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C BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct PNAS a is article paper.y This the wrote M.L.B., interest.y and J.E. competing J.H., and no research; Goihl, Gluza, declare M. M. authors designed J.B.-V., M.L.B., The J.H., research; J.E. M.L.B., performed and J.E. data; and analyzed and Gluza Gluza, Gluza, M. M. J.B.-V., M. J.H., J.B.-V., Goihl, M. M.L.B., contributions: Author matrix as such methods, sophisticated more the Other in exponentially size. scale system required but resources systems, physical computational a different the many yield for can frustrated results useful diagonalization for powerful of Exact problems plethora a sign (9–11). strong systems is by fermionic Carlo affected phase and Monte is particular it quantum a but example, of method, For properties interest. the understand of to val- needed expectation go. quantum ues important can most the we of some far calculating how know high-T magnets we or quantum the frustrated 8), higher-dimensional in in (7, true example, theory For be complexity to of believed simulation, field classical assumptions a certain is considering resource is the and what as long clas- of As with boundaries surpassed algorithms. be sical the cannot class pushing boundary complexity complexity physical slowly the well-determined possible, field these cases) the of which many Despite problem one (in (6). computational mod- each a a of that to to class belongs fact connected narrow be the simulation, a can from for efficient problems to arises efficient an constrained just This obtain be are els. to to algorithms sites need the atomic sizes or system dozen the couple cases, a many in ful, transition it. phase quantum from a away across a it or of be results experiments, the to compare directly simulation to experimental one critical allows a which can access signature we can yet, simulators everything quantum understand how not show do we which about system owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To fectto ofieeti ogrne oesfrwiha which lacking. for is models understanding long-ranged comprehensive signatures the in exhibit confinement can excitation DQSs in of DSFs the to partic- that In show At systems: compared we spin ular, materials. long-ranged directly explore we quantum be time, same of can the properties DQSs exploring from experiments results we how efficiently setups, show classically experimental state-of-the-art be complexity- on cannot Based provide computed. they and that evidence compute what theoretic can than larger simulations considerably sizes classical system by for structure (DSFs) dynamical accessible simulate factors into to method is a insights as quantum (DQSs) what dynamical simulators propose new We beyond computers. classical offer of matter means to correlated promise strongly simulators Quantum Significance hl urn ueia ehiusaesileteeyuse- extremely still are techniques numerical current While c uecnutr,w aen eei fcetwyof way efficient generic no have we superconductors, b,c ae Gluza Marek , b . almCne o ope Quantum Complex for Center Dahlem y raieCmosAttribution-NonCommercial- Commons Creative . y b https://www.pnas.org/lookup/suppl/ n esEisert Jens and , NSLts Articles Latest PNAS d emot-etu Berlin Helmholtz-Zentrum b,d | f12 of 1

PHYSICS product states (MPS), projected entangled Pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA), etc., are efficient for one-dimensional short-range systems, but these methods are constrained by the amount of entanglement present in the system. In this work, we propose dynamical analog quantum simula- tors (12, 13) as an alternative method to simulate low-energy excitations of strongly correlated matter. In particular, we sug- gest that dynamical structure factors (DSFs), which provide key physical insights into quantum matter, can be accessed with quantum simulators, while at the same time are a quantity which is significantly less accessible with classical computers. Large-scale analog quantum-simulation platforms are unique systems in that they show exceptionally strong quantum effects and allow for measuring expectation values of microscopic observ- ables (14–21). Among other platforms, the propagation of exci- tations in XXZ models (14, 15), Lieb–Robinson bounds (16), relaxation dynamics (22), and phase diagrams of Fermi–Hubbard Fig. 1. State-of-the-art exact numerical algorithms to time evolve two- models (17) have been probed with ultracold atoms beyond capa- body observables for the long-range TFIM. The larger system sizes for which bilities of current classical algorithms. At the same time, quantum unequal time correlators (and by extension DSFs) can be accessed are shown. simulations with trapped ions and Rydberg arrays have also seen At 19 sites, the Pauli operators at different sites have been time-evolved via several breakthroughs, as, for example, the quantum dynamics of exact diagonalization (ED) (26). Using Krylov-space methods, the system size has been extended to 25 (27). System sizes up to 27 (ED) and 128 [Lanczos the long-range transverse-field Ising model, which has recently and time dependent density matrix renormalization group (t-DMRG)] sites been studied in systems of over 50 atoms via time-dependent have been obtained, but only single-site observables have been accessed expectation values of single spin observables (18–21). Though a (28). Here, we show that our proposal offers a leap forward in terms of great body of observations has been assembled, a particular ques- the system sizes that can be employed to study DSFs of long-range mod- tion arises: Can quantum simulators provide qualitative dynam- els via quantum simulation. Please note that using variational methods, ical quantities of systems relevant in the condensed-matter con- entanglement entropies up to 125 sites can be obtained (29). text, for which there is evidence that in the regime discussed they are inaccessible to classical algorithms? We propose an answer to this question in form of the DSF, measurement of the DSF for the long-range TFIM as a practical a widely attainable experimental observable which gives infor- application of quantum simulators in a quantity relevant for both mation regarding dynamical properties of a given system. In condensed matter and material science. materials, it is experimentally measured by inelastic neutron scat- We study the long-range TFIM under the same imperfections tering (23) and resonant inelastic X-ray scattering (24). Given as for the short-range model. We show that the experimental the relative ease of measuring the DSF experimentally, an effi- imperfections currently present in quantum simulators do not cient way to simulate this quantity becomes imperative. We argue affect the DSF in a significant way and that the scaling of these that the DSF can be accurately accessed with quantum simula- errors in the DSF is well controlled in the full range of system tors within the experimental level of accuracy currently available sizes studied here. in the different architectures and for system sizes beyond what We also study the computational hardness of evaluating the current classical algorithms can achieve, as we show in Fig. 1. It DSF for general systems. We find that the DSF can be likened to is worth pointing out that in the quantum-simulator setting, there a bounded-error quantum polynomial time (BQP)-hard problem, have been previous measures of DSFs (25). The major difference meaning that any classical algorithm calculating it for general between those studies and ours is that while those proposals rely Hamiltonians efficiently would also efficiently solve all of the on a modification of the simulator architecture to measure the tasks that a quantum computer can tackle efficiently. The latter DSF (what would be considered a hardware solution), we pro- is regarded in the quantum-computing community as a highly pose a software solution in which the setup does not need any unlikely scenario. As such, realizing our proposal in practice specific modification to be able to detect DSFs. would tackle a task hard for classical computers in a field of prac- The DSF is a quantity which can be considered stable to tical importance in condensed-matter physics. While the specific small perturbations of the microscopic model whose excitations proposal of this work is centered on a specific model, it is worth it probes, given that the qualitative features of the DSF already pointing out that the proof of hardness is valid for a wide range provide a lot of information regarding those excitations. In this of Hamiltonians. It is our aim in this work to highlight a spe- sense, we expect to see an inherent robustness in the DSF, cific case in which the DSF can be experimentally achieved in finding that observing the signatures of low-energy excitations the near-term, but the protocol employed here, together with the is possible with state-of-the-art setups in the presence of mod- error analysis and the study of the different architectures, can be erate experimental imperfections. As a proof of principle, we easily applied to other models, as, for example, the XY model investigate the short-and long-range transverse field Ising model in superconducting chips (31) or Rydberg atoms (32). As such, (TFIM). The short-range model is integrable (30) and allows us future advances in the field, where analog quantum simulators to study relatively big system sizes comparable to those achiev- implement further models in higher dimensions, can make use of able in trapped ions and Rydberg atoms simulators. We first the study performed in this work to show a practical application study in detail the effects of experimental imperfections in the of quantum simulators through the DSF in those models. short-range model and the associated Fourier transform involved in the calculation of the DSF to give us an intuition of those DSF in Quantum Simulators effects. Once the short-range model is well understood, we move In order to employ quantum simulators to study the DSF of to our application proposal. The classical numerical calcula- solid-state systems, we want to probe the fluctuations of their tion of unequal time-correlation functions in long-range systems ground states or thermal states via unequal time correlation func- is constrained to system sizes much smaller than what current tions. For a spin system with lattice sites i, j ∈ Λ (where Λ is the quantum simulators can achieve (18, 19). Thus, we propose the collection of lattice sites), these are defined by

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TFIM the of solution exact the with accordance in space, ω and simulator quantum the of initialization the in resides then Eq. in tity observables given to form related those the particular are of sensitive a is it by which to described excitations system, given a Hamiltonian of excitations Protocol. energy systems Measurement of DSF class (33). proposed wide been and has a as atoms, for XXZ, and many Rydberg and Ising beyond in chips, ions, implemented qubit trapped be superconducting as offer can we architectures, context, which this different protocol In exhibit assumes. measurement not 33 do ref. a as which symmetries systems many to pro- as spin methods spectroscopy single-site this tomographic a extend via where We tocol setup implemented. any be in can ref. employed rotation by be to proposed can protocol which the access of 33, generalization Eq. gives a propose in which we as ing, existing protocol functions, the measurement time-correlation beyond unequal a supplemented is be to techniques needs that ingredient quantum state-of-the-art from obtained be can 2 simulations. Fig. a critical- in with from one continuum, the away cosine-shaped work, a at observe this we gap in figure, detail the in In ity. study will we models many-body quantum a to applied lacking. still is is 33 system ref. of itself, DSF proposal the the and fea- when effects the physical important of observing understanding of clear sibility a however, using fluctuation- devices spectroscopy; functions trapped-ion the Ramsey and via Green’s atoms equilibrium cold retarded in in theorem) DSF dissipation measure the to to related are how (which on (33) proposal where DSF the sites by real-space from matrices quantities momentum these Pauli of transform denote Fourier we And i.2. Fig. aze al. et Baez dpnettopril otnu xedn vrteetr reciprocal entire the over extending continuum two-particle -dependent ooti uhaDFi unu iuaos h crucial the simulators, quantum in DSF a such obtain To the of one TFIM, the for DSF typical a show we 2, Fig. In B S = a N ,b S o h FM eso h S wyfo h criticality, the from away DSF the show We TFIM. the for DSF q .W bev h a rudthe around gap the observe We 1.4. (q, 0 = stenme fltiests hr a enarecent a been has There sites. lattice of number the is ω h olo hswr st hwta S like DSF a that show to is work this of goal The . q = ) H ∈ ie h ento fteDFi Eq. in DSF the of definition the Given . R N 1 3 n iet frequency-domain time-to and C i X ,j i a ∈Λ ,j ,b (t h rtse ooti uhaquan- a such obtain to step first The 1. Z −∞ = ) ∞ h S fetvl rbslow- probes effectively DSF The hσ e dt i a (0)σ −iq·( σ j b q x (t a i = −x )i, 0, with j ) ω e iω = ntefollow- the In 1. t a on,adthe and point, 4 π/ C = i a ,j ,b ω x (t , ∈ y ), , R z The . the 2, yields x J i [1] [2] = to 1 iuao ilb ntaie nteuiu rudsaevco of vector ground-state quantum unique the the that H in simplicity, initialized of be sake will of the simulator state for assume, ground will the we ideally tion, state, low-energy a in at xml fti da osdrteuiayrpeetn a representing unitary the Consider idea: this of has example idea] tant, the of of pulse study the operation, detailed time unitary a initial a of for at part operator 34 the ref. that also insight been [see main The 33 trans- operators? ref. unitary spin of perform local unequal only measure two-point can and measure we formations we if can functions correlation How time hand: at question cial Correlations. Time Unequal temporal Measuring and spatial a via obtained operators transform. be spin Fourier can DSF local are the correlators measure time measured, unequal the we Once resolution. evolution, single-site with the is the after state exciting Finally, After the rotation. locally, spin single system a applying by particular excitations this with achieved be can which architecture. extending times effectively ions, evolution experi- compari- trapped the in in the evolutions preparation adiabatic reduce that ground-state to out considerably son for pointing required to time worth shown mental is been ground It have the Hamiltonians. QAOAs of nontrivial approximation good of experi- very state trapped-ions a achieving in (35), reported ments been recently have algorithms proposed recently the algorithms optimization time, same the spectroscopy At Ramsey the with 34 equilibrium ref. in it in be technique. exemplified state, as initial any not, with or used be can employ we h rttr ntels ieo Eq. of line last the in term first The rce rmtedt fi snneo h attr,o h other the on sub- term, be last The can nonzero. hence, is it and, if step data the excitation from the tracted omitting simply by function Green retarded with vec- low-energy-state a of excitation an as tor it use to like would We fw esr h xetto au of value expectation the thus, measure and, we unitary, If system are the operations of both evolution single-qubit that time subsequent unitary a and appropriate 3) an by obtained rotation be can which vector (33)]. state state initial and spin Hamiltonian the of of number symmetries odd the an from of value for vanishes let expectation operators simple, the discussion that the assume keep us To investigated. being system ing time π 4 rtto fasi tsite at spin a of -rotation hc erfrt as to refer we which , ebgnb h ai,ada h aetm otimpor- most time same the at and basic, the by begin We netegon tt sotie,w hnidc low-energy induce then we obtained, is state ground the Once evolution. adiabatic by achieved be can state a such Preparing aigteetoigeinsa ad ecncnie the consider can we hand, at ingredients two these Having hψ |ψ | t 0 R σ oendb h aybd aitna fteinteract- the of Hamiltonian many-body the by governed hc hni rbdb usqetevolution subsequent by probed is then which i, i x (i G U | , ψ x ret j ( i ,x j , ) = t (i = ) htlclyectstesse a h n nEq. in one the (as system the excites locally that , 2 1 j , hψ 2 1 t = ) hψ 0 | U |ψ σ 0 |ψ − | QOs a lobe mlyd These employed. been also can (QAOAs) i x ( i σ j 0 (t |ψ 2 ) = i j i x = ) hσ j σ a stecs o h FMadarises and TFIM the for case the is [as 0 U |ψ i x ∈ huhi rnil,teprotocol the principle, in though i, i √ x evolved (t (t 1 0 (t Λ 2 i )σ )U )σ (1 + ln the along j x j x ( G − |ψ j (0) ) x e snwdsustecru- the discuss now us Let ret 5 ihteHamiltonian the with |ψ 0 iσ ,x σ σ and i, a emaue directly measured be can − NSLts Articles Latest PNAS (i j i 0 i x x a asyinterferometry. Ramsey i ). (0) , nti tt,w obtain we state, this on σ , unu approximate quantum j x j x , |ψ (0)σ axis t a eotie as obtained be can + ) i G sasaevector. state a is i x R x ret U H (t ,x (i (t nti sec- this In . )i (i , Observe ). 0 , j . j , , t U | t ), ) (t f12 of 3 ) the [6] [3] [5] [4] H to .

PHYSICS hand, has a nontrivial unequal time dependence and, hence, aφ’s as rows, and, in a corresponding fashion, we can collect the must either vanish due to, e.g., symmetry arguments or has to measured bφ’s into a vector b. be reconstructed. The retarded Green’s function can be reconstructed by The case considered in ref. 33 is the one in which the Hamil- noticing that tonian Hˆ has a unitary symmetry P, such that the product of ? T −1 T v = (A A) A b, [14] an odd number of Pauli operators vanish. From this, it fol- lows that R(i, j , t) must vanish. As such, whenever a symmetry gives the value of v that minimizes the least-square residue of this kind is present (as in the TFIM), we obtain the iden- tity hψ| σx |ψi = Gret (i, j , t). Calculating this for all spin pairs i x,x min kAv − bk2. [15] ret v (i, j ), we obtain the retarded Green function Gx,x (i, j , t), and we can perform a Fourier transform in real space and time to ret Here, we assume that one can choose the excitation angles φ in obtain Gx,x (q, ω). Finally, we can relate the retarded Green T function, when linear response theory holds, to the DSF via the such a way that the matrix A A is well conditioned, as is done fluctuation-dissipation theorem in typical tomographic schemes. In order to measure the DSF, this procedure must be performed for all pairs of excitation and xx 1 ret measurement positions i, j ∈ Λ, and the Fourier transform of S (q, ω) = − [1 + nB (ω)]Im[Gx,x (q, ω)], [7] ? ret π the collection of reconstructed values v1 = Gx,x (i, j , t) will yield the DSF. ω/T where nB (ω) = 1/(e + 1). This way, we get direct access to the DSF by measuring the retarded Green’s function via the On the Computational Complexity of the DSF above measurement protocol. Once we have formalized how dynamical quantum simulators There are two points which need to be made before we move can access the DSF, we will concentrate on answering the ques- on: First, while we study the zero-temperature DSF, finite, but tion: In what specific way is the calculation of the DSF a small, temperatures will broaden the features of the DSF, but not computationally hard problem? In the following, we formalize change the overall behavior, provided that T is smaller than the the statements about classical hardness and show that a practi- smallest coupling of the model. Second, note that the fluctuation- cally accessible variant of the DSF is hard for the complexity class dissipation theorem holds when linear response theory is a good BQP. To this end, we show that the building blocks of the DSF, approximation, and its validity or lack of thereof away from equi- a,b the unequal time correlators Ci,j (t), are BQP-hard to compute. librium is a highly researched topic to the date (13, 36, 37). As To start with, and without loss of generality, we show that such, this measurement protocol for the DSF will be accurate σz (t)σz := hψ| σz (t)σz |ψi is BQP-hard to compute for when the system is close to thermal equilibrium in a practical i j ψ i j sense. product-state vectors |ψi and for ground states. Then, we use these observations to consider the DSF over a finite (but Tomographic Recovery Methods for Unequal Time-Correlation Func- arbitrarily large) interval of time tions. If the symmetry argument can be relaxed, we can show Z t1 how the term R(i, j , t) can be extracted. Let us define a modified z,z 1 X −iq(x −x ) iωt z z S (q, ω) = e i j e σ (t)σ dt, [16] Ramsey state vector which reads t0,t1 N i j ψ i,j t0 (j ) |ψφi = U (t)U (φ) |ψ0i , [8] where N is the system size. In particular, we prove the following.

where now we excite the ground state |ψ0i with a φ-rotation around the x axis Theorem 1 (Hardness of Computing the Approximate DSF). For t1 − t0 = poly(N ), product states |ψi, and two-local Hamiltoni- (j ) −iφσx x z,z U (φ) = e j = cos(φ)1 − i sin(φ)σ ans it is BQP-hard to approximate St ,t (q, ω) within an error j [9] 0 1 1 x ε = poly0−1(N ). = : cφ − isφσj . z,z We consider the quantity St0,t1 instead of the full Fourier For an analogous measurement to the case in the previous transform, as it is the practically accessible one: Any time obser- section, we obtain vation will necessarily be finite in practice. What is more, from a conceptual perspective, the latter is not even computable x 2 x ret hψφ| σi |ψφi = cφ hψ0| σi (t) |ψ0i + 2cφsφGx,x (t) on a Turing machine due to arbitrarily large errors that are 2 [10] introduced by the Fourier transform: The continuous Fourier + sφR(i, j , t). transform is not Turing computable. Hardness for estimating correlators on ground states. For hard- We now notice that we can directly measure the left-hand side z,z ness of ground states, we observe that computing Ci,j (t) = and the first term on the last line of the expression above. For a z z z,z hσi (t)σj iψ for any t is at least as hard as computing Ci,j (0) = fixed angle φ, we can write z z hσi σj iψ. First, computing correlators up to constant additive x 2 x errors on ground states of quasi-local Hamiltonians is BQP- bφ = hψφ| σi |ψφi − cφ hψ0| σi (t) |ψ0i . [11] hard by the Feynman–Kitaev construction (38). Furthermore Now, we can rewrite Eq. 10 as this remains true for several classes of local observables and local Hamiltonians, including one-local observables measured T aφ v = bφ, [12] on ground states of nearest-neighbor two-local Hamiltonians on qubits (39, 40) and two-local observables measured on where v is the vector we want to reconstruct, given by ground states of translation invariant nearest-neighbor two-local Hamiltonians with local dimension three (41). ret T v = [Gx,x (t), R(i, j , t)] , [13] Hardness for out-of-time correlators. For the product states, we start with a general observation: Consider an arbitrary cir- 2 and aφ = [2sφcφ, sφ]. If an experiment measures bφ using vari- cuit Cn = Un ... U1 consisting of k-local gates Ui . Evaluating φ A σz (t)σz |ψi ous angles , then we can build a matrix using the different the quantity i j ψ for product-state vectors within

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∆ i Com- i. z 16 ∆ ,j ψ ,z t . t )σ (t )∆t [19] [20] [18] [17] with H = ) j z h 2),

. j , . , where tpfnto prxmto ie stecmltderror cumulated the t us gives approximation step-function rudsaei ie yapoutsaeo pn ul polarized. fully spins of state product a by given is state ground two-kink a of terms in DSF the of in excitations model. signature low-energy their the and describe TFIM detail briefly the in will studied we been Here, (30). has TFIM TFIM. nearest-neighbor Short-Range short-range, the of Properties Universal directly platforms. those be on can experiments which with of data compare to provides study used thus Our model short-range. short-range of behaves values the effectively high system at the model the long-range where access the can or arrays long- model, Rydberg the short-range or study lattices optical several to Second, as on proposal. architectures move practical DSF. our can evaluate the we and of model completed, range simulation is quantum task effects this the expected After on the way, imperfections of This these understanding case. of of sufficient this study in provide imperfections can detailed evolution we a the perform of we effects instance which easy the problem, protocol an time-evolution measurement is model the the short-range of of the Since models, accuracy expected. imperfection help- the be different provide can on the can as long-range of DSF well effects its the as the of than on study insights understood a ful such, better as much and, short- short-range counterpart, the is First, the reasons. model for main two simulation range for importance quantum great of via is DSF case the to access (30). fermions noninteracting to field magnetic the by of given strength The is site-dependent. are be term can Ising ciple, the of parameters coupling The and function the inadotclltie,oecnsuytesotrnemodel short-range the study can one with lattices, (17) typically optical where and ions, tion trapped or arrays α Rydberg in simulations form long-range the take chain, the throughout those of presence the defined is in robust- 1D-TFIM of TFIMs invariant as translational degree long-range study The the will and imperfections. on we assess and short- imperfections, To experimental DSFs the against task. obtain DSFs this of to ness of employed complexity mea- be mentioned the previously can the protocol how on surement concentrated we far, Simulators So Quantum in DSFs of time Realization for Practical universal short-range, be to nearest-neighbor, expected the not exam- evolution. is of For model evolution sense. Ising transverse theoretic time not complexity the are the problem ple, this in in cases of hard all subclasses necessarily solves general, that In algorithm worst-case time. an so-called polynomial out a is How- rules this only time. that it out polynomial result—i.e., point to in important is DSFs it compute ever, that algorithms sical from errors controlled. possible and the behaved that well shown are have this we axis, time the discretiza- a of require tion quantum and classical both simulations since that proven have we essence, In error. arbitrar- constant within small approximation ily an for suffices small and constant 0 )L ∈ o h hr-ag FMi h ermgei hs,the phase, ferromagnetic the in TFIM short-range the For the model, long-range the on focused is proposal our While hrns rvdseiec gis h xsec fclas- of existence the against evidence provides BQP-hardness 1 6] [1, 0 S ∆t a ∆t ,b (t = (q .I h aeo iia simula- digital of case the In I). section Appendix, (SI J = , i ,j ω B 1 t H 1 e ) = − i −iq n nti ok ewl osdri uniform it consider will we work, this in and , − (J are J t 0 t δ ( 0 , ) x i 2 /M B i ,j hr naseicsne Furthermore, sense. specific a in BQP-hard −x L ±1 = ) 0 /M j nertn vrteerrmd ythe by made error the over Integrating . ) hc seatyslal yamapping a by solvable exactly is which , B e X i ∈Λ iω i where , J = t i C B ,j B i i z = ,j σ h pnsi neato can interaction spin–spin The . ,z i z J (t − /| L ec,choosing Hence, ). 0 i i X ,j steLpht osatof constant Lipshitz the is − ∈Λ NSLts Articles Latest PNAS j | J α i ,j o nlgquantum analog for σ h hsc fthe of physics The i x J σ i j x ,j . n,i prin- in and, , M C | i a f12 of 5 obe to ,j (t ,b [21] 1 (t α, − )

PHYSICS When the magnetic field and Ising coupling are at a finite value, In the case of the long-range TFIMs, while the ground-state fluctuations are induced in the system, in the form of fermionic phase diagram has been explored (50–52), much less is known pseudo-particles γ. These excitations can be seen in the spin pic- about its dynamical behavior (53–55). Recent studies (27, 29, ture as spin flips, or kinks over the fully polarized state. Once 56) concentrate on the entanglement growth and the spread a spin is flipped, it is free to move along the chain and cre- of correlations in this model as a function of the interaction ate a domain. The walls of this domain can be regarded as the length, α, or in the thermalization of different initial states under kinks (or, equivalently, the γ-fermions) that interpolate between this Hamiltonian (28). Analyzing the light cones and possible the two possible ground states connected by the Z2 symmetry Lieb–Robinson-like bounds in the long-range TFIM at zero tem- of the model. When a domain is formed, the domain walls or perature, these studies separate the dynamical behavior of this kinks behave as free fermionic particles that propagate through model in three regions. For α > 3, the system obeys the general- the chain. Since to create a domain, we need at least two kinks ized Lieb–Robinson bound (57), and the behavior of the system (particles), the first contribution to the excitation spectrum will mimics that of a short-range model. Via semiclassical arguments, come from the two-particle states, which will be described by the dispersion relation of excitations in the ground state (what we

their energy and momenta, E = q1 + q2 and q = q1 + q2. For a study here via the DSF) is found to approximately be a cosine, fixed q, the values of q1 and q2 can be chosen arbitrarily, which which coincides with the short-range behavior. From this, we can generates a continuum of excitations. say that for quantum simulators, the behavior of the DSF in the The spectrum of excitations will manifest in the DSF: Study- regime α > 3 is expected to be very close that of the short-range ing the longitudinal xx-structure factor, S xx (q, ω), we observe model. On the other hand, in the range 1 < α < 3, a broad light the gap, and the continuum of excitations (the so-called two- cone is observed and an excitation dispersion which is bounded. particle continuum) that corresponds to the two-particle states This case is of special interest in this work, since trapped-ion we mentioned previously. This observations have been shown, experiments can implement long-range TFIMs in this range, but both numerically (43) and experimentally via neutron scattering also given that it has recently been shown (26, 42, 58) that in this (23). In Fig. 2, we show the xx-DSF for the short-range TFIM, regime the long-range interactions introduce an effective attrac- as obtained from our free-fermionic calculation for J = 1 and tive force between a pair of domain walls. This attractive force B = 1.4, for 50 sites. We clearly observe the two-particle contin- confines the excitations in bound states analogous to the con- uum which characterizes the low-energy fluctuations, as well as finement of mesons in high-energy physics (26, 42, 58). Since the excitation gap at the point q = 0, ω ∼ π/4 (in units of J ). this exotic physics can be probed studying the confinement sig- natures in both the unequal time correlators and DSFs, our work Long-Range TFIM. We can now concentrate on the case which opens the door to the study of these effects in quantum simula- is our test of a practical application: the long-range TFIM. tors. Finally, we mention that for α < 1, the light cone completely Models with these kind of long-range interactions present con- disappears and a virtually instantaneous spread of correlations is siderable challenges to numerical studies. The long-range inter- observed. In Fig. 3, we show the DSF of the long-range TFIM, as actions severely constrain the system sizes which can be studied obtained numerically from a full exact diagonalization of a sys- with exact diagonalization techniques based on sparse matri- tem of 14 spins at zero temperature, for the interaction lengths ces. Furthermore, studies of these systems employing finite-size α = 1 (Fig. 3A), 2 (Fig. 3B), and 3 (Fig. 3C). In this figure, we MPS-based techniques (44, 45) are affected by severe finite-size see that for α = 1, the DSF shows no ω-dependence, which hints effects arising from the entanglement cutoffs required by these at the possibility of excitation confinement (26, 42, 58) being approaches. Recently, however, there has been success in study- evidenced through the DSF. For α > 2, the ω dependence is ing the statics of long-range models employing MPS algorithms, recovered, slowly approaching the short-range behavior as α is which directly act in the thermodynamic limit, such as iDMRG increased. (44, 46). At the same time, several algorithms which can time-evolve Imperfection Models. Three basic ingredients are needed to sim- an MPS with long-range interactions have been proposed (47, ulate DSFs on near-term devices: First, we need to be able to 48) to study the long-range TFIM (49). With the advent of prepare the ground state of the target Hamiltonian in a con- these new techniques, and the state-of-the-art quantum simula- trolled way, and ideally with as high-state fidelity as possible. tors capable of implementing long-range TFIMs, the question of Second, we need to be able to control the time evolution of the whether the DSF of this class of models can be accessed with system, in such a way that the physics we desire to investigate these experimental architectures naturally arises. is not severely mitigated by experimental imperfections. And,

A B C

Fig. 3. DSF for the long-range TFIM, for the cases α = 1 (A), 2 (B), and 3 (C). For the case α = 1, we see that the DSF does not vary in frequency, which is a possible signature of excitation confinement, in accordance with refs. 26 and 42. For α > 2, the two-particle continuum is noticeable, and the gap is lowered. As the value of α is increased, the interactions become shorter-range, and the gap approaches the value for the short-range model. Accordingly, the continuum changes shape, from the absence of ω dependence for α = 1 toward the cosine form at α = 3. This cosine shape corresponds to the short-range TFIM, obtained in the limit α → ∞. For comparison with the short-range TFIM, please refer to Fig. 2.

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Ising case, the the of first affects modulation the time-dependent plings In a how Hamiltonian. study TFIM will we the over effects the damental not function is Green’s state retarded imper- the prepared this recover Eq. the still via by if can modified we even state, not that ground one, such is than model, protocol with smaller fection measurement fidelity state a The ground has 1. state the prepared to the respect when will DSFs we measuring of which imperfections, own one their Every carries separately. functions. proto- consider Green steps measurement time these proposed unequal of the the determine employ to to col want we finally, etlipretosi nteDF ewl nlz w particu- two analyze will we DSF, the on is imperfections mental Imperfections. Quantifying model given a is. the of behavior their via excitations what low-energy simulation and the are quantum of a some from what and DSF, extrapolate, change, not safely does (when DSF can the control one of form below experimental overall set the of Thus, is behaved. level studied parameter current errors imperfection the the our behave, at they how that, excita- and of low-energy show system, behavior about given information overall a one of quantita- the gives tions and Since what qualitative is experiments. DSF the such the both of trust TFIM results can long-range tive and one short- and both small, for are DSF a in the state- produced in of be present would measurement which control errors of the levels architectures, experimental of-the-art the at that, show a is (59) simulation certifica- quantum the the task. such; of separate correctness as actual impact the their of and discuss tion imperfections we of Here, even long- role imperfections. simulation and experimental the quantum of short- a presence from the and the recovered both in qualitative be for the can DSF TFIM that the range demonstrate of will features we quantitative following, the In the of DSF TFIM the Short-Range on Imperfections Experimental of Effect disorder 100 and 50 point. between data employ per We realizations interval 0.4. and the 0.1, on 0.04, 0.01, distribution uniform a from cases, all In the in fields, While transverse respectively. random models, of long-range case and short- the for form the take the couplings cases, Ising the these interactions, Eq. In in form imperfections. the lattice takes Hamiltonian to related fields, netic frequencies different and 0.5, amplitudes, and modulation several study will We with J i ntecs feouinipretos ewl td he fun- three study will we imperfections, evolution of case the In effect the study will we imperfections, preparation the For opeoeooial n reysmaietersls We results: the summarize briefly and phenomenologically To mag- and interactions random of case the study also will We ,j J = 0 = (0) |i 7. J − (0) j J J | ξ i α o h og n hr-ag oes respectively. models, short-range and long- the for , ,i 1+ (1 sdanidpnetya adma ahsite, each at random at independently drawn is ±1 (J = A sin(wt + B Aξ oass htteefc fexperi- of effect the what assess To i = )), i ,j B ), + J ω u o h aeo random of case the for but 21, Aξ i J ,i ewe .5ad25. and 0.05 between i ±1 ,j i . = = J (i ,teDFi well is DSF the 5%), J + 0( + (0)(1 − ,1 [0.0, A Aξ j 0 = ) i α F ,j .5 0.1, 0.05, .01, .0) , = A hψ sin(wt with σ | ψ [24] [23] [22] 0 A )), i < = ie,we h nlfil au sfraa rmtequantum the from away evolution far different is by value affected field point, is critical final DSF the the when how times, an here it via study prepare We state will (35). can initial techniques optimization ions the approximate Rydberg quantum trapped prepare through Furthermore, and can them evolution. ions of adiabatic trapped Both architectures, arrays. DSF. atom two the on on Imperfections concentrate State-Preparation of Influence as the indicated of as is average defined space robustness the are (where and the to space correlators over assess DSF the maximum to the in and error as of The correlators, error transform. the the Fourier compare in will error we the correlators, the over small toward errors see would on we not gap, but the frequencies, close to given quantities were the models of shape their the Eqs. what studying assess by in can DSF we the small, on not is entire are effect errors the errors these over if these hand, flat then other and DSF, small the be account change should to not the If us do separately. allows models space reciprocal way imperfection and this Eq. frequency in obtains in effects one the imperfections for value, the that of at study error The maximal the selects one frequency given a of at value given error, a integrated at frequency consid- the error absolute are the imperfections on space, cut when reciprocal a DSF Eq. makes one way: the If following of ered. the error in absolute understood the be can errors These fixed for by error, in denoted absolute be models the will imperfection of different maximum The for Appendix. error average the and show evolution, time the of discretization where ino ieado h ag fteinteractions the of range by the given error is of error DSF long-range and integrated integrated size the the of of study tion properties will scaling we the model, determine to Finally, imperfections. of absence the in TFIM the where define We DSF. the as of error error absolute absolute the the on based quantities lar fw nert vrfeuny(eirclsae,w banthe (frequency), obtain space we reciprocal space), in (reciprocal error frequency average over in integrate models. perturbations we imperfection various If different the with from arising solution Hamiltonian, exact the the from obtained ic h ore rnfr spromda aaprocessing data as performed is transform Fourier the Since ∆ S S (q N 26 x ω = ) ,x ∆C stenme ffeunis hc eed nthe on depends which frequencies, of number the is (q and N , ∆S J 1 ω ω (t max 1 = ) ∆ o xml,i n fteeimperfection these of one if example, For 27. q X = ) (q q ∆ steDFotie rmteeatslto of solution exact the from obtained DSF the is ω S n nertsi vrfeuny n obtains one frequency, over it integrates and , , [∆S C and = ω ∆S L ω r 1 = ) 2 (t oobtain to , N q (q 1 X B (q = ) ω -space. r |S , 1 = , L ω 1 ω x |C ∆C 2 ] max )], ,x ,∆S ), X .4 r (q x ∆ ω r ,x (t , S (t ∆S X ω (q ,max ), q ω ) ) − − hsi qiaetfrcuts for equivalent is This ). (ω (ω ∆S [∆S C S ˜ NSLts Articles Latest PNAS ˜ = ) f nteohrhand, other the on If, ). L r x r q x (q ,x ,x (q stesse ie We size. system the is S ˜ and (q [∆C , L (t 1 x , ω 2 ,x ω , )|, ). ω X (q )]. ω r q )|, (t ∆S , ag.O the On range. ω hr the where α, ∆S )]. ) steDSF the is safunc- a as (q r ewill We = | , ω ω i f12 of 7 or ), 25 [26] [27] [30] [25] [29] [28] − 27. SI j q is ) ,

PHYSICS Adiabatic time evolution. The question that motivates us is how The adiabatic evolution has been performed for different evo- the features of the DSF change when the system is prepared lution times, ranging from τQ = 0.005 to τQ = 3000. In Fig. 4, we for a time τQ (the preparation time) from an initial polar- show the error analysis for preparation times τQ = 0.5, 1, 10, and ized state (which corresponds to the B → ∞ limit) to the final 100. Following the calculations of ref. 61, we estimate that the state B = 1.4. In the thermodynamic limit, this preparation time fidelity of the prepared state (assuming no other error sources) diverges when one approaches the quantum critical point. For a will correspond to F ∼ 0.043 for a preparation time of τQ = 0.5, finite system, it can be shown that the finite size gap destroys to F ∼ 0.59 for τQ = 10, and F ∼ 0.99 for τQ = 100. Our numer- the divergence, and a finite bound on the preparation time ical error analysis of the DSF and correlators coincides with can be obtained (60, 61) within which the evolution remains these fidelity estimates. We tackle the DSF first: In Fig. 4, Left, adiabatic. we show a typical DSF for a preparation time τQ = 0.5, and in Considering our previous discussion, it is imperative that we Fig. 4, Right, we show the maximum error of the DSF over fre- study how the properties of the DSF change when the prepa- quency (main image) and reciprocal space (Inset). For τQ = 100, ration is not adiabatic. In this case, if the preparation time is the maximum error is below 5% and mostly flat over the entire not large enough to be in the adiabatic regime, the quantum (ω, q) space, indicating that this τQ is enough to obtain an accu- simulation can still be able to obtain results which are close to rate DSF. This can be confirmed by comparing Fig. 4, Left and the physics that one desires to study. This can be understood, Fig. 2. From these figures, we notice that the discrepancy in the and generalized to preparation protocols beyond adiabatic evo- DSF between the exact case and the one studied in this section lutions, in the context of prepared state fidelity. If the fidelity is appears around the point (q = 0, ω = π/4), which corresponds to high, then the properties of the DSF evaluated over the prepared the position of the gap in the clean case. Even for τQ = 0.5, most state will remain close to the properties of the DSF evaluated of the error is constrained around the gap, indicating that the over the exact ground state. Furthermore, if the preparation evo- main contribution to the error in the DSF is a qualitative change lution is not adiabatic, but does not surpass the quantum critical in the overall broadness of the low-q, low-ω sector, even though point, then there exists a time interval in which the transition the overall shape of the DSF does not change (as is seen in probabilities toward excited states is sufficiently small, such that Fig. 4, Left). the main contribution to the state of the system is the ground In the case of unequal time correlators, shown in Fig. 4, Center, state (60–64). At the parameter values studied here, the fidelity we see that the maximal (average in Inset) error in this case, for of the prepared state F can be thought of as the probability that τQ = 100, is also below 5% (1%), but we can see how the aver- no extra domain walls (or, equivalently, kinks) have been created age error increases over time. While looking at the correlators during the preparation. Ref. 61 calculates the fidelity of the final directly could also be a way to study the ground-state fluctua- state with respect to the vacuum of excitations for a linear ramp, tion of the system (given that the effect of the imperfections is effectively probing the probability that no excitations have been small), the interpretation of the data as a function of time can created during the preparation. We find that the fidelity takes be much more challenging, especially for long times. This can be the form understood by considering the propagation of errors as a func- tion of time, which takes place with a maximal velocity consistent 2 with the Lieb–Robinson bounds. We show in SI Appendix, Fig. ln(1 − F ) = −π∆ /(4B/τQ ). [31] 5 the propagation of errors in the correlators as a function of time. With this in mind, we can note that the Fourier transform With this in mind, we can simply ask the question of: How large leading to the DSF allows one to account for all of the spatial does τQ need to be such that F ∼ 1, and what are the effects on and temporal data of the correlators, as well as understand and the DSF when F < 1? deal with errors arising from this imperfection model in a much We quantify the robustness of the DSF to preparation imper- simpler way. fections employing the error measures shown before. For this, we numerically calculate the DSF of the short-range TFIM, when Influence of Evolution Imperfections on DSF. In trapped-ions archi- we prepare the state by a total time τQ , starting with the field at tectures, the spin–spin interactions are created by coupling the Bini → ∞ and finishing at Bfinal = 1.4. spin states to the normal modes of motion of the ions by laser

Fig. 4. Effect of finite preparation times τQ on the DSF of the short-range TFIM. We show numerical results for the DSF and unequal time-correlation functions of the short-range TFIM subject to different preparation time. (Left) DSF of the TFIM for a preparation time τQ = 0.5. This can be directly compared to the imperfection free case shown in Fig. 2. (Center) Cuts of the absolute error for the unequal time-correlation function, depicted for a quantitative comparison. At short preparation times τQ = 0.5 and τQ = 1, we find significant deviations of local correlation functions. Center, Inset shows the averaged error in the correlators, which indicates that the deviation seen in the maximal error persists, and in fact increases, at all times on the level of uniform real-space average. (Right) Cuts in frequency and reciprocal space (Inset) of the DSF absolute error. We quantitatively verify the intuition given by Left. The DSF indeed encodes the correct physical information, despite the deviations in real space and the deformation of the low-ω sector. The low error intensity away from ω = π/4 and q = 0 indicate that the gap remains open for τQ = 100. Absence of errors for long preparation times at q > 0 and ω > π/4 indicate that the two-particle continuum is not affected by these preparation times.

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PHYSICS with time. In this sense, long measurement times can lead to very DSF is negligible, for all imperfection models, as it was found for large errors, rendering the results analyzed purely via correlation the short-range TFIM. functions highly unreliable. In Fig. 6, we show the integrated error (30) as a function of We note that the regime in which randomness is large is inter- the interaction range α for the models corresponding to evo- esting in itself, as it offers the chance to directly probe the lution imperfections. Fig. 6A shows the error for the case of effect of random disorder in spin chains via time-dependent laser-intensity fluctuations, while Fig. 6 B and C show the random observables, and the DSF in particular, in near-term quantum fields and random interactions, respectively. In both cases, we see devices. two regimes, where the error drastically changes for 1 < α < 3, while it stabilizes for α > 3. For the laser-intensity fluctuations, Influence of Experimental Imperfections on the DSF of the the error monotonically increases in the first regime and satu- Long-Range TFIM rates in the second. On the other hand, the opposite behavior is In the previous section, we assessed the accuracy of a DSF mea- observed for the lattice imperfections, where the error decreases surement for the short-range TFIM using quantum simulators as a function of α. and showed that the DSF is well behaved in the presence of While the errors change as the value of α is modified, at the experimental imperfections, even at large system sizes. Now, we imperfection levels present in the current architectures, the inte- will put forward the idea that quantum simulators can recover grated error is negligible, indicating that the DSF at all values the DSF of long-range models in such a way that the deviations of α can be probed by using these setups, and the measurement induced by the error sources do not obscure the overall features would yield accurate results. of the DSF and that this can be done for system sizes larger than what state-of-the-art classical simulations can treat. To show this, System-Size Scaling for Long-Range Models. Now, we concentrate we will numerically study the long-range TFIM in the presence on the scaling properties of the DSF of the long-range TFIM. of the same experimental imperfections as for the short-range We study system sizes ranging from L = 9 to L = 14 sites employ- TFIM. ing full exact diagonalization and analyze how the integrated In Fig. 3, we show the DSF in the absence of imperfections for error changes with size. Since current architectures can simu- three different values of α. In SI Appendix, section IV, we show late up to approximately 50 sites (18, 19), this is a playground the results for α = 2, 3, and 6. Furthermore, in SI Appendix, sec- in which the DSF can be employed to explore the potential tion V, we show the heat maps for the unequal time correlations of dynamical analog quantum simulators. We show the scaling x,x with respect to the middle of the chain, Ci,5 (t). In the regime properties of the error for α = 1.5 and α = 6. In Fig. 7, we show 1 < α < 2, our results exhibit a particular signature in the DSF the integrated error originating from the evolution imperfec- which has no ω−dependence. For higher values of α, remnants tions, as a function of system size, for the two aforementioned of this behavior are noticed, but an ω−dependence is recovered. values of α. For α > 3, we recover the cosine-shaped two-particle continuum. For imperfection levels below 10%, the integrated error is These results, especially the absence of ω-dependence for α < 2, relatively constant over the full range of system sizes studied indicate that the signatures of excitation confinement, which here. When the imperfection level is reduced further, below 5% have been recently proposed (26, 42, 58), can be observed in (within current experimental capabilities), it becomes negligible dynamical quantum simulators via the DSF by employing our for all system sizes and interaction ranges. Our data suggest that proposed method. the error remains constant and small at even the smallest sizes In SI Appendix, section IV, we show a typical case for the max- studied here, indicating that the integrated DSF error is inten- imal error as a function of frequency (reciprocal space) Eq. 26 sive with respect to the system sizes, and, as such, we expect that (Eq. 27) in the case of random transverse fields. There, we can the scaling properties will be maintained for larger chains. Fur- see that the overall behavior of the error is very similar to that thermore, our data indicate that a dynamical quantum simulator for the short-range TFIM. There is a large error around the gap, can measure the DSF of the long-range TFIM accurately, even in with small fluctuations at other values of ω for strong imperfec- the presence of realistic experimental imperfections, for system tions. For small imperfection levels (1 to 5%), the error in the sizes considerably bigger than what is currently achievable with

AB C

Fig. 6. Effect of experimental imperfections in the DSF for the long-range TFIM. Average error, ∆S, as a function of the interaction range, α, for L = 14 sites is shown. (A) Effects of Ising couplings harmonically modulated (Eq. 23). The error is minimal at α = 1, within the confined phase, and saturating at α > 3, when the system approaches the short-range TFIM. (B and C) Effect of lattice imperfection for random fields (B) and random interactions (C). For both these cases, the effect is the same: The DSF is highly susceptible to randomness at low values of α, and it monotonically becomes more robust as α is increased, recovering the short-range behavior for α → ∞. A physical interpretation of the effect on the excitations of these imperfection models on the DSF is beyond the scope of this work, as the presence of excitation confinement in the long-range model has been proposed recently (26, 42, 58). At the experimental levels of control currently available, the integrated error is minimal, and the overall shape of the DSF is unchanged, indicating that quantum simulators can probe the regime of interactions studied in this work, 1 < α < 6, and obtain accurate DSFs for system sizes bigger than what state-of-the-art classical algorithms can achieve.

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A F vrg ro nteDFaiigfo admtases ed.Fralteecss tteeprmna ee fcnrlover control of level experimental the at cases, these all For fields. transverse random from arising DSF the in error Average ) and < tbecomes it 20%, below is level imperfection the When sizes. of range whole the along constant and small is error the 5%, eso h fet fhroial ouae sn neatos ( B interactions. Ising modulated harmonically of effects the show We D) EF B A–C α ag mlydi hswork. this in employed range eso h eut o h neato exponent interaction the for results the show we , rn 546adUiest fGaaaRsac n nweg Transfer Knowledge and Research Skłodowska-Curie Horizon Granada Marie of Union’s Fund–Athenea3i. the University European and under 754446 the program Quan- Grant innovation by Large-Scale and supported research Atomic was 2020 innovation (Programmable J.B.-V. was and work Simulation). 817482 research This tum 2020 Grant MATH+. Horizon and under Union 2724; program European FOR by and EI supported 183, Grants also CRC discus- Forschungsgemeinschaft 519/15-1, EI Deutsche for Foun- 519/14-1, (Taming Institute; the Boche Council Foundation; Questions Research Templeton Holger European dational the the thanks Systems); by Quantum supported J.E. Non-Equilibrium was Thierry work discussions. and This Bauer, fruitful sions. Bela Budich, for Carl Jan Giamarchi Silvi, Pietro Roushan, Pedram Luitz, ACKNOWLEDGMENTS. information. Availability. Data time a for chain, boundary to up in described as to nique, up sizes system for physical Method. other Methods and in Materials quantity this of quantum stim- contexts. assessments work of present further the realm that the ulates hope We into and (71). devices simulators physics technological quantum condensed-matter in place time-dependent assess importance further to key tool of useful quantities a provides long- simulators the tum in confinement excitation the sufficient be and TFIM. would DSF range times the coherence sys- the probe the for atoms, even to as However, 100 of (19). of grows ratio atoms sizes of TFIM the tem number ions, the long-range of trapped the root for square for while emission 70), the (69, spontaneous with atoms inversely reported approximately of been scales has number it time atoms, coherence Rydberg the of implemented. that case sizes the system in exper- the example, the on For of depends time directly coherence achieved which be the iment, can on that depend times will evolution experimentally the that out pointing worth eteeoeageta h esrmn fDF nquan- in DSFs of measurement the that argue therefore We L = eepoe xc ignlzto osuyteln-ag TFIM long-range the study to diagonalization exact employed We 0sts nbt ae,tetm vlto a oei nopen an in done was evolution time the cases, both In sites. 50 l td aaaeicue nteatceadsupporting and article the in included are data study All IAppendix SI L ...tak at ens aksHy,David Heyl, Markus Kennes, Dante thanks M.L.B. = ∆S C 4sts hl eepoe refrinctech- fermionic free employed we while sites, 14 t rsn rmtedfeetipreto models, imperfection different the from arising , and ∈ [0, E L/2] osuytesotrnecs o systems for case short-range the study to , vrg ro nteDFfrtecs of case the for DSF the in error Average ) . 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PHYSICS 1. W. M. C. Foulkes, L. Mitas, R. J. Needs, G. Rajagopal, Quantum Monte Carlo 38. J. Kempe, A. Kitaev, O. Regev, The complexity of the local Hamiltonian problem. SIAM simulations of solids. Rev. Mod. Phys. 73, 33–83 (2001). J. Comput. 35, 1070–1097 (2004). 2. R. M. Noack, S. R. Manmana, Diagonalization and numerical renormalization group 39. D. Gosset, B. M. Terhal, A. Vershynina, Universal adiabatic quantum computation based methods for interacting quantum systems. AIP Conf. Proc. 789, 93–163 (2005). via the space-time circuit-to-Hamiltonian construction. Phys. Rev. Lett. 114, 140501 3. R. Orus, Tensor networks for complex quantum systems. arXiv:1812.04011 (10 (2015). December 2018). 40. S. Lloyd, B. M. Terhal, Adiabatic and Hamiltonian computing on a 2D lattice with 4. E. J. Chaisson, E. Chaisson, Cosmic Evolution (Harvard University Press, Cambridge, simple 2-qubit interactions. New J. Phys. 18, 023042 (2016). MA, 2001). 41. A. Ciani, B. Terhal, D. DiVinzenzo, Hamiltonian quantum computing with supercon- 5. D. L. Cosmogenesis, The Growth of Order in the Universe (Oxford University Press, ducting qubits. Quant. Sc. Tech. 4, 035002 (2019). Oxford, UK, 1991). 42. A. Lerose, B. Zunkoviˇ c,ˇ A. Silva, A. Gambassi, Quasilocalized excitations induced by 6. S. Aaronson, NP-complete problems and physical reality. arXiv:quant-ph/0502072 (12 long-range interactions in translationally invariant quantum spin chains. Phys. Rev. B February 2005). 99, 121112(R) (2019). 1 7. S. Aaronson, BQP and the polynomial hierarchy arXiv:0910.4698 (25 October 2009). 43. O. Derzhko, T. Krokhmalskii, Dynamic structure factor of the spin- 2 transverse Ising 8. R. Ran, A. Tal, “Oracle separation of BQP and PH” in Proceedings of the 51st chain. Phys. Rev. B 56, 11659–11665 (1997). Annual ACM SIGACT Symposium on Theory of Computing (Association for Computing 44. G. M. Crosswhite, A. C. Doherty, G. Vidal, Applying matrix product operators to model Machinery, New York, 2019), pp. 13–23. systems with long-range interactions. Phys. Rev. B 78, 035116 (2008). 9. M. Marvian, D. A. Lidar, I. Hen, On the computational complexity of curing the sign 45. D. Jaschke, K. Maeda, J. D. Whalen, M. L. Wall, L. D. Carr, Critical phenomena and problem. Nat. Commun. 10, 1571 (2019). Kibble–Zurek scaling in the long-range quantum Ising chain. New J. Phys. 19, 033032 10. J. Klassen et al., Hardness and ease of curing the sign problem for two-local qubit (2017). Hamiltonians. arXiv:1906.08800 (20 June 2019). 46. S. N. Saadatmand, S. D. Bartlett, I. P. McCulloch, Phase diagram of the quantum Ising 11. D. Hangleiter, I. Roth, D. Nagaj, J. Eisert, Easing the Monte Carlo sign problem. model with long-range interactions on an infinite-cylinder triangular lattice. Phys. arXiv:1906.02309 (5 June 2019). Rev. B 97, 155116 (2018). 12. J. I. Cirac, P. Zoller, Goals and opportunities in quantum simulation. Nat. Phys. 8, 264– 47. J. Haegeman et al., Time-dependent variational principle for quantum lattices. Phys. 266 (2012). Rev. Lett. 107, 070601 (2011). 13. J. Eisert, M. Friesdorf, C. Gogolin, Quantum many-body systems out of equilibrium. 48. M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore, F. Pollmann, Time-evolving a Nat. Phys. 11, 124–130 (2015). matrix product state with long-ranged interactions. Phys. Rev. B 91, 165112 (2015). 14. T. Fukuhara et al., Microscopic observation of magnon bound states and their 49. T. Hashizume, I. P. McCulloch, J. C. Halimeh, Dynamical phase transitions in the two- dynamics. Nature 502, 76–79 (2013). dimensional transverse-field Ising model. arXiv:1811.09275 (22 November 2018). 15. T. Fukuhara et al., Quantum dynamics of a mobile spin impurity. Nat. Phys. 9, 235–241 50. Z. Zhu, G. Sun, W.-L. You, D.-N. Shi, Fidelity and criticality of a quantum Ising chain (2013). with long-range interactions. Phys. Rev. A 98, 023607 (2018). 16. M. Cheneau et al., Light-cone-like spreading of correlations in a quantum many-body 51. D. Amit, J. K. Bhattacharjee, Phase transitions in the quantum Ising and rotor models system. Nature 481, 484–487 (2012). with a long-range interaction. Phys. Rev. B 64, 184106 (2001). 17. J. Simon et al., Quantum simulation of antiferromagnetic spin chains in an optical 52. T. Blanchard, M. Picco, M. A. Rajabpour, Influence of long-range interactions on the lattice. Nature 472, 307–312 (2011). critical behavior of the Ising model. EPL 101, 56003 (2013). 18. H. Bernien et al., Probing many-body dynamics on a 51-atom quantum simulator. 53. G. Piccitto, B. Zunkoviˇ c,ˇ A. Silva, Dynamical phase diagram of a quantum Ising chain Nature, 551, 579–584 (2017). with long-range interactions. Phys. Rev. B 100, 180402 (2019). 19. J. Zhang et al., Observation of a many-body dynamical phase transition with a 53- 54. M. Kastner, Diverging equilibration times in long-range quantum spin models. Phys. qubit quantum simulator. Nature 551, 601–604 (2017). Rev. Lett. 106, 130601 (2011). 20. J. G. Bohnet et al., Quantum spin dynamics and entanglement generation with 55. I. Homrighausen, N. O. Abeling, V. Zauner-Stauber, J. C. Halimeh, Anomalous dynami- hundreds of trapped ions. Science 352, 1297–1301 (2016). cal phase in quantum spin chains with long-range interactions. Phys. Rev. B 96, 104436 21. R. Islam et al., Onset of a quantum phase transition with a trapped ion quantum (2017). simulator. Nat. Commun. 2, 377 (2011). 56. A. S. Buyskikh, M. Fagotti, J. Schachenmayer, F. Essler, A. J. Daley, Entanglement 22. S. Trotzky et al., Probing the relaxation toward equilibrium in an isolated strongly growth and correlation spreading with variable-range interactions in spin and correlated one-dimensional Bose gas. Nat. Phys. 8, 325–330 (2012). fermionic tunneling models. Phys. Rev. A 93, 053620 (2016). 23. R. Coldea et al., Quantum criticality in an Ising chain: Experimental evidence for 57. M. B. Hastings, T. Koma, Spectral gap and exponential decay of correlations. emergent E8 symmetry. Science 327, 177–180 (2010). Commun. Math. Phys. 265, 781–804 (2006). 24. C. J. Jia et al., Persistent spin excitations in doped antiferromagnets revealed by 58. R. Verdel, F. Liu, S. Whitsitt, A. V. Gorshkov, M. Heyl, Real-time dynamics of string resonant inelastic light scattering. Nat. Commun. 5, 3314 (2014). breaking in quantum spin chains. arXiv:1911.11382 (26 November 2019). 25. R. Landig, F. Brennecke, R. Mottl, T. Donner, T. Esslinger, Measuring the dynamic struc- 59. J. Eisert et al, Quantum certification and benchmarking. Nat. Rev. Phys. 2, 382–390 ture factor of a quantum gas undergoing a structural phase transition. Nat. Commun. (2020). 6, 7046 (2015). 60. J. Dziarmaga, Dynamics of a quantum phase transition: Exact solution of the quantum 26. F. Liu et al., Confined quasiparticle dynamics in long-range interacting quantum spin Ising model. Phys. Rev. Lett. 95, 245701 (2005). chains. Phys. Rev. Lett. 122, 150601 (2019). 61. W. H. Zurek, U. Dorner, P. Zoller, Dynamics of a quantum phase transition. Phys. Rev. 27. D. J. Luitz, Y. Bar Lev, Emergent locality in systems with power-law interactions. Phys. Lett. 95, 105701 (2005). Rev. A 99, 010105(R) (2019). 62. L. Cincio, J. Dziarmaga, M. M. Rams, W. H. Zurek, Entropy of entanglement and corre- 28. K. R. Fratus, M. Srednicki, Eigenstate thermalization and spontaneous symme- lations induced by a quench: Dynamics of a quantum phase transition in the quantum try breaking in the one-dimensional transverse-field Ising model with power-law Ising model. Phys. Rev. A 75, 052321 (2007). interactions arXiv:1611.03992 (12 November 2016). 63. L. Cincio, J. Dziarmaga, J. Meisner, M. M. Rams, Dynamics of a quantum phase transi- 29. P. Hauke, L. Tagliacozzo, Spread of correlations in long-range interacting quantum tion with decoherence: Quantum Ising chain in a static spin environment. Phys. Rev. systems. Phys. Rev. Lett. 111, 207202 (2013). B 79, 094421 (2009). 30. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, UK, 64. D. M. Kennes, C. Karrasch, A. J. Millis, Loschmidt-amplitude wave function spec- ed. 2, 2011). troscopy and the physics of dynamically driven phase transitions. arXiv:1809.00733 31. P. Roushan et al., Spectroscopic signatures of localization with interacting photons in (3 September 2018). superconducting qubits. Science 358, 1175–1179 (2017). 65. K. Kim et al, Quantum simulation of frustrated Ising spins with trapped ions. Nature 32. D. Barredo et al., Coherent excitation transfer in a spin chain of three Rydberg atoms. 465, 590–593 (2010). Phys. Rev. Lett. 114, 113002 (2015). 66. E. E. Edwards et al, Quantum simulation and phase diagram of the transverse-field 33. M. Knap et al., Probing real-space and time-resolved correlation functions with many- Ising model with three atomic spins. Phys. Rev. B 82, 060412(R) (2010). body Ramsey interferometry. Phys. Rev. Lett., 111, 147205 (2013). 67. S. Schneider, G. J. Milburn, Decoherence in ion traps due to laser intensity and phase 34. B. T. Yoshimura, J. K. Freericks, Measuring nonequilibrium retarded spin-spin Green’s fluctuations. Phys. Rev. A 57, 3748–3752 (1998). functions in an ion-trap-based quantum simulator. Phys. Rev. A 93, 052314 (2016). 68. A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, T. Schaetz, Simulating a quantum 35. G. Pagano et al., Quantum approximate optimization with a trapped-ion quantum magnet with trapped ions. Nat. Phys. 4, 757–761 (2008). simulator. arXiv:1906.02700 (6 June 2019). 69. M. Ebert, M. Kwon, T. G. Walker, M. Saffman, Coherence and Rydberg blockade of 36. M. Baiesi, C. Maes, B. Wynants, Fluctuations and response of nonequilibrium states. atomic ensemble qubits. Phys. Rev. Lett. 115, 093601 (2015). Phys. Rev. Lett. 103, 010602 (2009). 70. H. Levine et al, High-fidelity control and entanglement of Rydberg-atom qubits. Phys. 37. E. A. Calzetta, B.-L. B. Hu, Nonequilibrium Quantum Field Theory (Cambridge Rev. Lett. 121, 123603 (2018). Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 71. A. Acin et al, The European quantum technologies roadmap. New J. Phys. 20, 080201 2008). (2018).

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