Many-body physics in the NISQ era: quantum programming a discrete time crystal
Matteo Ippoliti, Kostyantyn Kechedzhi, Roderich Moessner, S. L. Sondhi, Vedika Khemani
MMany-body physics in the NISQ era: quantum programming a discrete time crystal
Matteo Ippoliti, Kostyantyn Kechedzhi, Roderich Moessner, S. L. Sondhi, and Vedika Khemani Department of Physics, Stanford University, Stanford, CA 94305, USA Google Research, Venice, CA 90291, USA Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany Department of Physics, Princeton University, Princeton, NJ 08540, USA (Dated: July 24, 2020)Recent progress in the realm of noisy, intermediate scale quantum (NISQ) devices [1] representsan exciting opportunity for many-body physics, by introducing new laboratory platforms with un-precedented control and measurement capabilities. We explore the implications of NISQ platformsfor many-body physics in a practical sense: we ask which physical phenomena , in the domain ofquantum statistical mechanics, they may realize more readily than traditional experimental plat-forms. While a universal quantum computer can simulate any system, the eponymous noise inherentto NISQ devices practically favors certain simulation tasks over others in the near term. As a par-ticularly well-suited target, we identify discrete time crystals (DTCs), novel non-equilibrium statesof matter that break time translation symmetry. These can only be realized in the intrinsicallyout-of-equilibrium setting of periodically driven quantum systems stabilized by disorder inducedmany-body localization. While precursors of the DTC have been observed across a variety of ex-perimental platforms - ranging from trapped ions to nitrogen vacancy centers to NMR crystals -none have all the necessary ingredients for realizing a fully-fledged incarnation of this phase, andfor detecting its signature long-range spatiotemporal order . We show that a new generation ofquantum simulators can be programmed to realize the DTC phase and to experimentally detect itsdynamical properties, a task requiring extensive capabilities for programmability, initialization andread-out. Specifically, the architecture of Google’s Sycamore processor is a remarkably close matchfor the task at hand. We also discuss the effects of environmental decoherence, and how they canbe distinguished from ‘internal’ decoherence coming from closed-system thermalization dynamics.Already with existing technology and noise levels, we find that DTC spatiotemporal order wouldbe observable over hundreds of periods, with parametric improvements to come as the hardwareadvances.
I. INTRODUCTION
The quest to build a universal quantum computer hasfueled sustained progress towards the development of“designer” many-body quantum systems or analog quan-tum simulators, across a variety of platforms rangingfrom trapped ions to superconducting qubits [2, 3]. Whilethe ultimate goal of a fault-tolerant quantum computeris still far into the future, the possibility of harnessing thecomputational power of the quantum world with noisy,intermediate scale quantum (NISQ) [1] devices is alreadya reality. A notable milestone in this context was therecent announcement of “quantum supremacy” (moreaccurately, “quantum computational supremacy” ) inGoogle’s Sycamore device, a solid-state, Josephson junc-tion based platform with 53 qubits [5]. While the compu-tational task chosen for this purpose—simulating the out-put of random quantum circuits—may seem rather ab-stract and not useful in and of itself (though it does haveat least one application [6]), a very active search for high-impact applications of NISQ devices is underway. In this Much of nature routinely carries out processes that are not sim-ulable on a classical computer, but these are not recognizablycomputational tasks on highly controllable and thus recogniz-ably computational devices. See Ref. 4 for a discussion of thispoint. vein, two recent works demonstrated how to implementhighly structured circuits for quantum chemistry simula-tions [7] and combinatorial optimization problems [8] onSycamore.Now, a quantum computer is also necessarily a highlycontrollable many-body system [9], and so these advancesare also extremely tantalizing to many body physicistslooking to push the frontiers of their own discipline. In-deed, Google’s announcement, signifying a major break-through in computational science, also heralded the ad-vent of a new laboratory system with Hilbert spaces ofsignificant size, which can potentially be used to hostand discover new many-body physics.This paper is motivated, broadly, by asking what theNISQ era of tunable, programmable quantum systemsportends for many body physics; and, narrowly, by askingwhat interesting physics could be realized immediatelywith Google’s device. Which physical phenomena in therealm of quantum statistical mechanics can these devicesrealize, that have not yet been (as) crisply demonstratedin any other experimental setting? As with the randomcircuit problem, a first demonstration should perhapsexplore a landscape where some landmarks are alreadyknown and can be used to guide the search while leavingroom for discovery.Two conceptual challenges immediately present them-selves to the many-body physicist: (i) The natural time evolutions implemented on digital gate-based pro- a r X i v : . [ c ond - m a t . d i s - nn ] J u l grammable simulators (such as Sycamore) are quantumcircuits rather than Hamiltonians. This is quite far fromthe typical setting in which condensed matter theory op-erates, which concerns the low-energy, long-wavelengthemergent properties of equilibrium many-body systems.This is also distinct from regimes probed by analog simu-lators, such as cold-atom platforms, which generally tar-get specific model Hamiltonians [10, 11]. And (ii) thetradeoffs between unitary control and platform size in-evitably build some variation in individual circuit ele-ments, which presents an additional challenge for sim-ulating finely tuned model systems. We emphasize herethat we are not viewing these platforms as universal com-putational devices that can simulate any desired unitaryevolution [2, 12, 13] or allow computational investigationof the properties of particular Hamiltonians and quantumstates [14–16]. Instead, due to near-term limitations insize and coherence time, we are interested in identifyingphysical phenomena that these platforms can immedi-ately and naturally realize, as opposed to physics theycould realize universally and asymptotically [17].A parallel set of developments in quantum statisticalmechanics furnishes a domain where these specific chal-lenges turn into strengths: the study of non-equilibriumdynamics, and specifically the assignation of robust phasestructure to many-body systems out of equilibrium. Re-markably, even without the conceptual framework ofequilibrium thermodynamics, a possibility to identifyphases and phase transitions remains [18–20]. This line ofresearch has led to the discovery of new kinds of dynam-ical many-body phenomena that may otherwise be for-bidden by the strictures of equilibrium thermodynamics,a paradigmatic example being a time crystal phase thatspontaneously breaks time-translation symmetry [20–24].Combining these insights leads us to focus on dynam-ical phases in disordered, out-of-equilibrium quantummatter – specifically, many body localized (MBL) period-ically driven (or Floquet) phases – as natural candidatesfor the NISQ-era scientific program outlined above. In-eed, the quantum circuit structure which is Sycamore’s modus operandi lends itself naturally to implement var-ious Floquet drive protocols. Further, for these appli-cations, randomness in circuit elements is not only tol-erated, but is in fact necessary to stabilize the systemagainst heating, and thus for observing interesting phe-nomena. For these reasons, in this work we propose pre-cisely such a ‘physics-forward’ use of the Sycamore deviceand its relatives: to realize an MBL Floquet time crys-tal, a non-equilibrium many-body phase of matter thatdisplays an entirely new form of spatiotemporal order [20–22].Our choice has several desirable aspects: (i) the DTC isa genuine collective many-body phenomenon, and repre-sents the best known example of a new paradigm in quan-tum statistical mechanics, that of an out-of-equilibriumphase of matter; (ii) it is of clear fundamental and con-ceptual importance, given its distinctive pattern of spa-tiotemporal order; and (iii) despite promising precur- sors [25–28], a bona fide realization of this phase (orany many-body out-of-equilibrium phase, for that mat-ter) has proved elusive for differing reasons in each ofthe existing experimental platforms in which it has beenexplored. Indeed, as we explain below, there are funda-mental definitional aspects of the physics of this phase,specifically its central attributes of spatiotemporal orderand robustness to choice of initial state, that have notyet been observed [23]. Not only have these not beenobserved, detailed theoretical analysis has shown thatthese defining features are fundamentally absent in thestate-of-the-art experiments probing the DTC [23]. Thusthis proposal is not about repeating previous experimentswith incremental extensions to the scope of their obser-vations; rather, it is about realizing and demonstratingthe first genuine instance of this phase.There is much reason to be optimistic. The prior im-pressive experimental studies on DTCs have enabled adetailed understanding of the remaining obstacles to therealization of this phase, so that this goal appears emi-nently achievable in the near term. The resulting check-list contains several requirements that are hard to simul-taneously satisfy in the previous setups. But these aresufficiently well-defined to be individually addressed andsimultaneously realized on the Sycamore device. Indeed,as we show in this work, the existing capabilities, archi-tecture and gate-set in Sycamore satisfy all the desider-ata, and the platform seems almost tailor made for thisapplication!We flesh out our proposal as follows. Sect. II A con-tains a telegraphic account of the basics of DTCs to ori-ent the following discussion. Sect. II B presents a de-tailed account of the insights from previous experiments,from which we distill a list of experimental desideratain Sect. II C . Sec. III details how to meet these, andexplains how to address the implementation of the re-quired experimental protocol on a present-day quantumdevice, Google’s Sycamore processor. We then provideevidence that the phenomenon we are looking for is in-deed present for a range of experimentally achievable pa-rameters (Sec. III B), and present an analysis of noiseand other experimental imperfections to argue that itsobservation is possible despite present limitations of theNISQ platform (Sec. IV). We conclude by discussing ourresults and directions for future work in Sec. V. II. THE DISCRETE TIME CRYSTAL: THEORYAND EXPERIMENTS
We begin by briefly recapitulating the physics of theDTC phase in Sec. II A, which defines the model and no-tation. This provides a minimal set of facts about theDTC needed to render this article self-contained; it maytherefore be read diagonally by those with prior expo-sure to the field. Sec. II B discusses the state of the artin experimental efforts to engineer the DTC, followed inSec. II C by the enumeration of an experimental check-list of ingredients for realizing and observing this phase.These have not been simultaneously achievable in anysingle platform thus far. We refer the reader interestedin an in-depth account of these issues to a review on timecystals by some of the present authors [23].
A. Theoretical definitions and models
The canonical model of a (discrete) time crystal [20] isrealized in a Floquet system with a time-periodic Hamil-tonian, with discrete time-translation symmetry (dTTS) H ( t ) = H ( t + T ). A DTC spontaneously breaks thedTTS of the drive: observables in this phase show peri-odic dynamics with a period mT , with Z (cid:51) m >
1, cor-responding to a sharp subharmonic response in the fre-quency domain (for example, m = 2 for period-doubleddynamics).Period doubling (or multiplexing) is ubiquitous in clas-sical and quantum dynamical systems, in settings rang-ing from Faraday waves to parametric oscillators [29–31].However these examples arise in single- or few-body sys-tems, or in systems that are effectively few-body (in amean-field sense) [23]. On the other hand, defining atime crystal as a non-trivial, many-body phase of matterrequires us to consider macroscopic, strongly-interactingquantum systems. This is, in fact, the only setting inwhich time translation symmetry breaking is unexpectedfrom the viewpoint of equilibrium thermodynamics; one-or few-body systems, such as simple harmonic oscilla-tors, routinely exhibit oscillations and revivals in theirdynamics.A pervasive challenge with periodically driven many-body systems is their tendency to absorb energy from thedrive and thermalize to infinite temperature, maximiz-ing entropy in the absence of conservation laws [32, 33].One robust mechanism for escaping this “heat death”is many-body localization (MBL), wherein the dynamicsfails to establish local thermal equilibrium even at arbi-trarily late times due to disorder [34–39]. Thus preventedfrom heating to a trivial state, the system can supportvarious non-trivial non-equilibrium phases, of both thesymmetry-breaking and topological varieties [20]. TheDTC is one such example.We now turn to specific model realizations of thisphase. A standard model of a Floquet DTC is an Isingmodel periodically “kicked” by a rotation about the ˆ x axis [20]. The dynamics probed at ‘stroboscopic’ times, t = nT, n ∈ Z are captured by studying the proper-ties of the ‘Floquet unitary’, which is the time-evolutionoperator over one period, U F = e − ig (cid:80) i X i e − iT ( H z + H int ) , (1)where T ≡ X i ( Z i ) denote spin-1 / x ( z ) operators on site i , H z = (cid:80) i,j J ij Z i Z j is a diagonal Hamiltonian with Ising symmetry P = (cid:81) i X i , and H int represents additional generic interac-tions that may be present (examples include longitudi- nal fields H int = (cid:80) i h i Z i or XY interactions H int = (cid:80) ij J ⊥ ij [ X i X j + Y i Y j ]). Localizing the system to preventheating will require disorder in the couplings J ij .The model in Eq. (1) can potentially realize a discretetime-crystal phase in the regime g = ( π − (cid:15) ), with (cid:15) sufficiently small. This represents an imperfect ‘ π -pulse’ i.e. a nearly 180 ◦ rotation about the x axis. To un-derstand the properties of the phase, consider first thelimit (cid:15) = H int = 0. In this case, it easy to see thatstarting with a product state in the ˆ z basis, one actionof the unitary enacts a perfect 180 ◦ rotation and flipsall spins; these are then flipped back under a second ac-tion of U ( T ), thereby showing period doubled dynamics, (cid:104) Z i ( mT ) (cid:105) = ( − m (cid:104) Z i (0) (cid:105) .While the (cid:15) = H int = 0 limit is illustrative, defining theDTC as a phase of matter requires some degree of stabil-ity to the choice of parameters and interactions. Indeed,what is remarkable is that under suitable conditions (re-quiring the presence of MBL), the dynamics can remainrobustly locked at period doubling for infinitely long timesin an extended region of parameter space, i.e. even forimperfect rotations ( (cid:15) (cid:54) = 0) and in the presence of genericperturbing interactions ( H int (cid:54) = 0) [20–22]. We empha-size that this stability is inexplicable using any kind ofsemi-classical intuition; without quantum ordering, onewould expect a finite deviation in rotation angle ( (cid:15) (cid:54) = 0)to accumulate over consecutive cycles, destroying the pe-riod doubling over a finite time scale ∼ (cid:15) − .Instead, the rigid locking of the dynamics to perioddoubling follows from the presence of long-range order inspace that stems from spontaneously breaking Z Isingsymmetry, whence ‘spatiotemporal’ order [22]. This re-quires the Ising interactions H z to be the dominant partof the evolution during the first part of the drive. Atany stroboscopic time, spins are locked into a “frozen”pattern in space so that (cid:104) Z i Z j (cid:105) is nonzero for arbitrarilylarge | i − j | even in highly-excited states (but can havea random, “glassy” sequence of signs as a function of i , j ). This pattern then flips every period. Notably, theDTC phase is also stable to the addition of interactionsthat explicitly break Ising symmetry, such as longitudi-nal fields H int = (cid:80) i h i Z i [21, 22]. In this case, the long-range spatial order follows from spontaneously breakingan emergent Ising symmetry [22]. This is a manifestationof the fact that the DTC phase is, in fact, stable to all weak perturbations of the Floquet unitary (1), includingthose not encapsulated by H int or (cid:15) — a feature termed absolute stability by a subset of the present authors [22].In sum, the DTC is a robust, many-body phase ofmatter with spatiotemporal order (long-range order inspace + infinitely long-lived period doubling dynamics intime), realized in the intrinsically non-equilibrium settingof periodically driven, MBL quantum systems. Prob-ing spatiotemporal order requires measuring site-resolvedobservables, e.g. (cid:104) Z i Z j (cid:105) , and temporal autocorrelationfunctions, e.g. (cid:104) Z i ( n ) Z i (cid:105) . B. First Generation DTC Experiments
The DTC phase is particularly amenable to experi-mental detection due to its stability and its distinctivemeasurable dynamical signatures. Indeed, the theoret-ical prediction of this phase was rapidly followed by apair of experiments, one on disordered trapped ions in1D [26] and the other on disordered nitrogen vacancy(NV) centers in 3D diamond [25]. An experiment usingnuclear magnetic resonance (NMR) on a clean crystalline3D solid followed soon after [27, 28]. We will refer tothis set of experiments as “First Generation” (FirstGen)time-crystal experiments.Each of the FirstGen experiments simulates a modeldrive captured by Equation (1). The experiments dif-fer in various key details and, between them, realize avaried matrix of parameters such as spatial dimension,range and type of interactions, nature of disorder, statepreparation capabilities, microscopic controllability etc.Each one represents an experimental tour de force , andmanages to observe temporal signatures of DTC behav-ior (i.e. a signal locked at period doubling) over a fi-nite extent in parameter space for the (finite) coherencetime of the experiment. Despite the numerous differencesbetween the platforms, the observed signatures look re-markably similar. However, despite these encouragingresults, none of these platforms have all the ingredientsneeded for a genuine, asymptotic incarnation of the MBLDTC phase [23].A key challenge for all three experiments lies in stabi-lizing MBL. Despite this, all three platforms still observelong-lived precursors of DTC order. This is because, evenin cases where MBL is disallowed, it may neverthelessbe possible to engineer a separation of scales such thatthermalization happens on a parametrically slow scale –referred to as a ‘prethermal’ regime in certain cases [40–44]. Specifically: the diamond NV center experiment [25]is incompatible with MBL because of its long range ofinteractions, but instead realizes a ‘critical TC’ whichthermalizes in a power-law slow fashion [45]. Likewise,the NMR setup [27] has no disorder and hence no MBL,and the long-lived signal therein was later explained asa prethermal phenomenon associated with a global con-servation law [46]. Finally, the trapped ion setup [26] isthe smallest and most controllable, and has many of thenecessary ingredients for realizing MBL; however, it wasshown in Ref. [23] that, unexpectedly, the nature of dis-order in this setup is also not sufficient for localization,and the signal observed herein also turned out to be of aprethermal rather than asymptotic nature.Despite not realizing an asymptotic MBL DTC, allthree FirstGen experiments (and others [47], mentionedbelow) have greatly advanced our conceptual understand-ing of the DTC phase and led to new theoretical insights.These include the elucidation of a new mechanism forprethermalization [46] following the NMR experiment,and an understanding of the distinct types of disorderneeded to stabilize MBL phases with distinct types of quantum order [23]. These insights have enabled us toformulate a detailed checklist of desired experimental ca-pabilities for the next generation of DTC experiments.As an example, the eventual theoretical understandingof the FirstGen experiments as prethermal (or slowlythermalizing) phenomena – albeit of conceptually dis-tinct genres – emphasizes that a key experimental chal-lenge is to distinguish a genuine MBL DTC phase froma transient prethermal version. In principle, the maindifference between localized and prethermal DTCs lies inthe lifetime of their quantum order: infinite for the for-mer, transient for the latter [42]. However, the ubiquityof environmental decoherence makes this distinction voidin practice – measured
DTC signals will be transient nomatter what. Nevertheless, as we discuss below, fine-grained measurements of spatially resolved observableson a variety of initial states can discriminate betweenprethermal and asymptotic TCs, even within finite ex-perimental lifetimes.
C. Experimental checklist
In all, the FirstGen DTC experiments, with their var-ied strengths and limitations, have been instrumental indistilling a checklist of experimental ingredients neededfor the realization and detection of a bona fide DTCphase. These ingredients, and their presence or ab-sence in the various experiments, are summarized in Ta-ble I and articulated in more detail below; these serve toachieve two intertwined goals: • Realizing a genuine asymptotic
MBL DTC phase,i.e. engineering all the theoretical criteria forachieving MBL and DTC order, so that an ‘ideal’experiment (without external decoherence) wouldobserve an infinitely long-lived signal.This is a matter of principle - if internal decoher-ence (due to many-body quantum thermalization)in an ideal, noise-free incarnation of the platformdestroys the signal at late times, then the systemdoes not realize an asymptotic DTC phase (thiswould be true of all FirstGen experiments). On theother hand, if the lifetime is predominantly limitedby external decoherence, then this is an issue of en-gineering that will see sustained improvement withfuture hardware innovations. • Detecting the spatiotemporal order that is a defin-ing feature of the phase. This also entails exper-imentally discriminating between asymptotic (in-finitely long-lived) and prethermal (transient) vari-ants of DTCs, even within the constraints of envi-ronmental decoherence and finite experimental life-times.We now enumerate six desired experimental capabili-ties, grouped in three broad categories.
Requirements ExperimentsNV Trapped NMR Sycamorecenters ions crystal
Definitional
Long coherence time (cid:51) (cid:51) (cid:51) (cid:51)
Many-body (cid:51)(cid:51) ∼ (cid:51)(cid:51) (cid:51) Stabilizing MBL
Short-range int. (cid:55) (cid:51) (cid:55) (cid:51)
Ising-even disorder (cid:51) (cid:55) (cid:55) (cid:51)
Detection
Site-resolved meas. (cid:55) (cid:51) (cid:55) (cid:51)
Varied initial states (cid:55) ∼ (cid:55) (cid:51) TABLE I. Summary of experimental requirements for real-izing and observing DTC spatiotemporal order, and the rel-ative merits of different experimental platforms. The ‘dou-ble’ check-marks for the NV and NMR platforms in the‘many-body’ category are to emphasize that these setups,with > O (10 ) constituents, are operating in the thermody-namic regime, at a size that is orders of magnitude larger thanthe trapped ion experiment ( ∼
10 ions) and Sycamore ( ∼
1. Basic definitional requirements
As mentioned earlier, a DTC phase is characterizedby infinitely long lived, quantum-coherent oscillations ininfinitely large, macroscopic many-body systems. Whilean actual experiment will always be of finite size with afinite coherence time, non-trivial realizations still requireboth of these to be sizeable, with room for parametricimprovements with engineering advances. Thus two basicrequirements on the platforms are:(i)
Truly many-body.
The experimental systemsshould contain a number of qubits that does not qual-ify as “few-body”. While there is no sharp boundarybetween “few” and “many”, it is clear that the NVand NMR experiments satisfy this requirement ( > qubits), while the trapped ion experiment (10 qubits)may be considered border-line – a few tens to hundredsof qubits would more comfortably fit the description. Anadded bonus is if the platform permits one to vary thesystem size, which would allow for finite-size scaling anal-ysis of various order parameters. Another scenario ruledout by this requirement is that of effectively few-body sys-tems where, despite a nominally large number of qubits,the dynamics becomes few-body in a mean-field sense.Several recent TC experiments fall in this category [47–50], with Ref. [47] furnishing a particularly nice exampleusing NMR on ‘star-shaped’ molecules. We remark thatthis point is not about classical simulability, but specif-ically about physics. Time-crystals are only non-trivialfor macroscopic many-body systems; few-body systemsexhibit special phenomena (e.g. recurrences) that do notscale to the many-body limit, and could prove confound-ing to the observation of the desired phenomenon.(ii) Long Coherence time.
Experimental platformsaiming to exhibit dynamical phases clearly must be able to preserve quantum coherence for long enough, sothat the underlying dynamical phenomena can be distin-guished from short-time transients. Again, while thereis no sharp boundary, revealing DTC order requires acoherence time of at least multiple tens to hundreds ofFloquet cycles. We caution, however, that this may stillnot be enough to discriminate between MBL and prether-mal TCs without using additional fine-grained probes (cf.points (v) and (vi) below). All the FirstGen platformshad a lifetime on the order of 100 Floquet periods.
2. Requirements for stabilizing MBL
MBL is an essential ingredient for realizing a robustDTC phase in an extended region of parameter space,and in preventing periodic driving from heating the in-teracting system to infinite temperature. However, MBLis only stable under certain conditions sensitive to therange of interactions, and the scope for engineering dis-order:(iii)
Short-ranged interactions.
Long-ranged interac-tions are known to destabilize localization [51, 52]. In-teractions with strength scaling as 1 /r αij are incompatiblewith MBL if α > d , where d is the dimension of the sys-tem [52] . Out of the FirstGen experiments, the only one(marginally) satisfying this requirement is the one basedon trapped ions ( d = 1, α ≈ . d = α = 3 and are thusnot compatible with MBL.(iv) Dominantly Ising interactions with Ising-evendisorder.
While stabilizing MBL generically requiresdisorder in the drive parameters, the nature of the dis-order required to stabilize an MBL DTC is more spe-cific: one requires strong disorder in
Ising-even interac-tions H z = (cid:80) ij J ij Z i Z j [23] in a drive with dominantlyIsing interactions of the form (1). If, instead, the only op-erators coupled to disorder are odd under the Ising sym-metry P x = (cid:81) i X i (as is the case e.g. for on-site fields H int = h i Z i ), then the Floquet evolution over two cycles, U F , is only weakly disordered, and the dynamics is con-sequently not MBL. This is because the disordered fieldsare ‘echoed out’ by the approximate π -pulse, to leadingorder (see Appendix A for a discussion of this point).Of the FirstGen experiments, only the NV platform real-izes Ising-even disorder due to the random position of NVcenters in three-dimensional space; while this alone is notenough for MBL (because of the long-range interactions), Note that for the purpose of this article, we are not concern-ing ourselves with the open question of possible non-perturbative instabilities of MBL that may asymptotically destabilize localiza-tion in dimensions greater than one, or with power-law decayinginteractions with any power [53]. These effects, if they exist, willhappen for system sizes and time scales that are well beyond thecapabilities of any near-term simulators. the disorder still leads to algebraically slow themaliza-tion, giving a ‘critical time crystal’ in the NV setup. TheNMR system is clean and spatially ordered, and hencenot localized. Finally, the trapped ion setup featuresdisorder only in Ising-odd longitudinal fields, while theIsing-even interactions are non-random and well approx-imated as J ij ∼ J /r αij . In a finite lattice the displace-ments r ij (and thus the interactions J ij ) will includeweak inhomogeneities due to the interplay of Coulombinteractions with the confining trap; however these in-homogeneities are perfectly deterministic and reflection-symmetric, and turn out to be insufficient to stabilizeMBL [23]. In general, it is easier for many experimentalsetups to implement disorder in onsite fields rather thanIsing couplings, and this requirement is a key obstacletowards realizing DTCs on many such platforms.
3. Requirements for detection
Finally, we turn to the requirements of unambiguouslydemonstrating the DTC phase and distinguishing it fromits transient prethermal cousins – even within the realityof finite experimental lifetimes.The key idea of prethermal dynamics is that, in a suit-able reference frame, the system behaves for a long time as though it was governed by a static effective Hamilto-nian (although the temperature of the state slowly in-creases en route to infinite temperature) [40, 41]. If theeffective Hamiltonian has an ordered phase below a criti-cal temperature T c , then a low-energy initial state woulddisplay quantum order for a long time, before eventu-ally heating past T c thus causing the order to melt [42].However, a high-energy initial state would not show anyorder, even for short times. Thus, practically, a usefuldiscriminatory criterion is the dependence of the signalon the choice of initial state. In MBL DTCs there shouldbe no strong dependence (as the whole spectrum is lo-calized). On the other hand, certain prethermal DTCs(those associated with symmetry breaking) display long-lived oscillations for low-temperature ordered states butnot for others.Separately, another mechanism for prethermalizationis the emergence of a quasi-conserved quantity associatedto an approximate symmetry of the prethermal Hamil-tonian [41, 46]. This mechanism for slow thermaliza-tion can be at play even for very high-temperature initialstates. In this case, measurements of global observablessuch as the total magnetization are at risk of detectingthe slow relaxation of a quasi-conserved quantity ratherthan the DTC pattern of spatiotemporal order. Thus,one requires:(v) Widely tunable initial states.
To distinguish lo-calized and prethermal DTCs within a finite experimen-tal lifetime, one needs to test a variety of initial states(prethermal DTCs are highly sensitive to the choice un-like MBL DTCs). This cannot be done on platforms that only allow for the preparation of certain initial states,such as fully polarized ones. Only the trapped ion ex-periment has the capability to widely vary initial states,although this was not fully explored in Ref. [26]. Theexperiment only considered two states: a fully polarizedstate, | (cid:105) ⊗ L , and a state polarized on the left and righthalves, | (cid:105) ⊗ L/ | (cid:105) ⊗ L/ . However polarized or near polar-ized states are maximally ineffectual at distinguishing be-tween MBL and prethermal dynamics [23, 46]. Becausethe trapped ion experiment has long-range interactions,the effective Hamiltonian governing the prethermal dy-namics can have an Ising symmetry breaking transitionat a finite temperature T c even in one dimension, andnear polarized states are in the low-temperature sectorof the effective Hamiltonian. Indeed, detailed numeri-cal simulations of the trapped ion experiment on a widerclass of initial states found strong initial state depen-dence, with the DTC signal decaying much more rapidlyfor randomly picked initial product states, consistentwith prethermal DTC order [23]. Separately, a differentmechanism for prethermalization entails the long-livedquasi-conservation of a global operator such as the totalmagnetization. Again, polarized initial states have largetotal magnetization and can show slow dynamics due tothe quasi-conservation law, while randomly picked prod-uct states would not.(vi) Site-resolved measurements.
Detecting genuinespatiotemporal order requires measuring site-resolvedspatial correlation functions of the form (cid:104) Z i Z j (cid:105) , in ad-dition to temporal autocorrelators. This capability tolocally probe individual qubits is also necessary for dis-tinguishing MBL TCs from prethermal variants involvingglobal quasi-conservation laws. For instance, the NMRexperiment operates in an extremely hot regime, withvery high temperature initial states that would be wellabove the ordering temperature T c of the effective Hamil-tonian; but these can still show slow dynamics in globalobservables that couple to a quasi-conservation law, suchas the total magnetization [46]. In contrast, local auto-correlators would show a fast decay in this regime. TheNMR and NV center experiments (which involve > qubits) are limited to probing spatially averaged quanti-ties such as the total magnetization (cid:80) i Z i , which do notprovide the necessary resolution. Among the FirstGenexperiments, only the trapped ion experiment satisfiesthis requirement.We now turn to how the next generation (NextGen)of quantum simulators - such as the already operationalGoogle Sycamore processor - can be programmed to re-alize all these ingredients in turn, and hence to furnishthe first bona fide realization of the time-crystal phase.We should note that while the trapped ion experimenthas not yet demonstrated an MBL DTC phase, it maybe possible for future iterations of this platform to doso. The key engineering challenges entail scaling up thesystem to suitably larger numbers of ions, and addinguncorrelated disorder in the Ising couplings J ij (which G = (a) R z R z R z R z G (c) G
2, 3 G
4, 5 G
6, 7 G
1, 2 G
3, 4 G
5, 6 R x R x R x R x R x R x R x (b) FIG. 1. Simulating a 1D Floquet DTC on the Sycamore chip.(a) Modified transmon gate (cid:101) G in terms of the native gate G and single-qubit Z rotations. (b) Circuit for the DTC Flo-quet unitary: each Floquet cycle acts with (cid:101) G on each pairof neighboring qubits, followed by single-qubit X rotations,as depicted. (c) A closed loop through the Sycamore chip,simulating a 1D system. During each cycle, (cid:101) G gates act firston the blue bonds, then on the black bonds. All other bondsremain idle during the dynamics. is possible, in principle, with extensively many tuningknobs [54, 55]). These are achievable given enough timeand effort. Likewise, quantum simulators using Rydbergor dressed Rydberg atoms meet almost all the desired cri-teria, and are currently limited only by their coherencetime [56]. Future improvements will no doubt also en-able the observation of such phenomena on this versatileplatform. However, as we demonstrate next, currently existing capabilities in the Sycamore device already sat-isfy all the desiderata and, indeed, the platform seemstailor made for this application! III. NEXT GENERATION: REALIZING A DTCON THE SYCAMORE PROCESSOR
NextGen programmable quantum simulators are de-signed with quantum computing applications as a majordrive. These applications happen to require many of theitems of the above checklist. The preparation of arbi-trary computational-basis states and the capability forsite-resolved read-out are both key ingredients for quan-tum computing [9], so it is fair to assume their availabilityon a NISQ device. Moreover, these devices are designedto implement quantum circuit elements that are typicallyone- and two-qubit gates, which in the quantum many-body language means on-site fields and nearest-neighbor interactions; great care is taken to ensure that any cross-talk, i.e. longer-ranged coupling, is negligible within thecoherence time. Indeed, such finite-range interactions aremuch more suitable for MBL compared to the power-lawdecaying couplings native to many platforms [53]. Thusshort-ranged interactions (requirement iii), site-resolvedmeasurements (requirement vi) and tunable initial states(requirement v) are all at our disposal. Moreover, asthese devices enter the 50-to-200-qubit NISQ regime [1]they can be safely regarded as legitimate quantum many-body systems (requirement i) .According to the checklist in Section II C, The lasttwo points to be addressed are (a) whether the coherencetimes are long enough, given the eponymous noise inher-ent to NISQ devices and (b) whether the devices can im-plement a kicked Ising drive similar to the one in Eq. (1),with disorder in the Ising couplings, J ij . While a univer-sal fault-tolerant quantum computer can, of course, real-ize any drive with any set of couplings [12, 57], presentday NISQ devices may present obstructions due to theirfinite coherence time. Again, we are motivated by near-term applications that are immediately and naturally re-alizable on these platforms (as opposed to universallyand asymptotically). To address these points in a morespecific way, we focus on Google’s Sycamore processorfor the remainder of this work. In Sec. III A we lay outthe details of implementing the Floquet DTC as a quan-tum circuit with gates available on Sycamore, while inSec. III B we map out the phase diagram of this circuitmodel and present several diagnostics of the MBL DTCphase. All of the analysis for now assumes an ‘ideal’, i.e.decoherence-free realization; the analysis of noise whichinforms the coherence time is presented in Sec. IV. A. Floquet DTC circuit on Sycamore
We begin by noting that the Floquet unitary evolu-tion operator for the canonical model of a DTC, Eq. (1),can be naturally written as a sequence of gates when H int = 0, and when the J ij couplings are limited to near-est neighbors. We confine the dynamics to a one dimen-sional system, where the existence of MBL and thus ofthe DTC phase is on firmest ground [35, 53]. In this case,one first acts with a layer of Ising gates e − iJZZ on theeven bonds of the 1D subsystem, then a layer of Isinggates on the odd bonds, and then a layer of single-qubit X rotations, e − igX : U F = e − ig (cid:80) i X i e − i (cid:80) i J i Z i Z i +1 = (cid:89) i R xi (2 g ) (cid:89) i e − iJ i − Z i − Z i (cid:89) i e − iJ i Z i Z i +1 (2)where R xi ( α ) = e − iαX i / is a single qubit X rotation.This model has Ising symmetry and is exactly solvable,being mappable to free fermions. In this limit, the sys-tem is in the DTC phase (with period doubled dynamicsand spontaneously broken Ising symmetry) as long as theaverage J couplings obey [20] (cid:12)(cid:12)(cid:12) J i − π (cid:12)(cid:12)(cid:12) ≤ g − π g, J i ∈ [0 , π/
2] without loss of generalityas the phase diagram repeats symmetrically outside thissquare). As mentioned earlier, the DTC phase persistsfor a finite region in parameter space surrounding g = π ,even upon perturbing the drive in Eq. (2) with genericinteractions to make the model non-integrable, as long asthe disorder in J i is strong enough to stabilize MBL.On the Sycamore chip, a unitary evolution close toEq. (2) can be straightforwardly implemented. Singlequbit X rotations R xi are readily available [5]. For thetwo-qubit interaction, the Sycamore device allows imple-mentation of a continuously parameterized family of highfidelity ‘transmon’ gates of the form [58, 59] G , = R z ( h a ) R z ( − h a ) fSim , ( θ, φ ) R z ( h b ) R z ( h c ) , (4)where R zi ( α ) = e − iαZ i / is a single-qubit Z rotation, the h angles result from the frequency excursion of the singlequbits during the interaction , and fSim is the ‘fermionicsimulation’ two-qubit gate [60], fSim , ( θ, φ ) = e − i θ ( X X + Y Y ) − iφ Z − I Z − I , (5)defined by an ‘iSWAP angle’ θ and a ‘controlled-phaseangle’ φ . The latter provides the crucial ingredient forthe Floquet DTC unitary: the two-qubit Ising coupling e − iJZZ , with the identification J ≡ φ/ θ and the single-qubit Z rotations (coming both from fSim and from the h angles), represent deviations away fromthe solvable limit in Eq. (2), but these deviations canbe controlled and manipulated rather straightforwardly.Specifically, the angles θ ij , one for each coupler in theSycamore chip, can be independently tuned to arbitraryvalues (including zero) within calibration accuracy. Forthe purpose of this paper we will sample each θ ij out ofa normal distribution with variable mean θ and standarddeviation ∆ θ = π/
50, representing gate calibration errorof a few degrees ( π/
50 rad = 3 . ◦ ), a deliberately conser-vative upper bound. The ‘extra’ single-qubit Z rotationscan also be tuned and cancelled “by hand” (within cali-bration accuracy) with active Z rotations of appropriateangles on each qubit before and after each application of G , see Fig. 1(a). The result is a modified gate (cid:101) G i,j = R zi ( δh ija ) R zj ( − δh ija ) e − i θ ij ( X i X j + Y i Y j ) − i φ ij Z i Z j × R zi ( δh ijb ) R zj ( δh ijc ) (6) There are only 3 independent angles because Z + Z commuteswith fSim . where the δh are small residual rotation angles, taken tobe normal random variables of standard deviation ∆ h = π/
50. Note that the non-zero ∆ h , θ and ∆ θ make themodel genuinely interacting and non-integrable; the ∆ h terms also break the Ising symmetry. Both effects arenecessary for a nontrivial demonstration of the stabilityof the phase. Thus, even as calibration errors continue toimprove, these deviations can and should be deliberatelyincluded for a non-trivial demonstration of the phase.We have explicitly verified by numerical diagonalizationthat ∆ h = ∆ θ = π/
50 is large enough to visibly breakintegrability even when θ = 0.With the (cid:101) G gate defined above, it is now straightfor-ward to define our model Floquet circuit: U F = (cid:89) i R xi (2 g ) (cid:89) i (cid:101) G i − , i (cid:89) i (cid:101) G i, i +1 , (7)sketched in Fig. 1(b). This represents a generically per-turbed and non-integrable variant of the solvable modelin Eq. (2). Single-qubit rotations are widely and easilytunable on Sycamore, allowing for arbitrary values of theˆ x rotation angle 2 g (or equivalently the π pulse imper-fection (cid:15) = π − g ). The two-qubit gates act, in turn,on the even and odd bonds along a one-dimensional paththrough Sycamore, such as the one sketched in Fig. 1(c).All the parameters specifying the individual (cid:101) G ij gates( φ ij , θ ij , δh ija,b,c ) are drawn randomly for each gate (oneper spatial bond), but are time-independent : all thesechoices are fixed once per realization, and then repeatedin time so as to define an ideal time-periodic (Floquet)model . Again, we chose to use a one-dimensional paththrough Sycamore rather than the full 2D array of cou-plers in order to remain within the territory where MBLand the DTC phase are firmly established on theoreticalgrounds. However we note the extreme flexibility of thisplatform in potentially choosing different geometries –e.g. 1D paths of different lengths, with open or periodicboundary conditions, or 2D patches of various shapes –all on the same chip, simply by selecting which couplersto activate and which to leave idle during the dynamics.Having discussed the parameters g, θ ij , δh ija,b,c above,we now turn to the φ ij angles, which set the strengthof the ZZ coupling and address the final requirementof Ising-even disorder. From an engineering perspective,two-qubit gates are generally more demanding than sin-gle qubit rotations: each distinct gate acting on a givenbond (cid:104) i, j (cid:105) must be calibrated individually [58]. Thephases φ ij are thus drawn randomly from a discrete setof M values ( M ∼
10 appears realistic in the near term),rather than a continuous distribution as is usually as-sumed in studies of MBL. In this work we choose the Any temporally random fluctuations and/or additional decoher-ence due to the execution of the active Z rotations can be ac-counted for by increasing an effective ‘Pauli error rate’; we willreturn to them when we discuss the noise model in Section IV. discrete set of disordered couplings to be { φ + W cos( πm/ ( M − m = 0 , . . . M − } , (8)where φ sets the average coupling and W the disorderstrength. The use of a nonlinear function ensures thatthere are incommensurate spacings between the differentphases φ ij , thus limiting the effect of accidental reso-nances ; the choice of cos( x ) is otherwise arbitrary andis expected to yield generic results. For specificity, inthe following we fix the average controlled-phase angleto φ = π corresponding to J = π/
4. This choice is at thecenter of the DTC phase in the non-interacting model,and allows for the widest range of rotation angles g (cf.Eq. (3)). The disorder strength is set to W = π/
2; thisis fairly strong while also ensuring that all the φ anglesare far from 0 (where the experimental implementationcould be problematic in some cases [59]). Finally, we set M = 8 based on numerical results obtained via full diago-nalization of the Floquet unitary U F which indicate thatthat M = 8 disorder values are sufficient to qualitativelyreplicate the continuous disorder ( M → ∞ ) case.The quantum circuit so defined captures all the cru-cial aspects of the canonical Floquet DTC, Eq. (1),in a “Trotterized” form. It differs from the solvablelimit, Eq. (2), in specific ways: the nonzero iSWAP an-gles θ ij introduce interactions and make the model non-integrable; the nonzero longitudinal fields, ∆ h , also addinteractions and weakly break the Ising symmetry; andfinally the disorder in the φ ij couplings is discrete ratherthan continuous.In the following we confirm that these do not destroythe DTC phase, as expected from its absolutely stablenature [22]. By varying g and θ , with all other parame-ters fixed as described above, we obtain a phase diagramfor the model circuit, shown in Fig. 2. This was obtainedby combining various phase diagnostics, discussed in thenext section. It includes two MBL phases for sufficientlyweak θ : a DTC phase near g = π/ π -flip), and a paramagnetic phase near g = 0.These are separated by a large thermal region, whichexpands as the interaction strength θ is increased, even-tually destroying both MBL phases for θ (cid:38) π/
8. Thenext section presents a detailed discussion of the diag-nostics used to obtain this phase diagram and to detectthe different phases in an experimental setting. Localization is expected to be stable even with discrete (ratherthan continuous) disorder, provided the number of values M inthe discrete set is large enough ( M = 2 is pathological) [61].However, we note that discrete disorder falls outside the setof conditions required for a rigorous non-perturbative proof ofMBL [35], and may thus generate resonances that eventuallydestabilize localization. However, any such effects would appearon a parametrically long timescale, akin to concerns regardingthe stability of MBL in higher dimensions or with power-law in-teractions of any power [53]. These open issues are beyond thepurview of this work, and will be invisible at the system sizesand times accessible to near-term devices. g / MBLPM MBLDTCthermal g / r Poisson L FIG. 2. Phase diagram of the circuit Eq. (7) as a function ofthe pulse parameter g and the average iSWAP angle θ . Inset:level spacing ratio (cid:104) r (cid:105) , Eq. (9), vs g on the θ = 0 cut. (cid:104) r (cid:105) is averaged over eigenstates and over between 400 and 4000realizations of disorder (depending on L ). B. Diagnostics of the MBL DTC phase
Nonequilibrium phases and phase transitions are un-derstood as eigenstate phases [19, 62–64]; their theoreti-cally sharpest diagnostics involve properties of the many-body eigenspectrum and of individual many-body eigen-states of the Floquet unitary U F , which change in a singu-lar manner across phase boundaries. While theoreticallyuseful, these eigensystem diagnostics are not directly ac-cessible to experiment, and their numerical explorationis limited to the small sizes amenable to exact diagonal-ization of U F . Fortunately, these diagnostics translateto distinctive measurable signatures in dynamics fromgeneric initial states, that are both observable in experi-ment and accessible to numerics for much larger sizes.We now present various eigenspectrum and dynamicaldiagnostics for identifying both MBL and the DTC order,which were used to derive a phase diagram for the modelpresented in the previous section. Level Repulsion
Many-body localization, aside from itsdynamical signatures in the form of a persistent memoryof initial conditions, is characterized by the absence ofrepulsion between quasienergy levels in the spectrum of U F . The eigenvalues of U F are phases { e − iE n } ; thesecan be used to obtain the quasienergies { E n } , definedmodulo 2 π . The statistics of quasienergy levels has beena powerful tool in the numerical study of MBL on finitesystems, in particular the level-spacing ratio [65]: r = min( δ n , δ n +1 )max( δ n , δ n +1 ) (9)with δ n = E n +1 − E n , the n th spacing between thequasienergies of U F . In an MBL phase, the value of (cid:104) r (cid:105) averaged over eigenstates and disorder realizationsapproaches the Poisson value (cid:104) r (cid:105) Poisson (cid:39) .
39 with in-creasing system size, reflecting the lack of level repul-sion that arises from localization. In an ergodic phase it0should instead approach the Gaussian unitary ensemble(GUE) value (cid:104) r (cid:105) GUE (cid:39) .
60, characteristic of random-matrix behavior [66]. Finite size scaling of this quantityacross different cuts in parameter space was used to mapout the phase diagram in Fig. 2. The inset displays onesuch cut, at θ = 0, with two crossings separating thethermal phase ( (cid:104) r (cid:105) increasing with L ) from the two MBLphases ( (cid:104) r (cid:105) decreasing with L ). Notice the dip below thePoisson value near g = π/ g = π/
2, wherethe h fields are exactly ‘echoed out’ over two periods. Real-time oscillations
The level spacing ratio distin-guishes between MBL and thermal phases, but not be-tween different MBL phases. To do this, we need to con-sider specific features of the quantum order inherent inan MBL DTC. The hallmark of a DTC is spatiotempo-ral order: infinitely long-lived period-doubled oscillationof spins, in conjunction with long-range glassy order inspace. This is encoded in the behavior of a two-pointcorrelation function [20, 22] C ij ( n ) = (cid:104) Z i (0) Z j ( n ) (cid:105) ∝ ( − n s ij (10)at late times, where n counts Floquet cycles and s ij en-codes the “glassy” spatial order (i.e. is non-zero, but mayhave random sign as a function of i and j ). This meansmemory of an initial glassy configuration is preserved for-ever, with the configuration itself flipped at every cycle.Starting from a computational basis state | ψ (0) (cid:105) = | σ (cid:105) ( σ ∈ { , } L ), the statement in Eq. (10) simplifies to (cid:104) Z j ( n ) (cid:105) ∝ ( − n (cid:104) Z j (0) (cid:105) : each spin gets flipped at ev-ery cycle, while maintaining a finite fraction of its initial(maximal) polarization. In contrast, an MBL paramag-net will retain memory of the initial configuration, butthe spins do not get flipped.We perform exact numerical simulations of time-evolution under the circuit Eq. (7) on systems of up to L = 22 qubits starting from various computational basisstates (ranging from polarized states to pseudorandombitstrings). Representative plots for all three phases areshown in Fig. 3(a-c) for θ = 0 and one value of g ineach phase. We compute and plot C ( n ) = L (cid:80) i C ii ( n )which is the spatially resolved autocorrelator, Eq. (10),averaged over all sites i and over at least 10 disorderrealizations. In the DTC phase, all initial states showa persistent period doubled DTC signal C ( n ) ∝ ( − n up to at least n max = 10 Floquet cycles (Fig. 3(a)). Incontrast, the MBL paramagnetic phase near g = 0 showsa persistent signal C ( n ), but at frequency ω = 0 ratherthan ω = π (Fig. 3(c)). The large steady signal for a widerange of choices in initial states is a signature of MBLDTCs, which distinguishes them from prethermal DTCs.For example, a similar numerical simulation of autocorre-lators in the trapped ion experiment sees strong state-to-state dependence, with C ( n ) quickly decaying for mostinitial states [23]. Finally, the behavior of both MBLphases should be contrasted with that of the thermalphase (Fig. 3(b)) where the autocorrelator C ( n ) quicklydecays to zero for all initial states. The insets for panels (a-c) in Fig. 3 show space-timecolor plots of (cid:104) Z i ( t ) (cid:105) , visually depicting the oscillatingglassy order in the MBL DTC, frozen memory in theMBL paramagnet, and rapid thermalization in the ther-mal phase. Importantly, measuring such site-resolved space-time correlators for a wide range of initial states is well within the existing capabilities of the Sycamoredevice. As discussed in Section II C, such measurementsare essential for a detection of the spatiotemporal orderthat defines the MBL DTC, and for distinguishing be-tween MBL DTCs and prethermal variants. Frequency-space peaks
The real-time dynamics canalso usefully be examined in frequency space, and usedto probe how the DTC order melts and gives way to athermal phase as the π pulse imperfection (cid:15) = π − g is increased [20]. Fig. 4(a) shows data obtained fromdynamics simulations of L = 14 to 20 qubits at severalvalues of the pulse parameter g between g = π/ ◦ pulse, center of the DTC phase) and g = π/ C ( ω ) (obtained from Fourier-transforming the real-time signal C ( n ) collected out to n max = 10 ) shows a peak at ω = π in the DTC phase,as expected; its height drops smoothly as one exits thephase (Fig. 4(a)). While this is expected to sharpen withincreasing system size, the finite-time limitation turnsthis into a smooth crossover (Fig. 4(a) inset). Such ananalysis can, of course, also be done with experimentallymeasured dynamical signals.Given that real-time dynamics simulations are in-evitably limited to finite time n , a useful complementaryperspective is achieved by examining spectral functions,where – at the expense of more severe finite-size limita-tions – we can effectively probe infinitely long times by afull diagonalization of the Floquet unitary U F . The pe-riod doubled behavior in Eq. (10) corresponds to a sharpdelta-function peak at frequency ω = π in the spectralfunction C ij ( ω ) = 12 L (cid:88) µ,ν (cid:104) µ | Z i | ν (cid:105) (cid:104) ν | Z j | µ (cid:105) δ ( E µ − E ν − ω )(11)where µ, ν label the eigenstates of the Floquet unitary U F and E µ are its quasienergies, i.e. U F | µ (cid:105) = e − iE µ | µ (cid:105) .This function represents a Fourier transform of the auto-correlator, Eq. (10), over infinite stroboscopic times andaveraged over all initial states. It was used in conjunc-tion with the level statistics to map the phase diagramin Fig. 2, as described below.In a finite-size system, the spectral function C ij ( ω )must be regularized by integrating over a finite frequencywindow δω , ˜ C ij ( ω, δω ) ≡ (cid:90) ω + δωω − δω dω (cid:48) C ij ( ω (cid:48) ) . (12)A delta-function peak C ij ( ω ) ∼ δ ( ω − π ) in the infinite1 FIG. 3. Dynamics of the ideal (noise-free) circuit in the MBL DTC ( g/π = 39 / g/π = 19 / g/π = 1 /
80) phases ( θ = 0). (a-c) Position- and disorder-averaged temporal autocorrelator C ( n ) starting fromvarious initial bitstring states for L = 20 qubits. In the DTC phase the envelopes at even and odd times are highlighted. Insets:space-time color plots of expectation values (cid:104) Z x ( n ) (cid:105) for L = 16 qubits. (d-f) Disorder-averaged probability distribution of theHamming distance d from the initial bitstring in two consecutive Floquet cycles at late time, n = 10 , for L = 20 qubits. size limit translates to a finite limitlim δω → ˜ C ij ( π, δω ) = const. (cid:54) = 0 , as opposed to the generic non-DTC behavior˜ C ij ( ω, δω ) ∼ δω γ , γ > δω →
0. Fig. 4(b,c)show numerical results for ˜ C ( ω = π, δω ) at representa-tive points in the three phases. The onset of a plateauis clearly visible for increasing system size in the DTCphase, indicating the formation of a delta-function peakin C ( ω ) at ω = π . Both the thermal and MBL paramag-netic phases instead obey the scaling ˜ C ( ω, δω ) ∼ δω → δω → Glassy spatial order.
As discussed already, a keyfeature of the DTC phase is long-range spatial ‘spin-glass’ order which stems from spontaneously breaking an(emergent) Ising symmetry [20, 22]. This can be detectedfrom long-range spatial correlation functions measuredin the many-body eigenstates of the Floquet unitary (or,equivalently, from non-zero mutual information betweendistant subregions of the eigenstates [21]). It can alsobe detected in dynamics through autocorrelators of theform Eq. (10).Here we will use a classic diagnostic of spin-glasses re- lated to the Edwards-Anderson order parameter [67]: χ SG = 1 L (cid:88) i,j (cid:104) ψ | Z i Z j | ψ (cid:105) . (13)This quantity is extensive in a phase with glassy order(where all L items in the sum are finite); otherwise it isof order 1 (with only the i = j contributions being signif-icant). It can be examined in the many-body eigenstatesof a Hamiltonian or of U F [20, 68], and its finite-size scal-ing provides yet another mechanism to deduce the phasediagram in Fig. 2.Importantly, in a platform such as Sycamore with fullspatial resolution, this quantity can also be examined dy-namically starting from varied initial states. In Fig. 4(d), χ SG (averaged over late times and disorder realizations)is plotted as a function of g for θ = 0, and clearly shows acrossing with increasing system size, at a value of g con-sistent with the phase boundary in Fig. 2. Note that theeffective system size probed on Sycamore can be easilyvaried by choosing which couplers to activate i.e. con-sidering ‘snakes’ of various lengths (cf. Fig. 1). Thispresents a unique opportunity for experimentally con-ducting finite-size scaling studies of the novel phase tran-sition between the MBL and thermal phases, whose na-ture remains an active area of theoretical investigation.2 Hamming distance.
Finally, we present a diagnostic ofspatiotemporal order that, while quite unusual from thepoint of view of many-body physics, is tailor-made fordevices like Sycamore. The ‘quantum supremacy’ exper-iment [5] started with an initial bit string, time-evolvedit under a random circuit, and then probed the outputstate by sampling its probability distribution over all bit-strings. We present a diagnostic for the different phasesin our model that is in that vein, by considering the prob-ability distribution of
Hamming distances between theinitial and time-evolved states.Unlike ergodic dynamics, which quickly turns an initialbitstring state into a random state spread out over theentire computational basis, MBL prevents an initial statefrom veering too far from its initial condition. This factcan be quantified by the Hamming distance d [69, 70],which counts the minimum number of bit flips necessaryto turn a bitstring σ ∈ { , } L into another, σ (cid:48) : for ex-ample d = 0 ( L ) only for identical (flipped) bitstrings,while typically d = L/ | ψ (0) (cid:105) = | σ (cid:105) and itstime evolution after n Floquet cycles, | ψ ( n ) (cid:105) , we can de-fine the Hamming distance distribution P n ( d ) = (cid:104) ψ ( n ) | Π σ ( d ) | ψ ( n ) (cid:105) , (14)where Π σ ( d ) is the projector on bitstrings σ (cid:48) that area Hamming distance d away from σ . We note thatthe average of the Hamming distance distribution, d = (cid:80) d P n ( d ) d , is information that can also be extractedfrom local expectation values of Z , since 2 d = L − (cid:80) i ( − σ i (cid:104) Z i ( t ) (cid:105) ; in particular for the polarized initialstate this becomes a global observable, the total magne-tization (cid:80) i Z i . However the full distribution P ( d ) re-quires measuring the probabilities of entire bitstrings –a natural task for a programmable quantum simulatorsuch as Sycamore that may instead be impractical or im-possible on other platforms where such detailed read-outis unavailable.Fig. 3(d-f) show data for the Hamming distance distri-bution P n ( d ) (Eq. (14)) in consecutive Floquet cycles atlate times, n = 10 and n = n +1, in the three phases.In the DTC phase (Fig. 3(d)), P n ( d ) remains peaked near d = 0 (the initial bitstring) at even n and, symmetrically,near d = L (the globally flipped initial bitstring) at odd n . On the contrary, in the MBL paramagnet (Fig. 3(f)) P n ( d ) remains peaked near d = 0 at all times. The behav-ior of both MBL phases should be contrasted with thatof the thermal phase (Fig. 3(e)), where the Hamming dis-tance distribution quickly becomes peaked at d = L/ IV. EFFECT OF NOISE
The discussion in the previous Section shows that theSycamore device has, in principle, all the ingredients nec-essary to stabilize and detect a DTC phase. We now / C () (a) g / C () L g / g / S G (d) L ( , ) MBL DTC (b) L ( , ) t h e r m a l M B L P M (c) FIG. 4. Other diagnostics of DTC order. (a) Fourier trans-form of the temporal autocorrelator, C ( ω ), averaged overposition and disorder, for several values of g spanning theDTC and thermal phases. Data from dynamics simulationsof L = 18 qubits starting from a fixed bitstring state andevolving to n max = 10 Floquet cycles. Inset: height of the ω = π peak as a function of g . (b,c) Spectral function C ( π, δω )(see Eq. (12)) from exact diagonalization of U F on small sizes,averaged over disorder. The DTC phase develops a plateaufor δω → π peak in theFourier response, while in the thermal and MBL PM phaseswe find C ( π, δω ) ∼ δω . (d) Spin glass order parameter χ SG evaluated at late times, n max / ≤ n ≤ n max , from dynamicssimulations as in (a). A crossing for increasing system sizeindicates a transition consistent with the phase boundary inFig. 2 at θ = 0. address the important question of the robustness of theimplementation and diagnostics to errors (in the formof noisy gates, environmental decoherence, and spurioustime-dependence of the circuit parameters). These give asignal that will be decaying in time, in practice. As dis-cussed below, estimates of current noise thresholds pre-dict that the distinctive temporal signatures of DTC or-der should still be visible for multiple hundreds of drivingperiods. We emphasize again that spatial randomness isan inherent part of the DTC Floquet circuit, so smallcalibration errors between target gates and actual circuitelements are not a problem, provided these are reliablyrepeatable in time to give a Floquet circuit.We model noise by considering a one- and two-qubitdepolarizing error model [71], acting on the system’s den-3sity matrix ρ asΦ (1 q ) i ( ρ ) = (1 − p ) ρ + p (cid:88) α (cid:54) =0 σ α,i ρσ α,i Φ (2 q ) ij ( ρ ) = (1 − p ) ρ + p (cid:88) α,β (cid:48) σ α,i σ β,j ρσ α,i σ β,j (15)(the primed sum denotes ( α, β ) (cid:54) = (0 , i is followed by an ap-plication of the channel Φ (1 q ) i ; each two-qubit gate onbond ( i, j ) is followed by Φ (2 q ) ij . Conservative order-of-magnitude estimates for the depolarizing error rates withcurrent technology [5, 58] are p ≈ − and p ≈ − .The additional errors introduced by the active single-qubit rotations in the definition of (cid:101) G (Eq. (6)) can betaken into account approximately by enhancing the val-ues of p , p . In the following we set p = p , p = p/ p as the ‘Pauli errorrate’ unless otherwise specified.The channels Eq. (15) subsume the effect of fairlygeneric experimental errors, e.g. environmental decoher-ence, temporally random fluctuations of gate parameters,etc. In reality the errors may be anisotropic, e.g. Z Paulierrors (phase-flip) may be more or less frequent than X (bit-flip) errors. While this issue can be completely ne-glected in ergodic circuits [5], where each qubit’s Blochsphere is quickly scrambled and the error model is madeeffectively isotropic, in this MBL setting this need not betrue. Indeed, in structured evolutions that explore theirHilbert space unevenly, the effect of errors depends onthe details of the circuit. Nonetheless, in the absence ofmore detailed device-specific error modeling, the depolar-izing model is a reasonable choice in that it involves allPauli errors. We have additionally verified that our con-clusions do not change qualitatively under a non-Paulierror model (the single-qubit amplitude-damping chan-nel [71]).Quantum channels such as Eq. (15) can be “unrav-eled” into stochastic unitary evolutions [72, 73]. Let usfocus on the one-qubit channel Φ (1 q ) i for simplicity. Itseffect can be thought of as follows: after acting witheach single-qubit gate R xi from Eq. (7), the experimen-talists toss a biased coin; with probability p , they ap-ply an additional gate (“error”) drawn at random from { X i , Y i , Z i } ; otherwise they apply I (i.e. they do noth-ing). After n cycles they get a pure state | ψ r ( n ) (cid:105) , wherethe label r keeps track of the error record, i.e. whicherror gates were applied, where and when during the en-tire evolution. Iterating this stochastic process gives anensemble of pure-state unitary evolutions (“quantum tra-jectories” [74]) {| ψ r ( n ) (cid:105)} that can be used to recover thedensity matrix ρ ( t ) resulting from the real noisy evolu-tion: ρ ( n ) (cid:39) N r (cid:88) r | ψ r ( n ) (cid:105) (cid:104) ψ r ( n ) | , (16)where N r is the number of sampled trajectories (this be-comes exact in the limit N r → ∞ ). Thus at the expense n C ( n ) L = 20, p = 10 (a) / | C () | MBL DTC ( g / = 39/80)MBL PM ( g / = 1/80)Thermal ( g / = 19/80)noise limit pn ( ) n C ( n ) MBL DTC( g / = 39/80) (b) n ( ) n C ( n ) p = 10 p = 10 p = 10 p = 10 L = 16 L = 18 L = 20 L = 22 FIG. 5. Noisy dynamics. (a) Time evolution of the spatially-averaged correlator C ( n ) for a circuit with L = 20 qubits inthe presence of depolarizing noise ( p = 10 − ) for the MBLDTC, MBL paramagnetic, and thermal phases, starting froma fixed bitstring state. The dashed line is the noise limit e − γt (see main text). Inset: Fourier transforms of the signals show(broadened) peaks at ω = π for the MBL DTC and ω = 0 forthe MBL PM. (b) Dependence on system size L and error rate p in the DTC phase. The DTC signal decays exponentiallywith the product pn , proportional to the number of errors persite , independent of system size. Inset: same data vs numberof Floquet cycles n . of simulating multiple trajectories, one can evolve purestates instead of density matrices, greatly reducing thecomputational cost.Aside from their computational usefulness, quantumtrajectories also offer a conceptually appealing view ofthe underlying error process. By unraveling a channel asoutlined above, it is possible to think of the combinedeffect of all non-ideal processes taking place in the ex-periment as “digital”, with discrete errors taking placeat specific locations in spacetime during the circuit dy-namics. In the ‘quantum supremacy’ experiment, Ref. 5,it was argued that a single such digital error could com-pletely randomize the output state: only “error-free” cir-cuit realizations could contribute to the signal being mea-sured in that work, hence its decay as (1 − p ) Ln ≈ e − pLn (for p (cid:28) n (cid:63) ∼ / ( pL ). This argumenthowever need not hold for many-body localized (MBL)dynamics, where information propagates very slowly inspace. It is plausible to expect in this case that a “digi-tal” error at a given location will only affect observables4in its vicinity, rather than completely randomize the out-put state.This expectation is borne out by numerical simula-tions of quantum trajectories. Given the depolarizingerror model of Eq. (15), the autocorrelator C ii ( n ) = (cid:104) Z i (0) Z i ( n ) (cid:105) inevitably decays in time. Even under theideal DTC circuit (with perfect π -pulse (cid:15) = 0 and no θ ij couplings) one can see that Z operators decay exponen-tially: Z i is invariant under the 2-qubit gates but decaysunder the subsequent error, Φ (2 q ) ij ( Z i ) = (1 − p / Z i ;after two iterations of this (with its two neighbors), Z i picks up a minus sign under the π pulse, followed by thedecay under single-qubit noise Φ (1 q ) i ( Z i ) = (1 − p / Z i .Thus overall Z i (cid:55)→ − e − γ Z i over one Floquet cycle, with γ = − ln (cid:34)(cid:18) − p (cid:19) (cid:18) − p (cid:19)(cid:35) an effective decoherence rate. Introducing non-ideal ele-ments to the DTC drive ( (cid:15) (cid:54) = 0, θ ij (cid:54) = 0, etc) is not goingto counter this decay; rather, it will generically include a(finite, transient) amount of ‘internal decoherence’. TheDTC signal is thus expected to be bounded by ± e − γn .The data in Fig. 5(a) shows a DTC signal with amplitudeclose to the maximal level allowed by noise.Already with current hardware, this would yield adetectable DTC signal for hundreds of Floquet cycles.Indeed, the measurement task consists of resolving theexpectation (cid:104) Z i ( n ) (cid:105) (which is small, ∼ e − γn at latetimes) of a binary variable with standard deviation (cid:112) − (cid:104) Z i ( n ) (cid:105) (cid:39)
1; this requires repeating the same ex-periment N s (cid:29) N s = 10 sam-ples were obtained in a few minutes). Equating the signalto the statistical noise floor then gives e − γn ∼ / √ N s ;i.e. the signal can be resolved up to n ≤ n (cid:63) (cid:39) γ ln( N s ).Letting p = 10 − (a conservative estimate for the presenttechnology) and N s = 10 we obtain n (cid:63) = 303 Flo-quet cycles. This would improve logarithmically with thenumber of samples N s , but most importantly it would im-prove linearly in the inverse Pauli error, which is set to seesubstantial improvements in the future. Finally, we notethat we are ultimately interested in the quantity (cid:104) Z i ( n ) (cid:105) ,averaged not only over quantum measurements but alsoover independent disorder realizations of the circuit foreach starting state. Thus, the N s = N d N q experimentalruns will be divided in practice between N d disorder real-izations and N q separate runs of each circuit for quantumaveraging. Benchmarking and calibrating a given realiza-tion of U F is more experimentally demanding than mul-tiple runs of the same circuit; N q ∼ O (10 − ) and N d ∼ O (10 − internal decoherence, i.e. by the system itselfacting as a bath for the local observable [36], as shownin the error-free simulations of Fig. 3(b).Because the signal’s lifetime in the DTC is only lim-ited by external sources of error, future hardware im-provements would directly translate to potentially muchlonger-lived realizations of the time crystal phase, asshown in Fig. 5(b). Furthermore, the signal’s decay ratedoes not scale with size, suggesting that only errors inthe vicinity of a given qubit cause damage to the lo-cal DTC signal. To confirm this picture, we have alsosimulated the dynamics of a system where a single bond ( i, i + 1) is subject to decoherence, and we find that thelocal DTC signal C jj ( n ) decays as e − n/τ j with a timeconstant that diverges exponentially in the spatial dis-tance from the faulty bond, τ j ∼ exp {| j − ( i + 1 / | /ξ } (data not shown). Thus when all bonds are noisy, by farthe dominant source of decoherence for the signal at anysite j is the noise in its immediate vicinity, and the decayis to a very good approximation independent of L .In sum: conservative estimates of noise levels suggestthat Sycamore should already be able to observe a DTCsignal for ∼ O (100) Floquet cycles, which is on par withwhat was observed in FirstGen experiments, but withsignificant improvements expected as the hardware con-tinues to advance. Importantly, the signal decay timedoes not directly scale with system-size, so that the plat-form can be scaled up in size without a correspondingcost in experimental lifetime. V. DISCUSSION AND OUTLOOK
In this work we have considered the question: whatdoes the dawning age of NISQ devices and pro-grammable quantum simulators have in store for quan-tum many-body physics, focusing in particular onGoogle’s Sycamore platform. We have observed that,while these devices offer universal gate sets that can inprinciple simulate any quantum system, their limitationin coherence time practically favors certain simulationtargets over others in the near term. Thus, when think-ing of these devices as experimental platforms for many-body quantum mechanics, it is important to engage withtheir strengths and limitations, which are quite differentfrom, and in some ways complementary to, those of themore traditional arenas for quantum many-body physics.This requires developing physical insight and intuitionmatching those needed in materials physics (regardingthe choice of chemical compound, its synthesis, the selec-tion and optimization of the experimental platform, andits theoretical modeling) or in cold-atomic systems (re-garding the choice of atom or molecule, the cooling andloss suppression strategies, Hamiltonian engineering, andobservable readout).In the spirit of tailoring the application to what ismost natural for the device in the near-term, we notedthat unitary circuits implement various kinds of driven5quantum evolutions more straightforwardly than they dotime-independent Hamiltonians. We have thus focusedon out-of-equilibrium many-body phases in driven (Flo-quet) systems. Specifically we have pointed to the Flo-quet discrete time crystal as a candidate well suited asa ‘physics forward’ simulation task on Sycamore, as itis simultaneously intrinsically interesting, a good fit forSycamore’s capabilities, and not yet realized in any otherexperimental platform. We have shown through detailednumerical simulations that the Floquet DTC can be sta-bilized on Sycamore over a range of realistic parameters,even under very conservative assumptions about gatecalibration error, and that all facets of the DTC spa-tiotemporal order can be compellingly revealed using thedevice’s extensive capability for initialization and site-resolved read-out.We have also addressed the effects of noise and de-coherence on detecting the DTC spatiotemporal order.While all quantum simulators have to contend with theeffects of environmental decoherence, the Sycamore plat-form has an edge insofar as the noise has been calibratedwith great care in this platform. This would make it eas-ier, in practice, to disentangle the effects of ‘internal’and ‘external’ decoherence upon observing a decayingsignal in time. Further, the great control afforded by thisplatform also could also permit the use of various ‘echosequences’ (such as one used in the NMR experiment,Ref. [27, 28]) to further separate the effects of internaland external decoherence. We note that the former is amatter of principle: if even in an ideal, noise-free modelthe signal is eventually destroyed by internal decoherence(i.e. quantum thermalization), then the system does notrealize a DTC phase (this is true of all FirstGen DTC ex-periments). On the other hand, if the signal’s lifetime islimited by external decoherence (i.e. environmental noiseand control errors), then this is an issue of engineeringand, as such, will see sustained improvement with futurehardware innovations.Our proposal falls squarely in the latter category. Thesignal lifetime, already in the hundreds of cycles withcurrent technology, is predicted to steadily increase withhardware improvements. The prospects for increasingthe spatial size of the system are also promising. Wehave shown that the DTC order is sensitive to noise onlylocally, so that its lifetime is not negatively affected byincreasing system size. The main constraint on the num-ber of qubits thus becomes the geometry of the device.We conclude by mentioning interesting directions forfuture work along these lines. A set of mild variationsof the set-up proposed here can realize and probe a hostof other interesting questions. These include prether-mal time crystals [42], in particular in two dimensions,as well as other nonequilibrium phases, such as Floquetsymmetry-protected topological phases [75], which wouldrequire realizing an Ising symmetry to a good approxima-tion, and are thus a good target for future tests of high-precision many-body simulations. Separately, quantumcircuits are increasingly being studied as toy models for exploring a host of foundational questions in quantumstatistical mechanics ranging from quantum chaos [76–83] to the dynamics of quantum entanglement [79, 84, 85]to the emergence of hydrodynamics [79, 86]. Exploringsome of these issues experimentally could have transfor-mational impact on our understanding.Finally, a direction we leave for future study is that ofestimating the classical computing resources needed tosimulate the proposed circuits. Circuits implemented ona specific hardware platform in the presence of finite er-rors require careful estimates of classical computationalresources. In general, however, we note there are no ef-ficient classical algorithms for exploring the entire phasediagram in Fig. 2. Indeed the nature of MBL-to-thermalphase transition is still a largely open question, in nosmall part because of severe finite-size effects plaguingnumerical explorations of this question [87–93]. The pos-sibility of platforms like Sycamore allowing for exper-imental finite-size scaling studies of this transition onmuch larger sizes (cf. Fig. 4(d)) could lend importantinsights to some of these open questions.
Acknowledgments—
We thank Rahul Nandkishore andSiddharth Parameswaran for discussions. This work wassupported with funding from the Defense Advanced Re-search Projects Agency (DARPA) via the DRINQS pro-gram. The views, opinions and/or findings expressedare those of the authors and should not be interpretedas representing the official views or policies of the De-partment of Defense or the U.S. Government. MI wasfunded in part by the Gordon and Betty Moore Foun-dations EPiQS Initiative through Grant GBMF4302 andGBMF8686. The work was in part supported by theDeutsche Forschungsgemeinschaft through the cluster ofexcellence ct.qmat (EXC 2147, project-id 390858490).
Appendix A: Necessity of Ising-even disorder
Here we explain why stabilizing an MBL DTC phase inthe model Eq. (1) requires having disorder in the Ising-even couplings J ij Z i Z J , whereas disorder in the longi-tudinal fields h i Z i is insufficient. Considering the case θ ij = 0 for simplicity (small non-zero values do not qual-itatively change the argument), the time evolution overtwo consecutive periods is given by U F = P g e − iH z [ J , h ] P g e − iH z [ J , h ] , (A1)where H z [ J , h ] ≡ (cid:80) i J i Z i Z i +1 + h i Z i and P g ≡ (cid:81) i R xi (2 g ) = (cid:81) i e − igX i is the imperfect π -flip, with2 g ≡ π − (cid:15) . By using the fact that Z i anticommutes withthe Ising symmetry P π = (cid:81) i X i , we rewrite Eq. (A1) as U F = P − (cid:15) e − iP π H z [ J , h ] P π P − (cid:15) e − iH z [ J , h ] = P − (cid:15) e − iH z [ J , − h ] P − (cid:15) e − iH z [ J , h ] . The crux of the argument is the fact that the fields h i have opposite signs in the two consecutive actions of6 n C ( n ) g / = , W = 0 g / = , W = g / = , W = 0 g / = , W = FIG. 6. Temporal autocorrelator C ( n ) for L = 16 qubits,averaged over position, 100 disorder realizations, and 100 ini-tial states, with maximal disorder in the Ising-odd h fields( h ∈ [0 , π ]), and with disorder W in the Ising-even φ angles. e − iH z : to leading order in (cid:15) , their effects cancel (“echoout”). To see this in more detail, we may write U F as U F = P − (cid:15) e − iP (cid:15) H z [ J , − h ] P † (cid:15) e − iH z [ J , h ] = P − (cid:15) e − iP (cid:15) H z [ J , P † (cid:15) × e − iP (cid:15) H z [0 , − h ] P † (cid:15) e − iH z [0 , h ] e − iH z [ J , ;where we have decomposed e − iH z [ J , h ] into the (commut-ing) factors e − iH z [ J , e − iH z [0 , h ] . Now if we take the J couplings to be clean, J i ≡ J , the above expression can be rewritten by isolating the disordered part as U F = U (1)clean · (cid:89) i e ih i P (cid:15) Z i P † (cid:15) e − ih i Z i · U (2)clean . Straightforward algebra yields e ih i P (cid:15) Z i P † (cid:15) e − ih i Z i = e − i ˜ h i ˆ n i · σ i where ˆ n i is a unit vector and ˜ h i obeyscos ˜ h i = 1 − sin ( h i ) (1 − cos (cid:15) ) , (A2)hence when (cid:15) (cid:28) h i ≈ (cid:15) sin h i (cid:28)
1. Thusthe effective disorder strength in the fields h i is greatlyreduced precisely in the regime where DTC order shouldbe found (small (cid:15) ), posing a problem for the stabilizationof the MBL DTC. Note that this is not a problem atsmall g ( (cid:15) ≈ π ), where disorder in the onsite fields doesnot get echoed out and can stabilize an MBL paramagnet.Numerical simulations of the model confirm this scenario,giving only an MBL paramagnetic phase (at sufficientlysmall g ) and an ergodic phase in the rest of parameterspace.To illustrate this, we have performed dynamics simu-lations of the model realizable in Sycamore, Eq. 7, withmaximal disorder in the h ija/b/c angles (sampled uniformlyfrom [0 , π ]), both with and without disorder in the φ angles (again the identification between controlled-phaseangles and Ising couplings is φ = 4 J ). We use the samediscrete-disorder model as in the main text, with M = 8values, φ = π , and disorder strength W set to either W = π/ W = 0. Finally wetake θ = 0 and ∆ θ = π/
50. The results are shown inFig. 6. While the MBL PM phase ( g = π/
80) is fullystabilized by the h fields, with negligible effect of W , theMBL DTC ( g = 39 π/
80, i.e. (cid:15) = π/
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