the many-body eigenstates in the intermediate phase and find strong fluctuations in eigenstate entanglement entropy within a given energy window, which is inconsistent with the eigenstate thermalization hypothesis. Machine-learning on the entanglement spectrum also reaches the same conclusion. Our large-scale simulations suggest that the intermediate phase with vanishing butterfly velocity could be responsible for the slow dynamics seen in recent experiments.

1 aXu, Shenglong1 aLi, Xiao1 aHsu, Yi-Ting1 aSwingle, Brian1 aSarma, Sankar, Das uhttps://arxiv.org/abs/1902.0719901125nas a2200121 4500008004100000245003500041210003500076260001500111520080500126100001600931700001900947856003700966 2019 eng d00aChaos in a quantum rotor model0 aChaos in a quantum rotor model c01/29/20193 aWe study scrambling in a model consisting of a number N of M-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large N and large M. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to 1M. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a T4 dependence on temperature T.

1 aCheng, Gong1 aSwingle, Brian uhttps://arxiv.org/abs/1901.1044601228nas a2200145 4500008004100000245007500041210006900116260001500185520076500200100002100965700002100986700001901007700001901026856003701045 2019 eng d00aA characterization of quantum chaos by two-point correlation functions0 acharacterization of quantum chaos by twopoint correlation functi c02/28/20193 aWe propose a characterization of quantum many-body chaos: given a collection of simple operators, the set of all possible pair-correlations between these operators can be organized into a matrix with random-matrix-like spectrum. This approach is particularly useful for locally interacting systems, which do not generically show exponential Lyapunov growth of out-of-time-ordered correlators. We demonstrate the validity of this characterization by numerically studying the Sachdev-Ye-Kitaev model and a one-dimensional spin chain with random magnetic field (XXZ model).

1 aGharibyan, Hrant1 aHanada, Masanori1 aSwingle, Brian1 aTezuka, Masaki uhttps://arxiv.org/abs/1902.1108601585nas a2200145 4500008004100000245010600041210006900147260001500216520109100231100002101322700002001343700001901363700002001382856003701402 2019 eng d00aQuantum circuit approximations and entanglement renormalization for the Dirac field in 1+1 dimensions0 aQuantum circuit approximations and entanglement renormalization c05/21/20193 aThe multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories.

1 aWitteveen, Freek1 aScholz, Volkher1 aSwingle, Brian1 aWalter, Michael uhttps://arxiv.org/abs/1905.0882101364nas a2200157 4500008004100000245003000041210003000071260001500101490000800116520096500124100002101089700002101110700001901131700001901150856003701169 2019 eng d00aQuantum Lyapunov Spectrum0 aQuantum Lyapunov Spectrum c04/10/20190 v0823 aWe introduce a simple quantum generalization of the spectrum of classical Lyapunov exponents. We apply it to the SYK and XXZ models, and study the Lyapunov growth and entropy production. Our numerical results suggest that a black hole is not just the fastest scrambler, but also the fastest entropy generator. We also study the statistical features of the quantum Lyapunov spectrum and find universal random matrix behavior, which resembles the recently-found universality in classical chaos. The random matrix behavior is lost when the system is deformed away from chaos, towards integrability or a many-body localized phase. We propose that quantum systems holographically dual to gravity satisfy this universality in a strong form. We further argue that the quantum Lyapunov spectrum contains important additional information beyond the largest Lyapunov exponent and hence provides us with a better characterization of chaos in quantum systems.

1 aGharibyan, Hrant1 aHanada, Masanori1 aSwingle, Brian1 aTezuka, Masaki uhttps://arxiv.org/abs/1809.0167101664nas a2200157 4500008004100000245006600041210006200107260001500169520119400184100001801378700001901396700001701415700001901432700001801451856003701469 2019 eng d00aThe Speed of Quantum Information Spreading in Chaotic Systems0 aSpeed of Quantum Information Spreading in Chaotic Systems c08/19/20193 aWe present a general theory of quantum information propagation in chaotic quantum many-body systems. The generic expectation in such systems is that quantum information does not propagate in localized form; instead, it tends to spread out and scramble into a form that is inaccessible to local measurements. To characterize this spreading, we define an information speed via a quench-type experiment and derive a general formula for it as a function of the entanglement density of the initial state. As the entanglement density varies from zero to one, the information speed varies from the entanglement speed to the butterfly speed. We verify that the formula holds both for a quantum chaotic spin chain and in field theories with an AdS/CFT gravity dual. For the second case, we study in detail the dynamics of entanglement in two-sided Vaidya-AdS-Reissner-Nordstrom black branes. We also show that, with an appropriate decoding process, quantum information can be construed as moving at the information speed, and, in the case of AdS/CFT, we show that a locally detectable signal propagates at the information speed in a spatially local variant of the traversable wormhole setup.

1 aCouch, Josiah1 aEccles, Stefan1 aNguyen, Phuc1 aSwingle, Brian1 aXu, Shenglong uhttps://arxiv.org/abs/1908.0699301046nas a2200133 4500008004100000245003700041210003700078260001500115490000700130520069800137100002100835700001900856856003700875 2019 eng d00aThermalization and chaos in QED30 aThermalization and chaos in QED3 c04/11/20190 v993 aWe study the real time dynamics of NF flavors of fermions coupled to a U(1) gauge field in 2+1 dimensions to leading order in a 1/NF expansion. For large enough NF, this is an interacting conformal field theory and describes the low energy properties of the Dirac spin liquid. We focus on thermalization and the onset of many-body quantum chaos which can be diagnosed from the growth of initally anti-commuting fermion field operators. We compute such anti-commutators in this gauge theory to leading order in 1/NF. We find that the anti-commutator grows exponentially in time and compute the quantum Lyapunov exponent. We briefly comment on chaos, locality, and gauge invariance.

1 aSteinberg, Julia1 aSwingle, Brian uhttps://arxiv.org/abs/1901.0498401719nas a2200109 4500008004100000245005600041210005600097520138200153100001801535700001901553856003701572 2018 eng d00aAccessing scrambling using matrix product operators0 aAccessing scrambling using matrix product operators3 aScrambling, a process in which quantum information spreads over a complex quantum system becoming inaccessible to simple probes, happens in generic chaotic quantum many-body systems, ranging from spin chains, to metals, even to black holes. Scrambling can be measured using out-of-time-ordered correlators (OTOCs), which are closely tied to the growth of Heisenberg operators. In this work, we present a general method to calculate OTOCs of local operators in local one-dimensional systems based on approximating Heisenberg operators as matrix-product operators (MPOs). Contrary to the common belief that such tensor network methods work only at early times, we show that the entire early growth region of the OTOC can be captured using an MPO approximation with modest bond dimension. We analytically establish the goodness of the approximation by showing that if an appropriate OTOC is close to its initial value, then the associated Heisenberg operator has low entanglement across a given cut. We use the method to study scrambling in a chaotic spin chain with 201 sites. Based on this data and OTOC results for black holes, local random circuit models, and non-interacting systems, we conjecture a universal form for the dynamics of the OTOC near the wavefront. We show that this form collapses the chaotic spin chain data over more than fifteen orders of magnitude.

1 aXu, Shenglong1 aSwingle, Brian uhttps://arxiv.org/abs/1802.0080101804nas a2200157 4500008004100000245003600041210003600077520137400113100001701487700001801504700001901522700002401541700002501565700001901590856003701609 2018 eng d00aBlack Hole Microstate Cosmology0 aBlack Hole Microstate Cosmology3 aIn this note, we explore the possibility that certain high-energy holographic CFT states correspond to black hole microstates with a geometrical behind-the-horizon region, modelled by a portion of a second asymptotic region terminating at an end-of-the-world (ETW) brane. We study the time-dependent physics of this behind-the-horizon region, whose ETW boundary geometry takes the form of a closed FRW spacetime. We show that in many cases, this behind-the-horizon physics can be probed directly by looking at the time dependence of entanglement entropy for sufficiently large spatial CFT subsystems. We study in particular states defined via Euclidean evolution from conformal boundary states and give specific predictions for the behavior of the entanglement entropy in this case. We perform analogous calculations for the SYK model and find qualitative agreement with our expectations. A fascinating possibility is that for certain states, we might have gravity localized to the ETW brane as in the Randall-Sundrum II scenario for cosmology. In this case, the effective description of physics beyond the horizon could be a big bang/big crunch cosmology of the same dimensionality as the CFT. In this case, the d-dimensional CFT describing the black hole microstate would give a precise, microscopic description of the d-dimensional cosmological physics.

1 aCooper, Sean1 aRozali, Moshe1 aSwingle, Brian1 aVan Raamsdonk, Mark1 aWaddell, Christopher1 aWakeham, David uhttps://arxiv.org/abs/1810.1060101648nas a2200169 4500008004100000245006500041210006400106260001500170300000700185520110300192100001701295700002101312700002601333700002501359700001901384856007501403 2018 eng d00aEntanglement of purification: from spin chains to holography0 aEntanglement of purification from spin chains to holography c2018/01/22 a983 aPurification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.

1 aNguyen, Phuc1 aDevakul, Trithep1 aHalbasch, Matthew, G.1 aZaletel, Michael, P.1 aSwingle, Brian uhttps://link.springer.com/article/10.1007%2FJHEP01%282018%29098#citeas01255nas a2200133 4500008004100000245006300041210006100104260000900165490000800174520086800182100001901050700001501069856003701084 2018 eng d00aHolographic Complexity of Einstein-Maxwell-Dilaton Gravity0 aHolographic Complexity of EinsteinMaxwellDilaton Gravity c20180 v1063 aWe study the holographic complexity of Einstein-Maxwell-Dilaton gravity using the recently proposed "complexity = volume" and "complexity = action" dualities. The model we consider has a ground state that is represented in the bulk via a so-called hyperscaling violating geometry. We calculate the action growth of the Wheeler-DeWitt patch of the corresponding black hole solution at non-zero temperature and find that, in the presence of violations of hyperscaling, there is a parametric enhancement of the action growth rate. We partially match this behavior to simple tensor network models which can capture aspects of hyperscaling violation. We also exhibit the switchback effect in complexity growth using shockwave geometries and comment on a subtlety of our action calculations when the metric is discontinuous at a null surface.

1 aSwingle, Brian1 aWang, Yixu uhttps://arxiv.org/abs/1712.0982601923nas a2200109 4500008004100000245005100041210004900092520159800141100001801739700001901757856003701776 2018 eng d00aLocality, Quantum Fluctuations, and Scrambling0 aLocality Quantum Fluctuations and Scrambling3 aThermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators. Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs). However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large N models exhibiting exponential growth of OTOCs and a sharp chaos wavefront and other random circuit models showing a diffusively broadened wavefront. In this paper we propose a unified physical picture for scrambling in chaotic local Hamiltonians. We construct a random time-dependent Hamiltonian model featuring a large N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wavefront. We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a cross-over to a diffusively broadened wavefront. At small N we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques. To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a time-independent Hamiltonian model are also consistent with a diffusively-broadened wavefront.

1 aXu, Shenglong1 aSwingle, Brian uhttps://arxiv.org/abs/1805.0537601660nas a2200109 4500008004100000245006500041210006500106520130600171100001701477700001901494856003701513 2018 eng d00aProduct Spectrum Ansatz and the Simplicity of Thermal States0 aProduct Spectrum Ansatz and the Simplicity of Thermal States3 aCalculating the physical properties of quantum thermal states is a difficult problem for classical computers, rendering it intractable for most quantum many-body systems. A quantum computer, by contrast, would make many of these calculations feasible in principle, but it is still non-trivial to prepare a given thermal state or sample from it. It is also not known how to prepare special simple purifications of thermal states known as thermofield doubles, which play an important role in quantum many-body physics and quantum gravity. To address this problem, we propose a variational scheme to prepare approximate thermal states on a quantum computer by applying a series of two-qubit gates to a product mixed state. We apply our method to a non-integrable region of the mixed field Ising chain and the Sachdev-Ye-Kitaev model. We also demonstrate how our method can be easily extended to large systems governed by local Hamiltonians and the preparation of thermofield double states. By comparing our results with exact solutions, we find that our construction enables the efficient preparation of approximate thermal states on quantum devices. Our results can be interpreted as implying that the details of the many-body energy spectrum are not needed to capture simple thermal observables.

1 aMartyn, John1 aSwingle, Brian uhttps://arxiv.org/abs/1812.0101502365nas a2200145 4500008004100000245006700041210006000108260001200168490000600180520192900186100002802115700001902143700002002162856003702182 2018 eng d00aThe quasiprobability behind the out-of-time-ordered correlator0 aquasiprobability behind the outoftimeordered correlator c04/20180 vA3 aTwo topics, evolving rapidly in separate fields, were combined recently: The out-of-time-ordered correlator (OTOC) signals quantum-information scrambling in many-body systems. The Kirkwood-Dirac (KD) quasiprobability represents operators in quantum optics. The OTOC has been shown to equal a moment of a summed quasiprobability. That quasiprobability, we argue, is an extension of the KD distribution. We explore the quasiprobability's structure from experimental, numerical, and theoretical perspectives. First, we simplify and analyze the weak-measurement and interference protocols for measuring the OTOC and its quasiprobability. We decrease, exponentially in system size, the number of trials required to infer the OTOC from weak measurements. We also construct a circuit for implementing the weak-measurement scheme. Next, we calculate the quasiprobability (after coarse-graining) numerically and analytically: We simulate a transverse-field Ising model first. Then, we calculate the quasiprobability averaged over random circuits, which model chaotic dynamics. The quasiprobability, we find, distinguishes chaotic from integrable regimes. We observe nonclassical behaviors: The quasiprobability typically has negative components. It becomes nonreal in some regimes. The onset of scrambling breaks a symmetry that bifurcates the quasiprobability, as in classical-chaos pitchforks. Finally, we present mathematical properties. The quasiprobability obeys a Bayes-type theorem, for example, that exponentially decreases the memory required to calculate weak values, in certain cases. A time-ordered correlator analogous to the OTOC, insensitive to quantum-information scrambling, depends on a quasiprobability closer to a classical probability. This work not only illuminates the OTOC's underpinnings, but also generalizes quasiprobability theory and motivates immediate-future weak-measurement challenges.

1 aHalpern, Nicole, Yunger1 aSwingle, Brian1 aDressel, Justin uhttps://arxiv.org/abs/1704.0197101111nas a2200109 4500008004100000245004900041210004900090520079100139100001900930700001500949856003700964 2018 eng d00aRecovery Map for Fermionic Gaussian Channels0 aRecovery Map for Fermionic Gaussian Channels3 aA recovery map effectively cancels the action of a quantum operation to a partial or full extent. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel. As a by-product, we obtain a formula for the fidelity between two fermionic Gaussian states. We also discuss subtleties arising from the singularities of the involved matrices.

1 aSwingle, Brian1 aWang, Yixu uhttps://arxiv.org/abs/1811.0495601416nas a2200133 4500008004100000245004200041210004200083260001500125490000600140520105200146100001901198700002801217856003701245 2018 eng d00aResilience of scrambling measurements0 aResilience of scrambling measurements c2018/06/180 vA3 aMost experimental protocols for measuring scrambling require time evolution with a Hamiltonian and with the Hamiltonian's negative counterpart (backwards time evolution). Engineering controllable quantum many-body systems for which such forward and backward evolution is possible is a significant experimental challenge. Furthermore, if the system of interest is quantum-chaotic, one might worry that any small errors in the time reversal will be rapidly amplified, obscuring the physics of scrambling. This paper undermines this expectation: We exhibit a renormalization protocol that extracts nearly ideal out-of-time-ordered-correlator measurements from imperfect experimental measurements. We analytically and numerically demonstrate the protocol's effectiveness, up to the scrambling time, in a variety of models and for sizable imperfections. The scheme extends to errors from decoherence by an environment.

1 aSwingle, Brian1 aHalpern, Nicole, Yunger uhttps://arxiv.org/abs/1802.0158701302nas a2200121 4500008004100000245008600041210006900127520089100196100001901087700001801106700001901124856003701143 2018 eng d00aScrambling dynamics across a thermalization-localization quantum phase transition0 aScrambling dynamics across a thermalizationlocalization quantum 3 aWe study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with sub-ballistic operator spreading. The critical disorder strength for the ballistic to sub-ballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, we find that the operator dynamics is not very sensitive to the actual eigenstate localization transition. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.

1 aSahu, Subhayan1 aXu, Shenglong1 aSwingle, Brian uhttps://arxiv.org/abs/1807.0608601318nas a2200121 4500008004100000245004000041210004000081520097600121100002101097700002201118700001901140856003701159 2018 eng d00aSubsystem Complexity and Holography0 aSubsystem Complexity and Holography3 aWe study circuit complexity for spatial regions in holographic field theories. We study analogues based on the entanglement wedge of the bulk quantities appearing in the "complexity = volume" and "complexity = action" conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition about them. We find a promising qualitative match between the holographic action and what we call the purification complexity, the minimum number of gates required to prepare an arbitrary purification of the given mixed state. On the other hand, the holographic volume does not appear to match any of our definitions of mixed-state complexity.

1 aAgón, Cesar, A.1 aHeadrick, Matthew1 aSwingle, Brian uhttps://arxiv.org/abs/1804.0156102354nas a2200157 4500008004100000245007000041210006900111520185900180100001902039700002002058700002402078700001802102700001902120700002002139856003702159 2017 eng d00aEntanglement Wedge Reconstruction via Universal Recovery Channels0 aEntanglement Wedge Reconstruction via Universal Recovery Channel3 aWe apply and extend the theory of universal recovery channels from quantum information theory to address the problem of entanglement wedge reconstruction in AdS/CFT. It has recently been proposed that any low-energy local bulk operators in a CFT boundary region's entanglement wedge can be reconstructed on that boundary region itself. Existing work arguing for this proposal relies on algebraic consequences of the exact equivalence between bulk and boundary relative entropies, namely the theory of operator algebra quantum error correction. However, bulk and boundary relative entropies are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect. The framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture in addition to new physical insights. Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator. This formula can be interpreted as a noncommutative version of Bayes' rule that attempts to undo the noise induced by restricting to only a portion of the boundary, and has an integral representation in terms of modular flows. To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra

1 aCotler, Jordan1 aHayden, Patrick1 aPenington, Geoffrey1 aSalton, Grant1 aSwingle, Brian1 aWalter, Michael uhttps://arxiv.org/abs/1704.0583901605nas a2200121 4500008004100000245006800041210006800109260001500177520121600192100001901408700001901427856003701446 2017 eng d00aRobust entanglement renormalization on a noisy quantum computer0 aRobust entanglement renormalization on a noisy quantum computer c2017/11/203 aA method to study strongly interacting quantum many-body systems at and away from criticality is proposed. The method is based on a MERA-like tensor network that can be efficiently and reliably contracted on a noisy quantum computer using a number of qubits that is much smaller than the system size. We prove that the outcome of the contraction is stable to noise and that the estimated energy upper bounds the ground state energy. The stability, which we numerically substantiate, follows from the positivity of operator scaling dimensions under renormalization group flow. The variational upper bound follows from a particular assignment of physical qubits to different locations of the tensor network plus the assumption that the noise model is local. We postulate a scaling law for how well the tensor network can approximate ground states of lattice regulated conformal field theories in d spatial dimensions and provide evidence for the postulate. Under this postulate, a O(logd (1/δ))-qubit quantum computer can prepare a valid quantum-mechanical state with energy density δ above the ground state. In the presence of noise, δ = O( logd+1(1/)) can be achieved, where is the noise strength.

1 aKim, Isaac, H.1 aSwingle, Brian uhttps://arxiv.org/abs/1711.07500