Can one trust quantum simulators?

7/27/2019 Can one trust quantum simulators?

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Can One Trust Quantum Simulators?

Philipp Hauke,1, Fernando M. Cucchietti,1, 2 Luca Tagliacozzo,1 Ivan Deutsch,3, 4 and Maciej Lewenstein1, 5

1ICFO Institut de Cincies Fotniques, Parc Mediterrani de la Tecnologia, 08860 Castelldefels, Spain2Barcelona Supercomputing Center (BSC-CNS), Edificio NEXUS I,

Campus Nord UPC, Gran Capitn 2-4, 08034 Barcelona, Spain3Center for Quantum Information and Control (CQuIC)

4Department of Physics and Astronomy, University of New Mexico, Albuquerque NM 871315

ICREA Instituci Catalana de Recerca i Estudis Avanats, Lluis Companys 23, E-08010 Barcelona, Spain(Dated: July 27, 2012)

Various fundamental phenomena of strongly-correlated quantum systems such as high-Tc super-conductivity, the fractional quantum-Hall effect, and quark confinement are still awaiting a univer-sally accepted explanation. The main obstacle is the computational complexity of solving even themost simplified theoretical models that are designed to capture the relevant quantum correlationsof the many-body system of interest. In his seminal 1982 paper [1], Richard Feynman suggestedthat such models might be solved by simulation with a new type of computer whose constituentparts are effectively governed by a desired quantum many-body dynamics. Measurements on thisengineered machine, now known as a quantum simulator, would reveal some unknown or difficultto compute properties of a model of interest. We argue that a useful quantum simulator mustsatisfy four conditions: relevance, controllability, reliability, and efficiency. We review the currentstate of the art ofdigital and analog quantum simulators. Whereas so far the majority of the focus,both theoretically and experimentally, has been on controllability of relevant models, we emphasize

here the need for a careful analysis of reliability and efficiency in the presence of imperfections. Wediscuss how disorder and noise can impact these conditions, and illustrate our concerns with novelnumerical simulations of a paradigmatic example: a disordered quantum spin chain governed bythe Ising model in a transverse magnetic field. We find that disorder can decrease the reliabilityof an analog quantum simulator of this model, although large errors in local observables are intro-duced only for strong levels of disorder. We conclude that the answer to the question Can we trustquantum simulators? is... to some extent.

I. INTRODUCTION

In his 1982 foundational article [1], Richard Feynmansuggested that the complexities of quantum many-bodyphysics might be computed by simulation. By design-

ing a well-controlled system from the bottom up, onecould create a computer whose constituent parts aregoverned by quantum dynamics generated by a desiredHamiltonian. Measuring the properties of this nano-engineered system thus reveals some unknown or difficultto compute properties of a quantum many-body model,such as the nature of quantum phase diagrams. Feyn-mans machine is now known as a quantum simulator(QS).

Fueled by the prospect of solving a broad range oflong-standing problems in strongly-correlated systems,the tools to design, build, and implement QSs [13] haverapidly developed and are now reaching very sophisti-cated levels [4]. Researchers are making breakthroughadvances in quantum control of a variety of systems,including ultracold atoms and molecules ([517], for re-views see also [1820]), ions ([2127], for recent reviewssee [20, 2830]), photons ([3136], for a recent review see[37]), circuit quantum electrodynamics (CQED) and po-laritons ([3842], for a recent review see [43]), artificial

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lattices in solid state [44], nuclear magnetic resonance(NMR) systems [4550], and superconducting qubits (forreviews see [5154], for the current state of art [55, 56]and references therein). For a general overview see also[57]. At the current pace, it is expected that we will

soon reach the ability to finely control many-body sys-tems whose description is outside the reach of a classicalcomputer. For example, modeling interesting physics as-sociated with a quantum system involving 50 spin-1/2particles whose general description requires 250 1015complex numbers is out of the reach of current classicalsupercomputers, but perhaps within the grasp of a QS.

In a field brimming with excitement, it is important tocritically examine such high expectations. Real-world im-plementations of a quantum simulation will always faceexperimental imperfections, such as noise due to finiteprecision instruments and interactions with the environ-ment. Feynmans QS is often considered as a fundamen-tally analog device, in the sense that all operations arecarried out continuously. However, errors in an analogdevice (also continuous, like temperature in the initialstate, or the signal-to-noise ratio of measurement) canpropagate and multiply uncontrollably [58]. Indeed, Lan-dauer, a father of the studies of the physics of informa-tion, questioned whether quantum coherence was trulya powerful resource for computation because it requireda continuum of possible superposition states that were

analog in nature [59].

This contrasts with the operation of a universal digi-

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tal quantum computer as envisioned by David Deutsch,in which all operations are digitized into a finite set oflogic gates and measurements [60] [61]. The inventionof quantum-error-correcting codes showed that a quan-tum computer is in some sense both analog and digital.Through a discrete set of unitary transformations, we canget arbitrarily close to any superposition, and imperfec-tions can always be projected on a discrete set and thus

can be corrected [62]. When such a digital quantum sim-ulation operates with fault-tolerant quantum error cor-rection [63], we can trust its output to a known finiteprecision.

Universal digital quantum computers may serve as dig-ital QSs (DQSs) that mimic dynamics of some quantummany-body system of interest. Despite the fact that insuch a case error correction and fault tolerance is guaran-teed, the question of efficiency of such a device is highlynon-trivial. The number of resources needed for precisesimulation of continuous-time dynamics of a many-bodysystem by stroboscopic digital applications of local gatesmight be enormous [64]. One can also consider DQSs

that are experimental systems that have at their disposalonly a limited, non-universal set of gates. In such a sit-uation, the error correction and fault tolerance are notguaranteed and the question of efficiency is even morepertinent.

This raises the central problem of this key issue arti-cle: can we trust the results obtained with a real-worldanalog or digitalQS, and under what conditions are theyreliable to a known degree of uncertainty? Although ourmain discussion concentrates on analog QSs (AQSs), thearticle reports also on the state of art of DQSs. It is orga-nized as follows. Section II develops the general conceptof QSs in the spirit of the DiVincenzo criteria for quan-tum computing [65]. Here we present one of the mainresults of this article: a definition of the QS based onfour properties that a QS should have: relevance, con-trollability, reliability, and efficiency.

Section III is devoted exclusively to DQSs. It containsseveral subsections in which we review various proposalsfor DQSs, classify them, and discuss the present stateof knowledge concerning their controllability, reliability,and efficiency. Section IV is organized similarly, but fo-cused on AQSs.

Section V is perhaps the most important one from theconceptual point of view. Here, we formulate specificproposals to investigate the robustness of AQS and howto extend standard methods of validation and certifica-

tion of AQS. We illustrate these considerations in Sec-tions VI-VIII with calculations for a paradigmatic modelthat are not published elsewhere. Section VI describesthe investigated model, and the results concerning stat-ics and dynamics are presented in Sections VII and VIII,respectively. We conclude in Section IX. The paper con-tains also an Appendix describing technical details of themethods used.

Very recently, Nature Physics has published a focusissue with five articles devoted to QSs: a short article by

J.I. Cirac and P. Zoller [66], introducing the subject, andfour longer reviews on ultracold atoms [19], ions [30],CQED [67], and photons [37]. Our key issue article iscomplementary to these, in the sense that it addressesgeneral and universal problems of validation of quantumsimulations, their robustness, reliability, and efficiency problems that pertain to all kinds of QSs.

II. QUANTUM SIMULATORS

Before proceeding, we must establish a clear definitionof a QS. We consider here a QS to be a device which,when measured, reveals features of an ideal mathemati-cal model, e.g., the phase diagram for the Bose-Hubbardmodel on a specified lattice with specified interactions.This contrasts with, and is less demanding than, a fullsimulation of a real material, since typically a mathemat-ical model attempts to capture only the most relevantproperties of the real material. For example, the super-conducting properties of a cuprate might be shared, in

part, by a Fermi-Hubbard model [68, 69]. A QS may bea special purpose device that simulates a limited classof models, e.g, the Bose-Hubbard model simulated byatom transport in an optical lattice [70, 71], or a univer-sal machine that is capable, in principle, of simulatingany Hamiltonian on a finite-dimensional Hilbert space.

Based on this, we formulate the following workingdefinition of a quantum simulator in the spirit of the Di-Vincenzo criteria for quantum computing [65], providingsome more detailed explanations below (see also [3]):

Definition

A QS is an experimental system that mimics a simplemodel, or a family of simple models of condensed mat-ter (or high-energy physics, or quantum chemistry, . . . ).A quantum simulator should fulfill the following four re-quirements:

(a) Relevance: The simulated models should beof some relevance for applications and/or our un-derstanding of challenges in the areas of physicsmentioned above.

(b) Controllability: A QS should allow for broadcontrol of the parameters of the simulated model,and for control of preparation, initialization, ma-

nipulation, evolution, and detection of the relevantobservables of the system.

(c) Reliability: Within some prescribed error,one should be assured that the observed physics ofthe QS corresponds faithfully to that of the idealmodel whose properties we seek to understand.

(d) Efficiency: The QS should solve problemsmore efficiently than is practically possible on aclassical computer.


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digital analog

universal non-univ. open universal non-universal open

Realizations trapped ions,ultracold neutralor Rydberg atoms,circuit QED, super-

conducting qubits, . . .

same asuniversaldigital

as universaldigital (especiallytrapped ions,

ultracold neutral

or Rydberg atoms)

? many (trapped ions,ultracold atoms, pho-

tonic and polariton

systems, artificial

solid-state lattices, . . . )

same asnon-universalanalog

Control full (long-range inter-actions difficult ?)

partial partial full partial (but long-rangeinteractions easy)

partial

Errorcorrection(EC)

with exponentialoverhead(Trotterization issues)

notguaranteed

notguaranteed

nostandardEC

nostandardEC

nostandardEC

Reliability full notguaranteed

notguaranteed

? ? (partialvalidation

schemes

available)

? (partialvalidation

schemes

available)

Efficiency efficient without EC(for general class of

models); much lessefficient with EC(Trotterization issues)

at least asuniversaldigital, butmay not beprovable

can be betterthan universaldigital

? ? ?

Table I. In this table, we characterize the different classes of quantum simulators (digital vs. analog, universal vs. non-universal,Hamiltonian vs. open) focussing especially on the requirements (b) to (d). (Since the relevance (a) depends on the concretemodel simulated, we do not list it here.) Note specifically that little is known about reliability and efficiency of AQSs (questionmarks). Detailed descriptions are provided in Sections III and IV.

In Table I, we summarize to which extent existing

experimental proposals fulfill these requirements. Wewill characterize the different types of QSs in more detailin the following two sections, but before that, we wouldlike to make some general comments.

Comments ad a) We should demand that the mim-icked models are not purely of academic interest but thatthey rather describe some interesting physical systemsand solve open problems. This means also that thesimulated models should be computationally very hardfor classical computers (see also requirement (d)).

Comments ad b) and c) Regarding control over mea-surable observables, one should stress that very often theamount of output information required from quantumsimulators might be significantly smaller than one coulddemand from a universal quantum computer. Quantumsimulators should provide information about phase dia-grams, correlation functions, order parameters, perhapseven critical exponents or nonlocal hidden order param-eters. But a common assumption is that these quanti-ties are more robust than what is required for a univer-sal quantum computer, which typically relies on much

higher-order correlation functions than a QS.

Regarding control over model parameters, it is inparticular desirable to be able to set the parameters ina regime where the model becomes tractable by clas-sical simulations, because this provides an elementaryinstance of validating the QS. Furthermore, one of themain results of this paper is the proposal and analysis ofan even more sophisticated manner of validation, namelythe checking of the sensitivity of the quantum simulationwith respect to addition of noise and/or disorder. Sucha validation is only possible with sufficient control overthe system. Note, however, that there are other possi-bilities of checking the results, as pointed out to us byZ. Hadzibabic [72]. Namely, sometimes it is impossible

to simulate the system classically, but it might still bepossible by classical means to test the sensibleness of thequantum-simulation results. For instance, the measuredground-state energy should fulfill all known bounds,such as variational ones, and others.

Comments ad d) The notion computationally veryhard for classical computers may have several meanings:i) an efficient (scalable, with polynomial growth in re-sources as a function of problem size) classical algorithm


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to simulate the model might not exist, or might not beknown; ii) the efficient scalable algorithm is known, butthe required size of the simulated model is too large tobe simulated under reasonable time and memory restric-tions. The latter situation, in fact, starts to occur withthe classical simulations of the Bose or FermiHubbardmodels [73], in contrast to their experimental quantumsimulators. However, there might be exceptions to the

general rules. For instance, it is desirable to realize QSsto simulate and to observe novel phenomena that so farare only theoretically predicted, even though it mightbe possible to simulate these phenomena efficiently withpresent computers. Simulating and actually observingin the lab is more than just simulating abstractly on aclassical computer.

Comments ad c) and d) The requirements of relia-bility and efficiency are interrelated. In fact, we couldtry to improve the precision of a QS by averaging moreexperiments, but in hypersensitive regimes (like closeto quantum phase transitions) the necessary numberof repetitions can grow rapidly, bringing the overallefficiency of the QS down to the level of classical com-puters. A connection between (c) and (d) could also berelevant for the popular cross-validation approach [74].There, one compares the results of two different physicalrealizations performing a QS of the same model, andhopes to find universal features which then would beascribed to the simulated model. It may be, however,that the universal features shared by multiple platformsare robust only because they could have been predictedefficiently with some classical algorithm.

With our working definition at hand, one could ask What should a QS simulate? An important set of tasks

include:

1. Statics of the mimicked system at zero tempera-ture; this implies ground-state simulation and itsproperties.

2. Statics at thermal equilibrium, i.e., Hamiltoniandynamics at low energies or thermodynamics atnon-zero, typically low, temperatures.

3. Continuous-time dynamics of the system, in partic-ular Hamiltonian dynamics out of equilibrium.

4. Dissipative or open-system continuous-time dy-namics.

To understand which of these are most relevant, wenow discuss shortly which systems can be simulated effi-ciently classically and which systems are classically com-putationally hard. Classical simulations of quantum sys-tems are currently performed using one of the followingnumerical methods [3]:

Quantum Monte Carlo (QMC) Systematic perturbation theory

Exact diagonalizations Variational methods (mean field methods, density-

functional theory (DFT), dynamical mean-fieldtheory (DMFT), tensor-network states (TNS),density-matrix renormalization group (DMRG),tree tensor network states (TTN), multiscaleentanglement-renormalization ansatz (MERA),projected entangled-pairs states (PEPS), ...)

Each of these methods has its limitations. Let us firstfocus on points 1) and 2) of the previous list of possibleQS tasks. In these cases, QMC works for various largesystems, but fails for Fermi or frustrated systems dueto the famous sign problem [75]. Perturbation theoryworks only if there exists a small expansion parameter[76]. Exact diagonalization works only for rather smallsystems [75]. In the case of 1D systems, DMRG, MERAand TTN techniques scale favorably and can, in principle,treat very large systems [7779]. In 2D the situation ismore complex similar to exact diagonalization, DMRGand TTN work only for reasonably small systems [8082],whereas 2D tensor-network methods (PEPS, MERA) inprinciple work for arbitrarily big systems (bosonic, andeven fermionic [83] or frustrated [84]) but are biased to-wards slightly entangled states. Mean field [85], DFT[86, 87], or DMFT [88], finally, have other limitations,e.g., they are essentially designed for weakly-correlatedsystems.

Which are then the models that are computationallyhard for points 1) and 2) in the previous task list? Gen-erally speaking, computationally hard are those stronglyentangled models in more than 1D such as

Fermionic models, with paradigmatic examples be-ing the Fermi-Hubbard or tJ models for spin 1/2fermions [68].

Frustrated models, with paradigmatic examples be-ing antiferromagnetic Heisenberg or XY models ona kagom or anisotropic triangular lattice [89].

Disordered models, with paradigmatic models be-ing quantum, or even classical spin glasses [90].

When we move to points 3) and 4) of the task list, i.e.,studying dynamics, one can safely state that

Quantum dynamics on a long time scale is generi-cally computationally hard.

The latter statement implies that while it might be pos-sible to simulate with classical computers short-time dy-namics in a restricted class of 1D models, such attemptswill nearly always fail at longer time scales. Indeed, thisfact is related to correlation and entanglement spreadingaccording to the LiebRobinson theorem that states that,after a sufficiently large time, states can become stronglyentangled ([9196], see also [97]).

In the following two sections, we will explore in moredetail the state of the art concerning the four require-ments (a-d) of our definition, first for digital, then foranalog QSs.


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III. DIGITAL QUANTUM SIMULATORS (DQS)

In this section, we classify DQSs, discuss their generalproperties, various protocols for implementing such de-vices, and summarize state-of-art knowledge concerningtheir controllability, reliability, and efficiency.

A. Universal Digital Quantum Simulators (UDQS)

While the concept of QSs should be traced back toprophecies of Feynman [1], the ideas were made concreteby Lloyd who showed that any local many-body uni-tary evolution governed by a local Hamiltonian couldbe implemented by the control afforded by a universaldigital quantum computer [98]. For this reason, in thefollowing we will term Lloyds DQS a universal DQS(UDQS).

Lloyds UDQS is in fact a universal quantum computer,whose task is to simulate the unitary time-evolution op-erator of a certain quantum system described by a phys-

ical Hamiltonian, which can then be employed to extractquanties like energy gaps and ground-state properties.This is done by appropriate subsequent stroboscopic ap-plications of various quantum gates that mimic the actionof a global unitary continuous time evolution operator ofthe system. The mathematical basis for such a digital-ization is given by the TrotterSuzuki formula. In orderto realize Lloyds UDQS in a laboratory, the experimen-talist has to have to his/her disposal a universal set ofunitary quantum gates [99]. Let us list below some pos-sible realizations and properties of UDQSs:

Realizations: While implementation of a fully-functioning large-scale digital quantum computer is

still in development, there are several physical sys-tems for which the universal sets of quantum gatesare available, and for which realization of proof-of-principle UDQSs is possible. These systems in-clude ultracold ions [30], ultracold trapped atomsinteracting via cold collisions [19], or the Rydberg-blockade mechanism [100, 101], circuit QED [67],superconducting qubits (for reviews see [5154], forthe current state of art [56] and references therein).For a general overview see also [57]. The firstconcrete proposals for realization of UDQSs wheregiven in [102, 103], and the first experiments, per-haps, were performed in NMR systems [4547, 104].

Using a digital architecture and stroboscopic se-quence of gates, the quantum simulation of Ising,XY, and XYZ spin models in a transverse field wererecently demonstrated in a proof-of-principle exper-iment with up to six ions [24].

Controllability: In accordance with [98], a UDQSis perfectly controllable, i.e., with the help of a uni-versal set of gates sufficient control of the parame-ters can be achieved. This control allows for simula-tion of practically any local Hamiltonian evolution,

as well as for preparation, manipulation, and detec-tion of relevant states and observables of the systemin question. Further, Preskills group has provenrecently that the scattering amplitudes in the sim-ple relativistic quantum field theories can be effi-ciently (in polynomial time) simulated by UDQSs[105, 106]. Note, however, that not much is knownabout the possibility of quantum simulation of sys-

tems with long-range interactions like Coulomb ordipoledipole interactions using UDQSs.

Error correction: A UDQS is the only DQSwhich has guaranteed access to error correctionand fault tolerance [107, 108] (for the first proof-of-principle experiments see [56, 109113]).

Efficiency: So far, the community has mostly fo-cused on developing requirement (b) for suitablerelevant models, both theoretically and experimen-tally. The conditions (c) and (d) have receivedconsiderably less attention, especially their inter-relation. Most work is focused on efficiency in the

absence of errors. Lloyd showed that a TrotterSuzuki decomposition of a time-evolution operatoris efficient in that each logic gate acts on a scal-able Hilbert space associated with a small subsetof qubits and the total number of gates N scalespolynomially, N t2/, where t is the time of evo-lution to be simulated and is the error in the re-sult [98]. Aharanov and Ta-Shma showed that aUDQS is efficient when the Hamiltonian is sparse,i.e., the number of nonzero entries in any row is atmost poly(log(D)), where D is the dimension ofthe many-body Hilbert space [114]. In the absenceof errors, the computational complexity of such a

simulation has been well studied [115, 116].

Reliability: In the presence of errors, however, en-suring reliability to a desired precision has profoundimplications for efficiency even in a digital simula-tor on a fault-tolerant quantum computer [64, 117].In the digital approach with a finite universal gateset, one applies error-correction schemes that canmake the whole computation fault-tolerant whenthe error per operation is below a certain thresh-old thus digital simulators fulfill the reliabilityrequirement (c). The Trotter expansion, however,can scale poorly when error correction is included,as emphasized by Brown et al. [64] in studies ofan implementation of a quantum algorithm to cal-culate the low-lying energy gap in pairing Hamil-tonians [118]. Because the number of gates inthe expansion scales as 1/, in order to achieveM = log2() bits of precision, we must Trotter-ize the unitary evolution to be simulated into 2M

slices. In the presence of errors, each of the timeslices must be implemented with only a finite setof universal gates, according to the Solovay-Kitaevtheorem [119]; only then can they be implemented


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fault tolerantly. The result is that a fault-tolerantimplementation of the Trotter expansion requiresa number of gates and time to perform the simu-lation that grows exponentially with the degree ofprecision required, for a fixed number of particlesbeing simulated. Moreover, Brown et al. showedthat for a small number of qubits where one mightavoid error correction, analog control errors on the

logic gates can lead to faulty results, negating re-quirement (c), and robust control pulses becomeessential.

In a similar vein, Clark et al. [120] performed acareful analysis of the resources necessary to im-plement the Abrams-Lloyd algorithm [121] to cal-culate the ground-state energy of the one dimen-sional transverse Ising model (TIM) using a state-of-the-art fault-tolerant architecture for an ion-trapquantum computer. Again, the overhead in thenumber of time steps to fault-tolerantly implementthe quantum-phase estimation algorithm grows ex-ponentially with the degree of precision required.

They found that for 100 spins, in order to achieveb 10 bits of precision, at least two levels of con-catenated error correction are necessary, requiringat least 100 days of run time on the ion-trap quan-tum computer; for b 18, three levels are nec-essary, requiring at least 7.5 103 years! Theseresults assume a gate time of 10s. To recover the100 days limit with only one level of concatenatederror-correction coding, a gate time of 300ns seemsnecessary (as well as decreasing other parameterssuch as failure probabilities). On the other hand,for a fixed precision, the number of resources re-quired grows weakly with system size. So, if the er-

ror probability per gate can be reduced well belowthreshold to achieve the desired precision withoutmany layers of concatenated error-correction en-coding, then digital quantum simulation will scalefavorably with the number of particles.

Let us finally remark that, to increase their effi-ciency, digital-quantum-simulation algorithms often com-press the number of degrees of freedom that are necessaryto describe the many-body system, rather than directlymap the Hilbert space of the system to the Hilbert spaceof the simulator [117], an approach that has been bor-rowed from classical algorithms like MPS or PEPS. Cur-rently, there is a new theoretical development towardsa hybrid device, where the ground state of many-bodyHamiltonians is represented as a PEPS, but implementedon a quantum computer. This is efficiently possible whenthe gap between the ground and first excited state scalesas the inverse of a polynomial in the number of particles.Then, one can use the quantum computer to contracttensor networks and use that to calculate the expecta-tion value of any local variable, such as correlation func-tions [122]. In a similar spirit, Temme et al. developeda quantum-algorithmic version of the Metropolis Monte-

Carlo algorithm that allows one to efficiently sample froma Gibbs thermal state [123]. Such approaches point to ef-ficient DQSs for well defined classes of problems.

B. Non-Universal Digital Quantum Simulators(nUDQS)

A non-Universal Digital Quantum Simulator (nUDQS)is in many aspects similar to a UDQS, except that is it aspecial-purpose quantum computer. Its task is, however,the same as that of a UDQS: to simulate continuous-timequantum many-body dynamics of a certain quantum sys-tem described by a certain physical Hamiltonian. Theexperimentalist who realizes a nUDQS has to his/her dis-posal a non-universal set of unitary quantum gates. Letus list below some properties and possible realizations ofsuch a nUDQS:

Realizations: In all systems in which the univer-

sal sets of quantum gates are available, one canalso restrict the set of gates and realize a nUDSQ.For example, in some of the recent experiments ofBlatt [24], only a necessary subset of the availableset of universal gates was used. All of the systemsdiscussed above (atomic, superconducting, etc.) arepotentially platforms for implementing nUDQSs. Aseminal example of this approach goes back to theso-called Average Hamiltonian Theory in nuclearmagnetic resonance (NMR) [124, 125].

Controllability: nUDQSs are typically not per-fectly controllable, but in most experimental real-izations should allow for a wide control of parame-ters, which in turn should allow for simulations ofevolution for wide families of Hamiltonians of in-terest.

Error correction: For nUDQSs, it is not guaran-teed that error correction and fault-tolerant com-puting is possible.

Efficiency and reliability: All of the above dis-cussion concerning UDQSs applies also to nUDQSs.But, there are many novel, open problems associ-

ated specifically with nUDQSs, since, e.g., some-times giving up on universality can result in sub-stantial efficiency gains. For example, universalitycould be sacrificed in favor of a highly precise andfast gate [24] (a simple example is an external ho-mogeneous field, which in a UDQS might have tobe applied as a sequence of one qubit gates). Inparticular, it is possible that for some classes ofnUDQSs the problems of Trotterization are not assevere as in the case of UDQSs [101].


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C. Open-System Digital Quantum Simulators(OSDQS)

An Open-System Digital Quantum Simulator (OS-DQS) is a completely new concept, in principle very dif-ferent from DQSs aimed at Hamiltonian evolutions. OS-DQSs are designed to simulate open-system, dissipativedynamics described in the simplest situation by a Marko-

vian Lindblad master equation for the density matrix of amany-body system of interest. OSDQSs can be aimed ata continuous-time simulation of interesting open-systemdynamics, or at a designed dissipative dynamics towarda stationary state of interest, in particular a pure, highly-entangled state [126128].

The experimentalist who realizes an OSDQS, in con-trast to a UDQS or a nUDQS, needs to have at his/herdisposal some non-unitary, dissipative quantum gates,which mathematically correspond to Lindblad super-operators acting on the density matrix in the masterequation. This fact opens a plethora of new questions,e.g., what are the universal sets of gates for this type of

evolution. Note that in the case of unitary computing,the universal set of gates allows for realization of arbi-trary unitary transformations acting on the (pure) stateof the system. In the case of open-system dynamics, auniversal set of gates should allow for the realization of anarbitrary completely positive map (CPM) acting on thedensity matrix of a system. Moreover, for experimentalrealizations, we require the gates to be local.

While the conditions for controllability of an openquantum systems are under exploration [129], the ques-tion of a universal set of gates in this context remainsopen. A non-trivial reduction (cf. [130, 131]) of this ques-tion to the CPMs that correspond to Markovian evolu-tion, is also open. The problem of error correction in

this context is unsolved as well. All of these commentsimply that in the area of OSDQSs there are more openquestions than answers.

Realizations: In systems in which the universalset of quantum gates is available, one way to real-ize dissipative gates is by tracing out ancillas, thusallowing to realize an OSDQS. Good testbeds forexploring OSDQSs are provided by Rydberg atoms,atomic ensembles, NMR [132], or trapped ions. Infact, the first concrete proposals for open systemDQS concerned Rydberg gates [100, 101]. The firstexperimental realizations of these ideas, however,

have been achieved with trapped ions [25]. Controllability: OSDQSs are typically not uni-

versal since they are not usually controllable inthe sense of realizing an arbitrary quantum map[133]. Nevertheless, many experimental realiza-tions should allow for a wide control of parame-ters, which in turn should allow for simulations ofopen-system (Markovian) evolutions for wide fam-ilies of open systems of interest. As pointed outin [127, 128], due to the purely dissipative nature

of the process, this way of doing quantum informa-tion processing exhibits some inherent robustnessand defies some of the DiVincenzo criteria for quan-tum computation. In particular, there is a naturalclass of problems that can be solved by open-systemDQSs or AQSs: the preparation of ground states offrustration-free quantum Hamiltonians.

Error correction: For OSDQSs, it is not guaran-teed that error correction and fault-tolerant com-puting is possible in the sense defined above [134]

Efficiency and reliability: All of the above dis-cussion concerning UDQSs and nUDQSs appliesalso to OSDQSs. But, due to purely dissipativenature of the process, this type of simulation has acertain intrinsic robustness and built-in error cor-rection. A clear example is seen in the OSDQSimplementation of Kitaevs toric code [25, 101].However, as discussed in Ref. [101], errors in thegates result in effective heating. Also, the prob-lems of Trotterization are not as severe as in the

case of quantum simulators of Hamiltonian evolu-tion. Still, most of these general aspect concerningOSDQSs have not yet been investigated systemat-ically. The efficiency of OSDQSs for the case offrustration-free Hamiltonians depends on the sizeof the gap between the ground state and the ex-cited states, or more precisely on the real part ofthe first non-zero eigenvalue of the Lindblad equa-tion, which determines the rate of approaching thestationary (ground) state.

Currently, considerable attention has been devoted tothe problem of existence and uniqueness of the open-system preparation of ground states of frustration-free

Hamiltonians, and in particular, entangled states of in-terest. These states are annihilated simultaneously byall of the local frustration-free Lindblad superoperatorsentering the master equation. There is little known ingeneral about the many-body dissipative dynamics witha frustrated set of Lindblad superoperators competingwith Hamiltonian dynamics. For the first attempts tounderstand these kind of problems in the context of quan-tum diffusion-exclusion processes competing with Hamil-tonian evolution see, e.g., [135].

IV. ANALOG QUANTUM SIMULATORS (AQS)

In this section, we classify general properties of AQSs,discuss various proposals for such devices, and summa-rize the state-of-art knowledge concerning their control-lability, reliability, and efficiency. AQSs are experimentalsystems that are designed to mimic the quantum dynam-ics of interesting quantum many-body models, typicallyusing always on interactions between particles that areaugmented by fast local unitary control. While by defini-tion they operate in continuous time and thus the Trot-terization problems do not concern them, the standard


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error-correction methods and fault tolerance cannot beapplied.

A. Universal Analog Quantum Simulators (UAQS)

Sometimes known as Hamiltonian simulation, thegoal of a UAQS is to transform a given Hamiltonian

acting on a fixed Hilbert space into an arbitrary targetHamiltonian through a well-designed control sequence.While not conceived as a practical AQS device, the proto-col explores an abstract quantum-information-processingsystem capable of simulating unitary evolution for all (orat least all local) Hamiltonians.

Realizations: To our knowledge there are noconcrete proposals for experimental realizations ofUAQSs.

Controllability: While for UDQSs the issue isthe access to the universal set of quantum gates,for UAQSs the question is what the necessary re-

sources are (not necessarily quantum gates) thatallow for the simulation of all Hamiltonian evo-lutions of interest. Universal control sets (as op-posed to universal digital logic gates) that gener-ate an arbitrary Hamiltonian evolution have beenstudied [136, 137]. Typically, such an approach us-ing always on interactions is associated with morelimited control than is available in a universal dig-ital quantum computer.

Error correction: UAQSs do not allow for stan-dard error correction and fault tolerance.

Efficiency and Reliability: Dr et al. studied ahybrid construction of always-on interactions withstroboscopic digital control to achieve a universalHamiltonian simulator via the Trotter construc-tion [138]. They found that decoherence and analogtiming errors can make this inefficient for a Hamil-tonian simulator. Other issues concerning UAQSsare essentially the same as for non-universal AQSs,so we leave the discussion of them to the next sub-section. The only difference is that UAQSs, by def-inition, are capable of performing tests of robust-ness of the quantum simulations that we proposein the following section, i.e., tests based on addingdisorder or noise in a controlled manner to the sim-ulated Hamiltonian. For non-universal AQSs suchan addition requires additional resources.

B. Non-Universal Analog Quantum Simulators(nUAQS), or simply AQS

Non-universal AQSs constitute the most popular classof quantum simulators, but despite this fact, there is verylittle known about their reliability and efficiency. There-fore, we focus on them in the remainder of this paper,

where, for simplicity, we shall term them AQSs. AQSsare experimental systems that can mimic continuous-time unitary Hamiltonian evolution for a family of mod-els of many-body physics. Their characteristics are asfollows:

Realizations: The most advanced experimentswith AQSs have been with ultracold atoms in opti-

cal lattices [3, 515, 19, 71]. The degree of quantumcontrol is even better in ultracold-ion systems, butthese are so far limited to few ions [2123, 26, 27].The first step toward large-scale QSs with ions was,however, recently achieved [139]. Recently therehas also been substantial progress in investigationsof other possible candidates for AQS, such as pho-tonic systems [31, 32, 3436], photonic and polari-ton systems [33, 3842], artificial lattices in solidstate systems [44].

Controllability: Most, if not all of the propos-als for and realizations of AQSs allow for at leastpartial controllability. The paradigm examples areAQSs employing ultracold atoms in optical lattices(for more details, see Chapter 4 of [3]). Here, thetypical controls involve optical lattice parameters(laser intensity, wavelength, etc.), lattice geometry,lattice dimensionality, temperature and other ther-modynamical control parameters, as well as atomicinteraction strength and nature (van der Waals in-teractions are controlled via Feshbach resonances,while dipole interactions by the strength of thedipoles, lattice-site-potential shape, etc.). Further,tunneling can be laser assisted and can mimic arti-ficial Abelian or even non-Abelian gauge fields (cf.[16, 17]). Dipolar interactions may lead to non-

standard terms in Hubbard models, such as occu-pation dependent tunneling [140], or various effectsinvolving higher orbitals (see, e.g., [141]).

Error correction: AQSs do not allow for stan-dard error correction and fault tolerance.

Efficiency: The issues of reliability and efficiencyare essential for the usefulness of any QS, and AQSsin particular. In the context of AQSs, however,there has been little analysis of these problems.Firm criteria on computational complexity and ef-ficiency for AQSs are in general difficult to addressand have not yet been established. First of all,they require the knowledge of classical computa-tional complexity of the static or dynamical prop-erties of the considered quantum models. Unfortu-nately, in the realm of classical computation, thereare few proofs that a given computational problemis outside the class P, or even if there is a clear de-lineation between certain complexity classes. In re-cent years, there has been considerable progress inunderstanding that the ground states of 1D gappedsystems can be efficiently simulated by classical


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methods [142144], or that the quantum dynam-ics is in general computationally hard [96, 97]. Ifwe can set the parameters of our AQS to a regimewhere efficient classical simulation is possible, wecan assess the efficiency and reliability of the AQSin this case by direct comparison with classical sim-ulations (see below). However, there is no guaran-tee that such a calibration will hold in the truly

interesting regimes of parameters, where efficientclassical simulations are either impossible, or wedo not know how to perform them.

Reliability: So far, there exists no perfect andrigorous way to assess the reliability of AQSs, butthere are several complementary approaches. Oneapproach is by cross validation of a variety of differ-ent physical systems (e.g., atoms in optical lattices,ions in traps, and superconductors) [74]. The hopeis that since every platform has its own set of imper-fections, they will agree on the universal propertiesof the ideal quantum many-body model being sim-ulated. While it remains to be seen whether suchuniversal features would emerge, this approach hasa number of shortcomings. For example, theremay be models that have only one known imple-mentation, or different implementations may suf-fer in the same way from imperfections, hence con-sistently exhibiting features associated with noiserather than with the ideal model.

A more systematic approach is to validate resultsof a quantum simulator against analytical and nu-merical predictions in the regime of parameterswhere such comparison is possible. This was re-cently demonstrated in experiments with ultracold

bosonic and fermionic atoms [68]; amazingly, inone case numerical simulations helped to correctthe expected experimental temperature by up to30%. Relying solely on validation from classicalcalculations, however, would restrict QSs to modelsin regimes where these efficient classical algorithmsexist that means contradicting our relevance andusefulness requirement, point (a) of our definitionof a QS. In general, we desire to operate quantumsimulators in regimes whose properties are difficultto deduce by classical methods, e.g., near or at thecritical point of a QPT, or in genuine terra incog-nita regimes. In these regimes, however, many rel-evant models become hypersensitive to perturba-tions [145, 146], and even small levels of noise mayspoil completely the results of the quantum simula-tion. Indeed, the capability of an analog quantuminformation processor whose dynamics is charac-terized by quantum chaos (i.e., well described byrandom matrix theory) can be severely impactedby imperfections [147, 148]. More importantly, thisalso means that successfully validating a QS in aclassically accessible regime does not give certaintyabout its robustness in regimes which are classically

not accessible.

C. Open-System Analog Quantum Simulators(OSAQS)

Finally, let us mention Open-System Analog QuantumSimulators (OSAQSs). Similar to OSDQSs, OSAQSs are

supposed to simulate dissipative dynamics described inthe simplest situation by a continuous-time MarkovianLindblad master equation for the density matrix of amany-body system of interest. OSAQSs can be aimed ata simulation of interesting open-system dynamics, or at adesigned dissipative dynamics toward a stationary stateof interest. Many-body Lindblad master equations havebeen studied in the context of evaporative [149155], laser[156167] and sympathetic [168172] cooling of degener-ate atomic gases (see also [173]). Recently, there has beena revival of interest in such systems in the context of pos-sibility of using them for preparation of interesting pure,highly-entangled states [126128]. Open-system quan-

tum simulators employing superconducting qubits mayalso give insight into exciton transport in photosyntheticcomplexes [174].

The experimentalist who realizes an OSAQS, in con-trast to an AQS has to have to his/her disposal somenon-unitary, dissipative quantum mechanism: in a sense,all designed cooling or entropy-reduction methods are ofthis sort.

Realizations: All AQS systems can, in principle,be used as OSAQSs.

Controllability: OSAQSs are typically not uni-versal in a sense similar to OSDQSs; they allowneither for simulating arbitrary (local) Markoviandynamics, nor do they allow for preparation of ar-bitrary states. Nevertheless, in most of the propos-als [126128] or experimental realizations they al-low for a wide control of parameters, which in turnallows for simulation of open-system (Markovian)evolutions for wide families of open systems.

Error correction: For OSAQSs, it is not guaran-teed that error correction and fault-tolerant com-puting is possible in the sense defined in the sub-section on OSDQSs.

Efficiency and reliability: All of the abovediscussion concerning AQSs applies also OSAQSs.But, again due to the purely dissipative nature ofthe process, this type of simulation has a certainamount of intrinsic robustness and built-in errorcorrection - this is particularly clear for the OS-AQS of quantum kinetic Ising models or Kitaevstoric code [173]. Still, as in the case of OSDQSs,most of these general aspects concerning OSAQSshave not yet been investigated systematically.


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V. ROBUSTNESS OF ANALOG QUANTUMSIMULATORS

All of the above considerations clearly lead to the fun-damental question: Can we trust quantum simulators?From what we have said, the rigorous answer to this ques-tion is no, yet in practice we do tend to trust them, at

least to some extent.In order to gain more trust in the results of quan-

tum simulations, it is thus extremely important to designnovel tests and certification of reliability and validity ofQSs. In this section, which constitutes some of the mostimportant results of this paper, we propose such tests,which we call tests of robustness of quantum simulators.Our tests consist in checking the robustness of QSs withrespect to addition of imperfections, such as static dis-order or dynamical noise. This would then allow to (i)

judge how strong the reaction of the QS with respect tothese perturbations is, and (ii) might even open possibil-ities to extrapolate interesting observables to the ideal,

zero-disorder limit. Such tests can also be applied toDQSs, but are particularly suited for AQSs. For exam-ple, in an implementation with trapped ultracold atoms,disorder can be increased in a controlled manner [175].

In the following of this key issue article, we use theexample of the quantum Ising chain to substantiate ourdiscussion of the reliability of AQSs and the relationshipto the complexity/efficiency of the simulation. We studyhow imperfections affect the results of a AQS simulatingthat model, where, for simplicity, we assume quencheddisorder as the only possible imperfection. In the future,it will be in particular interesting to also investigate theeffects of dynamical noise, and the decoherence and re-

laxation that occurs due to coupling with an environment(see also Ref. [176]).

The quantum Ising model, which we describe indetail in Section VI, is exactly solvable, which allowsus to explore regions with universal behavior such assecond-order QPTs. In Section VII, we show that theground-state expectation values of certain local observ-ables appear fairly robust under disorder, while thisneed not be true for the global many-body state of thesimulator. In particular, disorder can have a significanteffect on relevant quantities that one could hope toextract from the simulator, such as critical points andexponents, or if the system is described by a conformalfield theory (CFT) its central charge [177]. Finally, webriefly address the relationship between robustness andcomplexity by studying the dynamics of different thermalstates after a quench of the Hamiltonian (Section VIII).We show evidence that QSs appear to work better inregimes that are classically easier to solve or simulate(high-temperature states), thus hinting at a connectionbetween the amount of quantum correlations and therobustness of a QS.

VI. THE MODEL

To illustrate the influence of disorder on an AQS, westudy the transverse Ising model (TIM)

H = i,j

Jijxi

xj

i

hizi , (1)

where x,zi are the usual Pauli spin matrices and i,jmeans sum over nearest neighbors. The system is sub-

ject to quenched disorder in both the interaction andfield terms. We denote the nearest-neighbor spin cou-pling and the transverse field by Jij = J(1 + rij) andhi = h(1 + ri), respectively, where ij and i are inde-pendent random variables with a Gaussian distribution ofmean zero and variance r. All details of our calculationsare presented in the Appendix.

The TIM, even under the presence of disorder, is effi-ciently solvable by which we mean that the eigenstatesand eigenenergies of the system can be found using a clas-sical computer, and that the cost of the algorithms (in

time and hardware) is polynomial with the size (numberof particles) of the system. The TIM, in particular, canbe solved by using a JordanWigner transformation to asystem of non-interacting fermions and the cost of solv-ing the non-interacting fermion system is the cost of di-agonalizing a matrix with rank equal to twice the numberof spins in the chain [178]. This model is well studied (seefor instance [179]): for low fields, the ground state is aferromagnet, while for large fields it is a paramagnet. Atzero temperature and disorder, the system undergoes aQPT when the dimensionless control parameter = h/Japproaches the critical value, c = 1, i.e., when the fieldintensity equals the interaction strength. The influence

of disorder can have dramatic effects on this phase dia-gram: imperfections can create new phases, or even de-stroy the ones we want to investigate. Indeed, in theTIM when the disorder strength is comparable with theinteractions, the critical point disappears and is replacedby a so called Griffiths phase [180], extending across a re-gion of size proportional to the disorder strength. Evenmore, in this Griffiths phase observables become non-self-averaging, i.e., fluctuations increase with system size, andhence dominate the thermodynamic limit. In this study,we consider small disorder strengths, which allows us toignore the Griffiths phase, especially in finite-size sys-tems. Moreover, a state-of-the-art AQS can achieve verylow levels of disorder, whence this is the experimentallyrelevant regime. Note that, while the TIM has been stud-ied extensively in the limit of very large disorder [181],there are few studies addressing directly the influence ofsmall disorder on the universal properties near quantumphase transitions. However, it is well known that evensmall disorder can lead to novel quantum phases such asBose [182] or Fermi [183] glasses (see also the section ondisorder in Ref. [184] and references therein). In the fol-lowing, we analyze the robustness of relevant observablesto disorder in static and dynamic situations.


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f1

f2

F

0.9 0.95 1 1.05 1.1

-1.5

-1

1

03

log(fn)/n

0 0.05 0.1 0.15 0.2

0.45

0.5

r

1.5

2

2.5

c

0

10

20

30

40

50

60

0.9 0.95 1 1.05 1.10

0.05

0.1

0.15

0.2

r

a

b c

Figure 1. A: The correlation length decreases with disor-der r, and its peak broadens (shown for a chain of 400 spins).The critical point (as extracted from a finite-size scaling ofthe energy gap ) moves to larger with increasing disorder(black line). B: For a chain of 400 spins, we show the meansingle- and two-site reduced simulator fidelities (f1 and f2),and the total simulator fidelity (F) for a fixed disorder levelr = 0.1. Local fidelities are more robust, which gives hopethat local quantities can be reliable even if disorder deteri-orates the overall ground state. As expected, disorder hasmore severe effects close to the QPT. C: The central chargec (full circles, left axis), extracted from a fit to the part-chain

entropy, and the critical exponent (open circles, right axis),extracted from a collapse of the correlations in different chainlengths. Both change with disorder, which can lead to erro-neously assigning the QPT to an incorrect universality class.However, the change begins relatively smoothly at low levelsof disorder.

VII. RESULTS STATICS

First, we investigate static properties of the AQS andtheir robustness to disorder (summarized in Fig. 1). Weaverage all the analyzed static quantities over many re-alizations of disorder.

One can evaluate the response of the AQS to disor-der using the simulator fidelity, which we define for purestates as the overlap between the state obtained with aperturbed simulator, |r(), and the ideal state |0(),

F(r, ) = | 0()|r() | . (2)Although we define the simulator fidelity for any pos-sible target state, we focus on the ground state. AsFig. 1B shows, this overlap is considerably suppressednear the QPT, reaching values as low as 55% (for r = 0.1

in a chain of L = 400 sites). When scaling to largersystems, F(r, ) will typically vanish exponentially fast,simply due to the exponential growth of the dimension ofthe Hilbert space (a kind of orthogonality catastrophe").In a universal quantum computation, the fidelity wouldhave to be very close to 1 for the quantum computer towork fault-tolerantly. However, QSs have the advantagethat we do not necessarily demand of the entire state to

be robust. Often, it is enough if we can distinguish therelevant phases by measuring faithfully local observables(local in the quantum information sense that few sitesare involved, although they may be physically far apart).Obviously, this is less demanding, yet very useful.

To quantify the robustness of local observables, we in-vestigate the single-site and (nearest-neighbors) two-sitesimulator fidelity f1(r, ) and f2(r, ), respectively. Asthe one- and two-particle density matrices will generallybe mixed when the overall pure many-body state is en-tangled, these are defined as the Uhlmann fidelity [185]between the single- or two-site reduced density matri-ces of the ideal state and the one at disorder strength r,f

Tr0r0. It can be assumed that fidelities ofthe reduced system decrease more or less monotonicallywith the number of sites involved. As seen in Fig. 1B,the reduced simulator fidelities are much more robust todisorder than the global one near the phase transition,f2(r, ) decreases to approximately 0.998, and f1(r, ) re-mains above 0.999. This gives optimism that local quan-tities are robust enough to allow a faithful distinctionbetween different quantum phases.

One step beyond local properties of the ground stateare the correlation lengths dictating the exponential de-cay of long-distance correlation functions. We investigatethe correlation length extracted from the correlationfunction

C(i, j) = r|(i)z (j)z |r r|(i)z |rr|(j)z |r ,(3)

where away from criticality C(i, j) exp( |i j| /).Without disorder and for infinite systems, diverges atthe critical point, because criticality is the emergence ofcollective phenomena involving infinite degrees of free-dom at all length scales. In practice, we can only dealwith finite systems so that we cannot observe real critical-ity but only smoothed out signatures of it, a phenomenonwhich one normally calls pseudo-criticality. For exam-ple, the correlation length is bounded by the systemsize. Still, its peak gives a reliable signature for the lo-cation of the critical point. As Fig. 1A shows, however,disorder suppresses correlations and broadens the peak of, thus making an extraction of the critical point muchless reliable.

Another criterion to locate the QPT is provided bythe energy gap between ground state and first ex-cited state. At criticality, the low-energy spectrum ofthe Hamiltonian is gapless in the thermodynamic limit.In finite systems, it presents non-vanishing gaps that de-crease in a systematic way with increasing system size.Due to this characteristic scaling of physical observables


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as a function of system size in pseudo-critical systems,criticality can be detected by studying a sequence of fi-nite but increasingly large systems, a technique calledfinite-size scaling [186]. We describe this technique forthe energy gap in the Appendix (see Fig. A1), and showin Fig. 1A (black line) the location of the critical pointextrapolated in this way. As can be seen, if one does notcorrect for disorder effects, one would locate the critical

point at values of that are too large.Perhaps of more fundamental interest than the ex-

act location of a critical point is its universality class.All models within a given universality class give rise tothe same collective behavior at large distances (typicallylarge with respect to the lattice spacing), irrespective oftheir microscopic details [177]. Therefore, all relevantthermodynamic quantities for all models within a classare characterized by the same small set of critical expo-nents which describe the power-law decay of the corre-lation functions of local observables in the large-distanceregime, a property that allows to differentiate among dif-ferent emerging collective behaviors. To investigate how

robust the universal behavior is, we compute the criticalexponent for the correlation length, , from a collapseof the correlations (as explained in the Appendix, seeEq. (A10) and Fig. A1). As shown in Fig. 1C, alreadyfor a few percent of disorder, increases strongly fromits ideal value 1. Therefore, if one simply neglects theinfluence of disorder, the extraction of critical exponentsyields wrong results.

If the QPT is described by a CFT (a specific subclassof one-dimensional critical systems), it is characterizedby a central charge c. The central charge appears ubiq-uitously [187]. It, e.g., governs the temperature depen-dence of the free energy (Stefan-Boltzmann law) and theCasimir effect in finite geometries, but also the scalingof the entanglement entropy of sub-regions of the groundstate of the corresponding quantum models. Models withdifferent central charge have different emerging collectivebehavior. For example, models whose collective behavioris that of a free Majorana fermion (as in the disorder-freeTIM) have central charge = 1/2, while models whosecollective behavior is that of a free boson have centralcharge = 1. Strictly speaking, the TIM has an under-lying CFT only in the disorder-free case, but there havebeen efforts to extract an effective central charge also forthe disordered model [181], for example, from the vonNeumann entropy S of the reduced density matrix of apart of the chain of size l. At criticality, this entropy

scales as c/6log(L/ sin(l/L))+ A [188190]. Figure 1Cshows that disorder decreases the effective central charge.Hence, ignoring the effects of disorder would give com-pletely erroneous results, since even a small deviation ofthe central charge indicates a completely different univer-sality class. Note also that the decrease ofc with disorderindicates the destruction of correlations by disorder.

Fortunately, for all the extracted quantities (except theglobal simulator fidelity), the levels of disorder for appre-ciable changes to occur are at least a few percent. If the

AQS can be operated below such a value, its results seemto be robust, at least in this simple model system. State-of-the-art experiments are good enough to fulfill this re-quirement. In fact, in many experimental situations onehopes to reach levels of disorder or noise that are belowa few percents, assuring the robustness of the AQS. Fre-quently, however, changing parameters from the regimewhere validation via classical simulation is possible to

the regime of terra incognita might lead to uncontrolleddisorder or noise. That is why checking sensitivity to dis-order and noise in those regimes where it can be checkedis of great importance.

VIII. RESULTS DYNAMICS

Efficient classical algorithms for computing static prop-erties of quantum systems are more developed than forcomputing dynamics (the difficulty arises mainly becauseentropy and correlations grow rapidly with simulatedtime). Therefore, one can assume that in the absence

of disorder, a quantum simulation of dynamics can muchmore easily outperform classical computers. Indeed, ina recent experiment based on ultracold bosonic atoms,the controlled dynamics ran for longer times than presentclassical algorithms based on matrix product states couldefficiently track [14]. We thus turn to the issue of howdisorder affects the reliability of quantum simulations ofdynamics. As with statics, we investigate the behavior ofthe simulator fidelity, but now also as a function of time,initial state, and external driving.

Typically, we expect that the simulator fidelity willdecay with time, and eventually reach an asymptotic fi-nite value. The effect of disorder in both the decay rate

and the asymptotic saturation value can, in general, beunderstood from established techniques such as FermiGoldens rule, and random matrices [191]. On the otherhand, the effect of the initial state and the external driv-ing is known to be nontrivial and of particular interestfor our purposes. For example, it is known that numeri-cal techniques such as the time-dependent density matrixrenormalization group (tDMRG) can simulate efficientlythe dynamics after a sudden quench of the field h, aslong as the quench is restricted to a few sites on thechain. However, if the quench is global, it has been shownthat the computational resources needed to keep a fixedamount of error grow exponentially with time [192, 193].Generically, solving for the dynamics of a quantum many-body system is a hard problem for classical algorithms.Our model is special because it can be solved exactlyfor all cases, although it remains hard for the tDMRGalgorithm. We use this to our advantage to study howthis class of classical algorithms behaves when solving forquantum dynamics.

We studied the behavior of the full simulator fidelityunder the following driving. As initial state we preparethe ground state of the Hamiltonian for a given value ofthe external field. At time zero, the field is quenched


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instantaneously to a larger strength, and the system isallowed to evolve. In panels B and C of Fig. 2, we com-pare the short- and long-time behavior of fidelity for thecase of a global and a local (single-site) quench. The AQSkeeps a high fidelity in the case of a local quench, whileit performs poorly for the global quench, with fidelitiesreaching lows of 0.8 even for small systems of 50 spins.We also observe that the AQS performs worse when the

quench crosses the critical point, as shown in Fig. 2C,where we fix the strength of the quench and vary theinitial field value.

The initial state can also have an effect on the effi-ciency of classical algorithms. Using the same setup witha global quench, but starting from a thermal initial state,tDMRG becomes efficient for high temperatures [192]where the state and its correlations are almost classi-cal. However, it becomes exponentially inefficient withtime for low-temperature initial states. For initial ther-mal states, we can still compute the dynamics exactly,although computationally it becomes too expensive tocalculate the full many-body fidelity between the evolved

states. In this case, therefore, we focus on the reducedsimulator fidelity. For the regimes of disorder that westudied, we observe that the time dependent fidelity de-cays with a rate roughly proportional to the strengthof the disorder squared (typical of a Fermi golden rule[191]). For this reason, we show in Fig. 2A a rescaledform of the fidelity, (1f1)/r2, that exemplifies the typ-ical behavior for all disorder strengths, as a function oftime and temperature of the initial state.

As with the classical algorithms [192], the AQS remainsfaithful when the state is almost classical (high temper-atures). The simulator fidelity decreases rapidly for lowtemperatures, although it saturates at a fairly high value.In terms of distinguishability, the values we find imply

that a fair observer would have only a 4% chance of dis-tinguishing the 1-spin reduced state of the AQS from theideal state. In the inset of the top panel we show theaverage asymptotic fidelity as a function of temperatureof the initial state. Again, for low temperatures fidelityworsens, but saturates to a few percent. For high tem-peratures, it is simple to perform an expansion of thefidelity which shows that f1 1 T2.

IX. DISCUSSION AND OUTLOOK

A key issue for future investigation is the relationshipbetween the robustness of an analog quantum simulatorand its computational power. For the models we haveconsidered here, the physically relevant correlation func-tions are robust for a reasonable degree of disorder. Thissuggests that such an AQS could perform well in a labo-ratory demonstration. But, the TIM that we consideredhere is simulatable on a classical computer. Is this con-nection between robustness and classical simulatabilitycoincidental, or does it reflect a deeper relationship?

Disorder reduces the correlation length of the spin

Figure 2. A: Evolution of the average reduced simulatorfidelity as a function of the temperature of the initial state.The system is an Ising spin chain of length 50, the initial stateis a thermal state at criticality ( = 1), and at time zero thefield is suddenly quenched to = 2. In the vertical axiswe show the infidelity (one minus fidelity) normalized by thedisorder strength r squared. For larger temperatures (wherethere are less correlations) the state is more robust. In theinset, we show the average asymptotic infidelity as a functionof temperature. For large temperatures it decays as 1/T2.B: Evolution of the full simulator fidelity for an initial stateequal to the ground state (zero temperature) at = 0.75 aftera sudden quench to = 1.25. For a local quench in a single

site, fidelity saturates rapidly at large values, but decreasesstrongly for a global quench. C: Asymptotic value of the totalsimulator fidelity as a function of the initial value of the field, with a fixed quench strength of = 0.25. The systemis much less robust for global quenches and near criticality( = 1).

chain. Because less-correlated quantum states can bedescribed with fewer parameters, there is reason to sus-pect that certain aspects of weakly disordered quantummany-body systems could actually be easier to simulateon classical computers than their clean idealized versions.This happens, for example, in the realm of digital quan-tum computation, where a quantum circuit becomes clas-sically simulatable for noise levels above a certain levelwhen quantum gates lose their entangling power [194196]. In the context of many-body physics, the successof DMRG in, e.g., 1D spin chains, is rigorously related tothe existence of efficient matrix-product-state represen-tations [197]. These take advantage of the small amountof quantum correlations in such systems, thus compress-ing the O(exp(n)) parameters needed to describe a gen-eral n-particle state to O(n) finite-dimensional matrices.


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In higher-dimensional lattices, states which obey the so-called Area law [97], where quantum correlations aresmaller than in generic states, may still be amenable to aclassical simulation using state-of-the-art techniques suchas tensor networks [77, 81, 82, 198, 199], Density Func-tional Theories [86], or Quantum Monte Carlo [75].

We thus arrive at the fundamental question: Do thefinite imperfections of an analog quantum simulator re-

duce the correlations, and thus the number of parame-ters needed to describe the system, so as to render the

device simulatable by classical means? We know thatfor noise above certain levels a digital quantum circuitis classically simulatable and for levels below a certainthreshold it can be rendered fault tolerant. Is there anintermediate regime for which noise is too great to al-low fault-tolerant universal quantum computation, butsmall enough that an AQS accesses physics beyond clas-sical simulation? The existence of an intermediate regimewould imply that there exists a whole class of problemsoutside P that we can access in the near future, evenwithout a fully functioning quantum computer.

The results we present here, in particular those for dy-namics, are an initial attempt albeit in a trivial model at understanding the above problem. We can see how ananalog quantum simulator works well when a classical so-lution is efficient, and worsens (but only in a limited way)when the problem becomes classically hard to simulate.Even though the underlying model is actually solvable,this may be positive evidence for the existence of an in-termediate regime of noise, and the efficiency of AQSs inmore complex situations.

Our main discussion focused on AQSs, but similar is-sues pertain to DQSs. Since to date there exists noknown way to fault-tolerantly error-correct AQSs, there

is a natural tendency to explore the advantages of DQSs,where error correction is possible. The above discussionshows, however, that a digital implementation of a quan-tum simulation does not, in itself, guarantee an efficientand more powerful simulation than one that is carriedout classically. As in any quantum algorithm, initial-ization, evolution of the state, and measurement mustbe performed efficiently, i.e., with a polynomial use ofphysical resources (space and time). Digital quantumsimulation is no exception. Indeed, as discussed above, afault-tolerant implementation of the standard approachbased on the Trotter expansion [98] comes at the cost ofan overhead in the number of gates and time requiredthat grows exponentially with the degree of precision re-

quired [64, 120]. If we can guarantee the reliability ofanalog quantum simulators while avoiding such exponen-tial costs, many open problems from all areas of physicscould suddenly come into the reach of being solved.

Finally, we can turn the problem of quantum simu-lation on its head and ask, what does Nature do? Forany real material, like a high-Tc cuprate, has imperfec-tions. Does Nature access highly correlated states thatcannot be efficiently simulated on a classical computer?Certainly, in some cases we believe it does, as for exam-

ple in high-Tc superconductors [69] or in certain groundstates of frustrated quantum antiferromagnets which arebelieved to carry topological order [89]. If noise is lowenough, does Nature protect quantum correlations to adegree that classical methods cannot efficiently representthe physically interesting quantities? And, can we exploitthis capability with a quantum simulator? If Nature doesit, we should take advantage of it!

Acknowledgements We gratefully acknowledgesupport by the Caixa Manresa, Spanish MICINN(FIS2008-00784 and Consolider QOIT), AAII-Hubbard,EU Projects AQUTE and NAMEQUAM, ERC GrantQUAGATUA, and Marie Curie project FP7-PEOPLE-2010-IIF ENGAGES 273524. IHD was supported byNSF grants 0969997 and 0903953. We also acknowledgefruitful discussions with Carl Caves, Dave Bacon, RobinBlume-Kohout, Rolando Somma, and Roman Schmied.

Appendix

Quadratic fermionic systems The transverse fieldIsing model, Eq. (1) of the main text, even with disor-der, can be solved by casting it into the form of non-interacting fermionic particles using the JordanWignertransformation,

+j = cj

j1m=1

eicmcm , (A1a)

j =

j1m=1

eicmcmcj , (A1b)

zj = 2c

jcj 1 . (A1c)

The cj , cj obey on commutation relations. This trans-

formation leads to

H =i,j

ciAijcj +

ciBijc

j + h.c.

1

2

j

Ajj ,

(A2)where

Aij = Jij (j,i+1 + j,i1) 2hij,i , (A3a)Bij = Jij(j,i+1 j,i1) . (A3b)

Hamiltonian (A2) can be diagonalized to

H =Nk=1

kkk + E0 , (A4)

where = (A B) is diagonal. , , and can be obtained from the singular-value decompo-sition of Z A B. The normal modes are k =N

j=1

gk,jcj + hk,jc

j

, where g = ( + ) /2, and h =

( ) /2. From this, we can compute the relevantground-state properties.


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15

Ground-state fidelity and correlations From thenormal modes obtained in the diagonalization of the pre-vious section, we can compute the observables we areinterested in: the simulator fidelity F (the overlap to thedisorder-free ground state), reduced simulator fidelities,the energy gap, and the ZZ-correlations.

In general, the overlap between the ground states of

two realizations Z and Z is [200]F

Z, Z = det 1 + T1 T2

, (A5)

with T =

11

Z. We define the simulator fidelityF as the overlap at fixed between the ideal, disorder-free state and the state at disorder strength r,

F(r, ) F(Z()r, Z()0) . (A6)This is a global quantity, but one can expect that localobservables are less affected by disorder. A measure forthe change of local quantitites is the single-site simulator

fidelity

f1(r, ) =Li=1

tr

(i)0 ()

(i)r ()

(i)0 () , (A7)

where (i)r = trj=i is the reduced density matrix of

site i under disorder r, and (i)0 is the equivalent in the

disorder-free case. The single-site reduced density matrixis completely determined by the expectation values ofi ,

= x,y,z, since one can expand (i) = 12

i .

Here, the sum runs over , = x,y,z, and (0) = 1.We also analyse the two-site simulator fidelity

f2 =Li=1

tr

(i,i+1)0 ()

(i,i+1)r ()

(i,i+1)0 () , (A8)

for nearest neighbors. Here, (i,i+1)r = trj=i,i+1 is the

reduced density matrix of sites (i, i + 1) under disorder r,

and (i,i+1)0 is the equivalent in the disorder-free case. We

compute all considered static quantities as the mean overa large number of disorder realizations; for the fidelitiesF,f1, and f2 displayed in Fig. 1B of the main text, weused 5000 realizations at chain length L = 400.

The correlations, finally, can be computed using thefact that the ground state

|

of Eq. (A4) is the vacuum

of the normal modes (i.e., k | = 0, k). For example,for the ZZ-correlations this yields

C(i, j) r|(i)z (j)z |r r|(i)z |rr|(j)z |r= 4r|cicicjcj |r 4r|ci ci|rr|cjcj |r= 4 (hh)ij (g

g)ij 4 (hg)ij (gh)ij . (A9)Away from criticality, the correlations decay as C(i, j) exp( |i j| /) with correlation length . In Fig. 1A ofthe main text, we display extracted from fits to part

of the wings of C(i, j) (for L = 400 and 10000 disorderrealizations).

Without disorder,

C(i, j)L2 f(|i j| /L) (A10)for some universal function f [177]. Hence, one can ex-tract the critical exponent for the correlation length from a data collapse of the correlations. In Fig. 1C ofthe main text, we show the erroneous values for , ex-tracted from Eq. (A10) if one naively neglects that thisrelationship is no longer true in the presence of disor-der. Figure A1 shows the best collapse achieved withEq. (A10) for disorder levels r = 0 and 0.2. The valueof for the best collapse increases with disorder. Hence,using Eq. (A10) on a disordered AQS yields a too largecritical exponent, compared to the ideal model. More-over, the qualitity of the collapse worsens with increas-ing disorder, demonstrating that a naive application ofEq. (A10) is unjustified if disorder is large. For this anal-ysis, we used L = 100 to 190 in steps of 10 with 106

disorder realizations, L = 200, 250, and 300 with 5

105

realizations, and L = 350 and 400 with 105 realizations.The correlations are intrinsically connected to the en-

ergy gap (L), since a gapped system necessarily has ex-ponentially decaying correlations [201]. Via a finite-sizescaling, the gaps at finite systems also allows to extractthe location of the QPT in the infinite system, as seen inFig. A1. There, we plot curves 1/(L(L)) for closebychain lengths L, where is the dynamical critical expo-nent, which for the disorder-free case equals 1. Thesecross at a series of pseudo-critical points which with in-creasing L tends rapidly to the critical point of the ther-modynamic limit [202]. Assuming that it does not changemuch for small disorder, we identify as an approximation

the critical point with the mean of the crossing pointscurves 1/(L(L)) for L = 100, 150, 200, 300, 350, 400with 10000 realizations of disorder each. As displayedin Fig. A1, the crossing point moves to larger values of with increasing disorder (see also Fig. 1A of the maintext). This means that applying this analysis to a real-world AQS without correcting for disorder can yield erro-neous results (compared to the ideal model) for the loca-tion of the QPT. Moreover, the crossing point becomesless well defined with increasing disorder showing thatthis analysis should be corrected for the presence of dis-order. Finally, at large disorder, a second crossing pointappears at lower . The two crossing points open up to aV-like structure with increasing disorder. This could beinterpreted as an indication of the Griffiths phase (thecrossing points are qualitatively consistent with the ex-tent of the Griffiths phase found in [203]). For a morequantitative analysis, however, one would need to ac-count for a change of with increasing disorder.

If a given model can be described by a conformal fieldtheory, its universal critical behavior (including all criti-cal exponents) is completely defined by a single number,the central charge c. To extract it, we compute the vonNeumann entropy S of the reduced density matrix of a


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16

0.5 0 0.5

0

10

20

30

i/L

log( C(L/2,L/2+i) L2 ) r=0; =1.0065

0.5 0 0.5

0

10

20

30

i/L

log( C(L/2,L/2+i) L2

) r=0.2; =2.4984

0.9 0.95 1 1.05 1.10

0.5

1

1.5

2

2.5

6/(L

)

r=0

0.995 1 1.0050.95

1

1.05

0.9 0.95 1 1.05 1.10

0.5

1

1.5

2

2.5

6/(L

)

r=0.2

1 .0 35 1 .0 4 1 .0 450.95

1

1.05

0.93 0.94 0.95

1.2

1.4

Figure A1. Upper row: For low disorder (r = 0, left),the curves C(i, j)L2 for different L plotted as functions of|i j| /L collapse perfectly for the physically correct value ofthe critical exponent = 1 (dots of different color correspondto different L). For increasing disorder (r = 0.2, right), thecollapse worsens, and the best collapse is obtained for some > 1. Lower row: For low disorder (r = 0, left), thecurves 1/(L(L)) cross perfectly at the location of the criticalpoint, = 1. With increasing disorder (r = 0.2, right), thecrossing point moves to larger values of and becomes lesswell defined. Also, at large disorder there appears a secondcrossing point below = 1. The insets show zooms on thecrossing points. The findings of both rows of figures meanthat, ignoring the effects of disorder in an imperfect AQSyields, compared to the ideal model, too large values for thecritical exponent as well as the location of the critical point.

part of the chain of size l for L = 300 with 10000 disorderrealizations. For systems with open boundary conditions,a fit to c/6log(L/ sin(l/L))+A (excluding small valuesof l) yields the effective central charge c [188190]. Thedisorder-free value is c = 0.5. Increasing disorder sup-presses this, indicating the decrease of entanglement inthe system (see Fig. 1C of the main text). Again, apply-ing the analysis that is correct in the disorder-free case(where the system is indeed described by a conformalfield theory) without adjustments to the disordered sys-tem, yields results which deviate from the desired idealcase.

Time dependent fidelities For time evolution, wedistinguish between the zero and finite temperature fi-delities, although the underlying technique is the same.We start by rewriting the fermionic Hamiltonian above

as

H =1

2 H , (A11)

where = (c1,...,cN, c1,...,cN) is a 2N length vec-

tor composed of all creation and annihilation operatorspresent in H, and H = A z + iB y is a 2N 2Nmatrix with complex coefficients.For computing fidelities, we use the convenient Lev-itovs formula [204, 205], which relates traces of oper-ators in the Hilbert space of the fermions to determi-nants of much smaller matrices (like H). For example,

let P = P and Q = Q be two operatorsin the space of fermions, with P and Q complex valued2N 2N matrices. Then,

Tr

ePeQ

= det

1 + ePeQ

. (A12)

Similar formulas hold for more or less operators.In the zero-temperature case, when the initial state

remains pure after evolution, the fidelity takes the formof an overlap

F = | 0(t)|r(t) |, (A13)

where |0(t) = eiH0t |0 is the initial state evolvedwith the target Hamiltonian of the simulation, H0, and

|r(t) = eiHrt |0 is the same state evolved with animperfect Hamiltonian Hr = H0 + rV. Rewriting thefidelity,

F = Tr 0eiHrteiH0t, (A14)

with 0 = |0 0|, we can use Levitovs formula andobtain

F = det

1 G0 + G0eiHrteiH0t

, (A15)

with G0 = 0|G |0, and G the correlation matrix ofthe original fermionic operators, Gi,j =

ij .

If the initial state is not pure, but a thermal state,the state remains mixed even if the evolution is unitary.In this case, we cannot compute the fidelity for the fullmany-body state, but only the fidelity of the reduced den-sity matrix for a few spins. For this we must evaluate thecorrelation functions of the Pauli operators at differentsites of the chain. For the case of a single spin, the sym-metry of the system ensures that at all times the reduceddensity matrix can be written as = (1+ < ix >

ix)/2.

Since < ix >= cici, we only need to compute the evolu-

tion of the diagonal terms in the G correlation matrix.

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