arxiv:2107.09676v1 [quant-ph] 20 jul 2021

Realizing a dynamical topological phase in a trapped-ion quantum simulator

Philipp T. Dumitrescu,1, ∗ Justin Bohnet,2 John Gaebler,2 Aaron Hankin,2 David

Hayes,2 Ajesh Kumar,3 Brian Neyenhuis,2 Romain Vasseur,4 and Andrew C. Potter3, 5, †

1Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA2Honeywell Quantum Solutions, 303 S. Technology Ct., Broomfield, Colorado 80021, USA

3Department of Physics, University of Texas at Austin, Austin, TX 78712, USA4Department of Physics, University of Massachusetts, Amherst, MA 01003, USA

5Department of Physics and Astronomy, and Quantum Matter Institute,University of British Columbia, Vancouver, BC, Canada V6T 1Z1

Nascent platforms for programmable quantum simulation offer unprecedented access to newregimes of far-from-equilibrium quantum many-body dynamics in (approximately) isolated sys-tems. Here, achieving precise control over quantum many-body entanglement is an essential taskfor quantum sensing and computation. Extensive theoretical work suggests that these capabilitiescan enable dynamical phases and critical phenomena that exhibit topologically-robust methods tocreate, protect, and manipulate quantum entanglement that self-correct against large classes of er-rors. However, to date, experimental realizations have been confined to classical (non-entangled)symmetry-breaking orders [1–3]. In this work, we demonstrate an emergent dynamical symmetryprotected topological phase (EDSPT) [4], in a quasiperiodically-driven array of ten 171Yb+ hyperfinequbits in Honeywell’s System Model H1 trapped-ion quantum processor [5]. This phase exhibits edgequbits that are dynamically protected from control errors, cross-talk, and stray fields. Crucially,this edge protection relies purely on emergent dynamical symmetries that are absolutely stableto generic coherent perturbations. This property is special to quasiperiodically driven systems:as we demonstrate, the analogous edge states of a periodically driven qubit-array are vulnerableto symmetry-breaking errors and quickly decohere. Our work paves the way for implementationof more complex dynamical topological orders [6] that would enable error-resilient techniques tomanipulate quantum information.

Understanding and categorizing new types of univer-sal dynamical phenomena — the dynamical analogs of(meta)stable phases and critical phenomena — that canarise in isolated quantum many-body systems poses afundamental scientific challenge. Early investigationshave already yielded deep insights into the quantum me-chanical underpinnings of thermalization and chaos [7],and shown how thermalization can be prevented by ar-tificial randomness and disorder through many-body lo-calization (MBL). MBL can protect long lived quantumcoherent dynamics in “hot”, dense, and strongly-drivenmatter, and can enable new classes of inherently dynam-ical quantum phases with properties that would be fun-damentally forbidden in static thermal equilibrium, suchas dynamical symmetry breaking and topology [6].

From a practical point of view, universal and quantumcoherent dynamical behaviors tantalizingly offer error-resilient methods to create, protect, and manipulatequantum many-body entanglement — the driving-forceof quantum computation. To perform a quantum com-putation, one faces a trade-off between the desire to iso-late qubits to preserve their coherence, and the need tostrongly interact qubits in order to perform computa-tions. Even in perfect isolation from environmental de-coherence, strong inter-qubit coupling inevitably leads toresidual, coherent errors — from stray fields, gate miscal-

∗ [email protected]† [email protected]

ibrations, cross-talk, etc. — that disrupt computations.Perhaps counterintuitively, coherent errors can be moredamaging than incoherent ones. In particular, the in-fidelity resulting from N gates with error-amplitude εcan grow as ∼ N2ε2 for coherent errors compared to∼ Nε2 for incoherent ones [8]. Despite their outsizeddetrimental impact on algorithm performance, coherenterrors are challenging to detect. Standard randomizedbenchmarking procedures, for example, combine both co-herent and incoherent errors into a single effective error-per-gate, which can dramatically overestimate the accu-racy of structured circuits relevant for computations.

Employing dynamical decoupling pulse sequences is atime-honored approach to mitigate certain types of co-herent errors associated with uncontrolled static strayfields. However, for traditional methods using global,single-spin control, slight imperfections ε in dynamicaldecoupling pulses accumulate and spoil the decouplingin time ∼ 1/ε. By contrast, recent progress in un-derstanding dynamical phases [6] has theoretically pre-dicted that local dynamical control of multi-spin inter-actions can enable self-correcting dynamical decouplingsequences that are inherently robust against large classesof coherent errors. The robustness of these schemes arisesfrom sharply quantized topological invariants of the dy-namics that cannot be altered by generic coherent per-turbations below a critical strength, in direct analogyto the stability enjoyed by equilibrium phases of matter.In these driving protocols, termed dynamical topologicalphases, engineered random couplings (“disorder”) pro-

arX

iv:2

107.

0967

6v1

[qu

ant-

ph]

20

Jul 2

021

mailto:[email protected]

mailto:[email protected]

2

vide a key stabilizing element yielding long-lived many-body localization that avoids thermalization and its as-sociated chaotic scrambling of quantum information. De-spite extensive theoretical progress in formally classifyingand theoretically characterizing dynamical topologicalphases in both periodically-driven (Floquet) and quasi-periodically driven systems, to date only non-topologicaltime-crystalline phases with classical (relying neither onentanglement or coherence) rather than quantum dynam-ical orders have been achieved experimentally [1–3].

In this work, we experimentally implement two mod-els of inherently-quantum dynamical topological phasesin (quasi)periodically-driven 1d array of ten 171Yb+ hy-perfine spins in Honeywell’s System Model H1 quan-tum charge-coupled device (QCCD) trapped-ion archi-tecture [5]. These two drive protocols, respectively imple-ment i) a quasiperiodically-driven Emergent DynamicalSymmetry Protected Topological Phase (EDSPT) illus-trated and defined in Fig. 1, and ii) a Floquet symme-try protected topological phase (FSPT) (Fig. 2). TheFloquet model was previously introduced in [9], and theEDSPT model is a “stroboscopic” version of the modelstudied in [4] that is more amenable to implementationon a gate based quantum processor (though this modifi-cation produces metastability on exponentially long-timescales beyond the experimental life-time as we discuss ex-tensively in Appendix C).

The hall-mark of these topological phases are robustedge-modes that can phase-coherently retain informa-tion despite strong and repeated interactions with otherqubits and external fields, and which slowly decay at arate set only by the incoherent errors and imperfect en-vironmental isolation of the trapped-ion qubits. The dy-namical topological protection of the EDSPT edge statesdoes not require any fine-tuned symmetry, but ratherstems from purely emergent dynamical symmetries thatare “absolutely stable” [10] to generic coherent pertur-bations. The quasiperiodicity of the drive is essentialfor achieving this property: topological order withoutsymmetry protection is fundamentally impossible in 1dbosonic (spin) systems with static or periodically time-dependent (Floquet) Hamiltonians [4]. In stark contrast,in the FSPT model, the edge spins rely on a fine-tunedmicroscopic symmetry that renders them fragile to co-herent symmetry-breaking errors, which naturally arisein all quantum simulation platforms. We show that thisvulnerability causes the FSPT edge spins to quickly de-phase after only a handful of drive cycles. We observethat the EDSPT edge states are insensitive to the samecoherent errors that destroy the FSPT edge states (andin fact even to much stronger intentionally-introducedcoherent errors).

While the FSPT realization only survives for shorttimes, our results suggest that its periodically repeat-ing drive protocol and symmetry sensitivity amplify co-herent errors. By contrasting decay rates of bulk andboundary spin correlations along different axes (whichhave different sensitivities to different types of coherent

and incoherent errors) these drive protocols can actuallyserve as a useful tool for diagnosing coherent errors alongselectable channels. Moreover, in Appendix B we demon-strate a novel many-body interferometric probe to detectthe dynamical topological order of the FSPT (which can-not be detected by any local measurements).

Weakly-open MBL – Before discussing results, webriefly remark on the expected behavior. MBL systemsare defined by the emergence of an extensive number oflocal conservation laws, and associated local integrals ofmotion (LIOMs) [7]. In an idealized, perfectly-isolatedMBL system, site- or disorder- averaged spin correlationswould exhibit a rapid, transient decay before saturatingto a long-time value set by the overlap of the single-sitespin operator with the LIOMs (which may vanish if dic-tated by symmetry). In practice, however, any experi-ment is partially open to its environment resulting in aslow melting of MBL and gradual decay of non-thermalcorrelations. We will refer to MBL-like dynamics that de-cay on a time scale set by gate-error rates (generally muchlonger than the interaction time-scale), as weakly-openMBL. We will contrast the observed behavior to noisysimulations with depolarizing noise channels with one-and two-qubit gate depolarization parameters p1q, p2q re-spectively, which we estimate from randomized bench-marking experiments. We note that, the weakly-openMBL dynamics is not sensitive to weak coherent errors,since MBL is stable to generic unitary-preserving pertur-bations, these only weakly perturb the LIOM structureand give minor quantitative corrections to correlations.

Topology from Quasiperiodic Driving – The ED-SPT model consists of a Fibonacci sequence of two dif-ferent types of circuit layers Ux,Uz [11] as defined andillustrated in Fig. 1(a). Specifically, the sequence of uni-taries at “Fibonacci times” (i.e. for number of circuitlayers, tn = Fn where Fn is the nth Fibonacci numberdefined by Fn+1 = Fn−1 +Fn, F0,1 = 1) is defined recur-sively through, U(t = Fn) = Un, with:

Un+1 = Un−1Un, U1 = Ux, U2 = UzUx (1)

This recursion relation generates a quasiperiodic se-quence of time-dependent unitaries. We emphasize thatthe random axis fields Bα in each layer completely breakall microscopic symmetries 1.

This model represents a discrete-pulse (“strobo-scopic”) version of the smooth drive that was examinedin [4] through numerical simulations and analytic tech-niques of [12]. There, it was shown that for pulse strengthJ close to π, the system exhibited a pair of emergent dy-namical Z2 symmetries despite lacking any true micro-scopic symmetries. This model exhibits edge states char-acterized by the same topological invariant as the equi-librium AKLT/Haldane spin chain [13–16], but with the

1 In Appendix D, we also implement an analogous drive with afine-tuned Ising symmetry axis analogous to the FSPT modelstudied below, and find similar results.

3

`

<latexit sha1_base64="ADQmBWBAoPLixuZPUlIHOFM0N78=">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</latexit>

XX✓

<latexit sha1_base64="J0P9+8ynFr+WovJCIO4Ctgi+keg=">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</latexit>

UXX✓= e�

i2 ✓�

x⌦�x

<latexit sha1_base64="glictdReuKPwAFj5PI108rmcvHY=">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</latexit>

B[i]x

<latexit sha1_base64="4Iw/i9HROuhEkGX60AVH5hetBlA=">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</latexit>

i<latexit sha1_base64="qBdDhnPQGo8lqF1Gz4tShGTDU1I=">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</latexit>

UB

[i]x

= e�i2Bx,i·�i


i<latexit sha1_base64="SZiCeEeEUd2WBFRxUVVCqsAvlOo=">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</latexit>

B[i]z

<latexit sha1_base64="dDD7OnJPYJofLz6YC7EtTjM5K98=">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</latexit>

UB

[i]z

= e�i2Bz,i·�i

<latexit sha1_base64="GXFWyg0avVYVRB8lXe9S4TzF+OY=">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</latexit>

UZZ✓= e�

i2 ✓�

z⌦�z<latexit sha1_base64="9UgrKJmqz/Dpek01urkEPZQnhDI=">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</latexit>

ZZ✓

<latexit sha1_base64="ZNxVltEEXdnfjtLpFsZQ04u+IdI=">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</latexit>Ux<latexit sha1_base64="AsGDRG5qpVWx6bztplvyfrfN+JM=">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</latexit>Uz<latexit sha1_base64="GIwhiqX1s8KH1vaZD4KkEydFKCk=">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</latexit>...

<latexit sha1_base64="GIwhiqX1s8KH1vaZD4KkEydFKCk=">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</latexit>...<latexit sha1_base64="GIwhiqX1s8KH1vaZD4KkEydFKCk=">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</latexit>...




<latexit sha1_base64="7x7BWAlWtlypYEp7oppIKHTEA0Y=">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</latexit>. . .

<latexit sha1_base64="wzi5DOHYT3zoMGpMxtbNRKLz/ss=">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</latexit>

(a)<latexit sha1_base64="0EzbcLI4Vz/Mzj/bEC26un8IBng=">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</latexit>

(b)

<latexit sha1_base64="/TM3Y6TR4HDFEwIZxGrjCy1Gv2k=">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</latexit>=<latexit sha1_base64="ZNxVltEEXdnfjtLpFsZQ04u+IdI=">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</latexit>Ux

<latexit sha1_base64="IbYV5YrwwzV/GJTI/Yuq0IrByG8=">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</latexit>

XXJ


XXJ

<latexit sha1_base64="glictdReuKPwAFj5PI108rmcvHY=">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</latexit>

B[i]x


XXJ


XXJ


XXJ

<latexit sha1_base64="QC2ebO0XvMmj6BC7bvSavgL9Uxw=">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</latexit>

XX[1]K

<latexit sha1_base64="S6vFT8K3/9CIAAxwg8oZJ7QhUbA=">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</latexit>

XX[2]K

<latexit sha1_base64="C83sC6baj77l95L4mSMoTrO1XK4=">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</latexit>

XX[3]K

<latexit sha1_base64="cL1Kix3b/zC1/NpsGWH+WrVsl44=">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</latexit>

XX[4]K

<latexit sha1_base64="/TM3Y6TR4HDFEwIZxGrjCy1Gv2k=">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</latexit>=<latexit sha1_base64="AsGDRG5qpVWx6bztplvyfrfN+JM=">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</latexit>Uz

<latexit sha1_base64="72+ZxwFaAfzGR/UmkRuM88oPXWY=">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</latexit>

ZZJ<latexit sha1_base64="siDVbr4DihVuNqaO8RnFy9IqgWU=">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</latexit>

ZZ[1]K

<latexit sha1_base64="KjRhFQG7pxa4IfI955oEGNr3AU8=">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</latexit>

ZZ[2]K

<latexit sha1_base64="j5P89YX8ICyRmL75UyuxLjmkDlk=">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</latexit>

ZZ[3]K

<latexit sha1_base64="U3m/EYmcJn730RdWqqTo18u5C+g=">AAAEMnicbZPNbtQwEMfdho+yfLXlyCVihbR7aLSpKugBVZW4IHEpEttWm4TKcSa7Vh07sp1sV1begis8BS8DN8SVh8DJ7qJtUktxRvObv2Y8Y8c5o0qPRj+3tp179x883HnUe/zk6bPnu3v750oUksCYCCbkZYwVMMphrKlmcJlLwFnM4CK+fl/zixKkooJ/1oscogxPOU0pwdq6JpPJ1ccvJjiKqqvd/sgbNcvtGv7K6KPVOrvac/bDRJAiA64Jw0oF/ijXkcFSU8Kg6oWFghyTazyFwJocZ6Ai05Rcua+tJ3FTIe3Htdt4NxUGZ0otsthGZljPVJvVzrtYUOj0ODKU54UGTpaJ0oK5Wrj1+d2ESiCaLayBiaS2VpfMsMRE2y71eiGHORFZhnliwneVCRnm0+Y0mwAsKLGEXFEmeAsWeVVvWEoxb6FEzHm1/K2xhM2AE0vlOuNtNKiLgVQPOmRYi+h0pocdFKxEQYdEa1HUQWUV+JEJ697Gqen7le2iG5buwYlb2t7ZmQ3idNg6mbJXYFaZxA1nys4CzMHIO4abKpQFg8D3DuEmMkuX8Xy7t1oTq2XWWLCkHrxgTeZ2bYktO6+vGGbt3gKzjOOY4a6KlmsWkkTolpQU8r821NTe0/a8c1U1+x3jzpcVYcZsAfYV+e030zXODz3/jed/Ouqfeqv3tINeoldogHz0Fp2iD+gMjRFBHH1F39B354fzy/nt/FmGbm+tNC/QreX8/QcTmXPa</latexit>

ZZ[4]K


ZZJ


ZZJ


ZZJ


ZZJ

<latexit sha1_base64="Kcq5B6x661/t6ZzwvpwMnm/8DSY=">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</latexit>

B[i]z

Quasiperiodic Drive

-1.0

0.0

1.0h�x(t)�x(0)i h�z(t)�z(0)i

edge bulk

1 2 3 4 5 6 7 8 9Fibonacci Time n

-1.0

0.0

1.0


0.95 0.90 0.85

(a) (b)

FIG. 1. Fibonacci drive (EDSPT) model (a) Schematic of a Fibonacci sequence of two elementary circuit layers Ux,zthat defines the quasiperiodic model, whose ideal implementation realizes am (exponentially long-lived, though ultimatelymetastable) EDSPT phase with edge states protected purely by emergent dynamical symmetries that are robust against

generic quasiperiodic perturbations. Couplings K [i] are drawn randomly and independently for each odd-bond from [0, 4π], and

on-site fields B[i] are drawn independently for each site, i with magnitude ∼ [0, 4π] and uniformly random direction, breaking allmicroscopic symmetries. (b) Edge and (site-averaged) bulk spin correlators sampled at Fibonacci times, tn = Fn up to maximumtime F9 = 54, for exchange coupling J = 0.95π (top row). Pale lines show ideal (noiseless) simulations, dashed lines showsimulations with depolarizing noise channel, and dots with 1σ error bars indicate experimental data. Bulk correlations rapidlydecay due to random fields. Edge correlators enjoy topologically-enchanced coherence and exhibit characteristic persistentperiod-three oscillations in the Fibonacci index n. This behavior persists over a range of couplings (bottom row).

crucial difference that this behavior is not fine-tuned tosymmetric drives, but rather automatically self-correctsagainst all generic coherent perturbations to the drive.

The characteristic behavior of the topological regimeof this model are topological edge states that undergocoherent quasiperiodic oscillations, which can be mosteasily understood by examining their behavior at Fi-bonacci times, tn = Fn, where the edge motion exhibits3n-periodic oscillations [4, 11] indicative of the periodic-ity of the even/odd parity of Fibonacci numbers. In theexperimental implementation, we observe weakly-openMBL versions of these topological edge-oscillations (seeFig. 1) over a wide range of pulse strengths J . The datamatches quantitatively to simulations using depolarizingnoise channels with randomized benchmarking (RBN)-measured error rates, despite the presence of coherentgate errors and intentional breaking of all symmetries.These results highlight the robustness of the emergent dy-namical symmetries protecting the EDSPT edge-modesagainst generic coherent errors.

While the pulsed version implemented here is muchmore convenient for implementation on a gate-basedquantum simulator, avoiding the need to discretize asmooth time-dependence at significant cost to circuitdepth and error accumulation, it does introduce some(in principle) important changes in the resulting dynam-ics [11, 12]. Namely: even in a perfectly-isolated sys-tem, the recursive drives would exhibit logarithmically

slow heating [11] with numerical simulations showingthat MBL dynamics eventually melting away into infinitetemperature incoherent at an exponentially-long ‘heat-ing’ time-scale th ∼ e1/δ [11] where δ quantifies the de-viation from an ideal (purely commuting) drive (see Ap-pendix C for detailed definitions). In Appendix C, weadapt the recursive Magnus expansion of [11] to treattopologically non-trivial drives, and demonstrate analyt-ically the presence of emergent dynamical symmetrieslasting up to parametrically long time scales t∗ ∼ δ−3,and provide numerical evidence that the emergent sym-metries and topological edge dynamics actually surviveall the way to th � t∗. We emphasize that this expo-nential dependence makes this metastability irrelevantin practice – as th can readily be pushed far beyond theexperimental lifetime by modest changes in δ.

An ersatz FSPT – The EDSPT model above providesan example of quasiperiodically driven spin chain whichexhibits robust edge states that do not rely on symme-try protection. To emphasize the importance of thisfeature, we now turn to a Floquet model which showsthat symmetry-protection requirements render topologi-cal edge states vulnerable to coherent errors (see Fig. 2).An ideal implementation of this drive respects a Z2

(Ising) symmetry generated by π rotations about x: g =∏i σ

xi . This Floquet model was previously introduced

by some of us [9] in the context of studying non-localstring order parameters for the FSPT phase. When the

4

c

<latexit sha1_base64="tNg9J1SVtgpIr0cRqB/y7rTAbOo=">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</latexit>

XYJ

<latexit sha1_base64="/kmK4t0itaPvClOL3H+xwArM97c=">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</latexit>

UXYJ= e�

i2 J(�x⌦�x+�y⌦�y)

<latexit sha1_base64="pqmAmm5+srhCBP/Jg77lS5sPoLg=">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</latexit>

Xhi


i<latexit sha1_base64="+HIntRoMc0kCPILnbK5xuggv1Q8=">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</latexit>

UX

[i]h

= e�i2 hi�

xi

<latexit sha1_base64="wzi5DOHYT3zoMGpMxtbNRKLz/ss=">AAAEK3icbZPNbtQwEMfdBuiyfLXlyMVihbR7aLSpBPSAqpW4cGwF21ZKospxJrtWHTuynWxXUR6BKzwFT8MJxBXEY+Bsumib1FKc0fzmrxnP2FHGmTbj8Y+tbefe/Qc7vYf9R4+fPH22u7d/pmWuKEyp5FJdREQDZwKmhhkOF5kCkkYczqOr9zU/L0BpJsUns8wgTMlMsIRRYqzr45CMLncHY3e8WrhreDfGYIJP//7p7bw+udxz9oNY0jwFYSgnWvveODNhSZRhlEPVD3INGaFXZAa+NQVJQYflqtYKv7KeGCdS2U8YvPJuKkqSar1MIxuZEjPXbVY772J+bpKjsGQiyw0I2iRKco6NxPXBccwUUMOX1iBUMVsrpnOiCDW2Pf1+IGBBZZoSEZfBu6oMOBGz1Wk2AVhQEAWZZlyKFsyzqt6IUnLRQrFciKr5rbGCzYBjS9U64200rIuBxAw7ZFSL2GxuRh3k34j8DgnXorCDisr3wjKoexsl5cCrbBdxUOCDY1zY3tmZDaNk1DqZtldgXpUxDubazgLKg7F7BNdVoHIOvucewnVYNq7S9ezeak2km6yR5HE9eMlXmdu1xbbsrL5ihLd7C9wyQSJOuipWrFlAY2laUpqr/9rAMHtP2/POdLXa7xh31lREOLcF2Ffktd9M1zg7dL03rndqn9MENauHXqCXaIg89BZN0Ad0gqaIohn6jL6gr84357vz0/nVhG5v3Wieo1vL+f0P3Ktzdw==</latexit>

(a)<latexit sha1_base64="7x7BWAlWtlypYEp7oppIKHTEA0Y=">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</latexit>. . .

<latexit sha1_base64="LAt9+859xVRVSbelaekjDqnkuGE=">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</latexit>U<latexit sha1_base64="GIwhiqX1s8KH1vaZD4KkEydFKCk=">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</latexit>...<latexit sha1_base64="GIwhiqX1s8KH1vaZD4KkEydFKCk=">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</latexit>...


<latexit sha1_base64="GIwhiqX1s8KH1vaZD4KkEydFKCk=">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</latexit>...

<latexit sha1_base64="0EzbcLI4Vz/Mzj/bEC26un8IBng=">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</latexit>

(b)Floquet Drive

-1.0

0.0

1.0h�x(t)�x(0)i h�z(t)�z(0)i

edge data

calc. (noisy) calc. (ideal)

0 10 20 30Time t

-1.0

0.0

1.0

0 10 20 30Time t

bulk data

calc. (noisy) calc. (ideal)

(a) (b)

<latexit sha1_base64="/TM3Y6TR4HDFEwIZxGrjCy1Gv2k=">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</latexit>=

<latexit sha1_base64="GEyjDocdKx/2HVPE2zTLe+59a7Y=">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</latexit>

X[i]h


XYJ


XYJ


XYJ


XYJ


XYJ


XYJ


XYJ


XYJ


XYJ

<latexit sha1_base64="LAt9+859xVRVSbelaekjDqnkuGE=">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</latexit>U

FIG. 2. Floquet model (a) Circuit implementation of the FSPT model proposed in [9] generated by sequence of repeatedcircuit layers U , each consisting of a brickwork of nearest neighbor XY-interactions followed by random fields along x that areindependently and identically distributed on each site ∼ [0, 4π] (maximal disorder strength). For |J − π| . 0.2π, numericalsimulations predict an FSPT phase with bulk MBL dynamics and topological edge modes protected by a combination ofsymmetry, topology, and dynamics. (b) Experimental spin auto correlation data (data points) atop theoretical calculationswith (dashed lines) and without (solid lines) incoherent depolarizing noise. The site-averaged bulk data show behavior consistentwith weakly-open MBL dynamics. Though σz correlations exhibit quantitative agreement with incoherent depolarizing noisesimulations, the characteristic period-doubled oscillations of the edge-spin symmetric (σx) correlators quickly dephase aftert & 15 oscillations. We attribute this behavior to the presence of coherent symmetry breaking phase errors as discussed in thetext. The vulnerability of the FSPT edge states to such errors highlights the comparative robustness of the EDSPT phase.

exchange coupling J is close to π: |J/π − 1| < δc ≈ 0.2,numerical simulations [9] predict an MBL FSPT phasewith bulk MBL dynamics and topological edge states(for open boundary conditions). When the protectingsymmetry is intact, the edge modes undergo coherentperiod-two oscillations that dynamically decouple themfrom bulk degrees of freedom despite repeated strong in-teractions between all neighboring spins. Formally, thetopological dynamics of the edge is characterized by localanticommuting action of time-translation and Ising sym-metry generators, corresponding to a non-trivial groupcohomology element of H2 (Z× Z2, U(1)) = Z2 [17–20],which can be physically understood as a quantized pump-ing of Ising symmetry “charge” (i.e. parity quantum num-ber for π-rotations about x) onto the boundary in eachdriving period [9, 17–20]. In Appendix B, we exploit thispumping picture to devise and implement interferometricprobe to directly measure this topological invariant.

Figure 2 shows experimental spin-correlations data forthis model with J = 0.9π up to 35 Floquet periods (seealso Appendix D for additional parameters), along sidenoisy and ideal simulations. The (site-averaged) bulkcorrelators exhibit characteristic short-time transient de-cay followed by slowly-decaying plateau for symmetriccorrelators (Cx) and near-zero value for (Cz), consis-tent with weakly-open and symmetry-preserving MBLdynamics, and quantitatively matches simulations withdepolarizing noise with RBN-determined parameters.

By contrast, the edge spins exhibit weakly-openperiod-doubled amplitude oscillations (Cz) but whichquickly dephase as indicated by the random behaviorof Cx for t & 15 Floquet periods. This behavior can-not be explained by incoherent errors alone, but ratheris consistent with significant 2 − 3% coherent error am-plitudes per two-qubit gate that appear to be predomi-nately phase errors that affect the Cx but not Cz edgecorrelators. We note that these errors translate to muchsmaller gate infidelities of ∼ 5×10−4 consistent with ran-domized benchmarking chracterizations [5]. We discusspossible physical origins for this error in Appendix A,and suggest that these coherent errors most likely stemfrom small inhomogeneities and drifts in magnetic fieldthat can accumulate up to ≈ 10◦ rotations about σz perFloquet period on each qubit, and which are stable overthe time-scale of individual circuits. These account qual-itatively and quantitatively for the observed dephasingof the FSPT edge states. While this error mechanism isspecific to trapped-ion qubits, similar types of coherenterrors are pervasive across other hardware platforms.

These results highlight that the symmetry protectionrequirement for FSPT edge states is a significant liabil-ity in practice, as discrete protecting symmetries are finetuned and broken by inevitable coherent control and cal-ibration errors. Interestingly, while these coherent errorsare difficult to detect by other means, the different sen-sitivity of bulk and edge correlators and symmetry-axis

5

resolved edge behavior give characteristic fingerprints ofthe magnitude and symmetry-structure of coherent er-rors. The question of whether other MBL (F)SPTs withmore complex symmetry groups could be used to diag-nose general multi-qubit coherent error channels is a po-tentially interesting line for future inquiry.

Discussion – The EDSPT implemented here representsthe first experimental realization of a purely dynamicaltopological phase that cannot arise in equilibrium, andof a one-dimensional bosonic topological phase that doesnot rely on symmetry protection. As we have demon-strated, the latter quality makes the resulting edge-statephenomenology considerably more robust than its morefragile SPT and FSPT counterparts, and opens the doorto new strategies for dynamically extending coherent in-formation storage in the presence of strong interactionsand cross-talk between many-qubits. Higher-dimensionaldynamical topological models have been predicted to of-fer even more dramatic capabilities to manipulate en-tanglement in a manner that is robust against coherenterrors, enabling for example chiral transfer of quantizedpackets of quantum information [21], and error-resilientimplementation of non-transversal operations on logicalqubits encoded in the boundary of topological codes [22].Our work paves the way for harnessing these capabilitiesfor practical quantum information processing.

Acknowledgements – We thank Yuxuan Zhang andMichael Foss-Feig for helpful discussions, as well as Do-minic Else, Aaron Friedman, Wen Wei Ho, and BraydenWare for prior collaboration on this topic. We thank theentire Honeywell Quantum Solutions team for their manycontributions. This work was supported by NSF Conver-gence Accelerator Track C award 2040549 (ACP), the USDepartment of Energy, Office of Science, Basic EnergySciences, under Early Career Award No. DE-SC0019168(RV), and the Alfred P. Sloan Foundation through SloanResearch Fellowships (RV and ACP). The Flatiron Insti-tute is a division of the Simons Foundation. Numericalsimulations were performed in part on the Lonestar5 su-percomputing system at the Texas Advanced ComputingCenter (TACC) at UT Austin.

Methods

Honeywell’s System Model H1 QCCD Architec-ture – Experiments were performed on Honeywell’s Sys-tem Model H1 trapped-ion quantum processor [5] basedon a Honeywell-fabricated planar chip trap operatingwith three parallel gate zones and 10 qubit ions. Qubitsare encoded in two clock states: {|0〉 = |F = 0,mF =0〉, |1〉 = |F = 1,mF = 0〉} of the S1/2 hyperfine manifold

of 171Yb+ ions, where F,mF are respectively the total in-ternal angular momentum and projection onto a ≈ 5Gmagnetic field axis. Accompanying, co-trapped 138Ba+

ions are used for sympathetic cooling of ion motionalmodes without affecting logical states. Ions are trapped

`

<latexit sha1_base64="3h60bR3IjcbS6eRxe5AWtyaa6lU=">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</latexit>

X✓

<latexit sha1_base64="9UgrKJmqz/Dpek01urkEPZQnhDI=">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</latexit>

ZZ✓<latexit sha1_base64="Br0Lj82GZzrUesPofFuATx4Oad4=">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</latexit>=

<latexit sha1_base64="V3qDFeazsdI1RrWhosIGO3TOKdY=">AAAEMXicbZNLj9MwEMe9Gx5LeeyDI5eICqk9bGkKgh7QaiUuHBeJ7kNJVDnOpLXWsSPbSbey8im4wqfg0+wNceVL4KQt6iZrKc5ofvPXjGfsKGNU6eHwdmfXefDw0eO9J52nz56/2D84PDpXIpcEJkQwIS8jrIBRDhNNNYPLTAJOIwYX0fXnil8UIBUV/JteZhCmeMZpQgnW1nV1NTVBRt+OyulBdzgY1sttG97a6KL1OpseOkdBLEieAteEYaV8b5jp0GCpKWFQdoJcQYbJNZ6Bb02OU1ChqSsu3TfWE7uJkPbj2q292wqDU6WWaWQjU6znqskq533Mz3UyDg3lWa6Bk1WiJGeuFm51fDemEohmS2tgIqmt1SVzLDHRtkmdTsBhQUSaYh6b4FNpAob5rD7NNgALCiwhU5QJ3oB5VlYbllIsGigWC16ufhssYTvgxFK5yXgX9apiING9FulXIjqb634L+WuR3yLhRhS2UFH6XmiCqrdRYrpeabvoBoV7fOIWtnd2Zr0o6TdOpuwVmJcmdoO5srMAczwcjOGmDGTOwPcGI7gJzcplBp7dG62J1CprJFhcDV6wOnOzttiWnVVXDLNmb4FZxnHEcFtFiw0LSCx0Q0py+V8baGrvaXPemSrr/Z5xZ6uKMGO2APuKvOabaRvno4H3YfDu6/vu6Xj9nvbQK/Qa9ZCHPqJT9AWdoQkiKEXf0Q/00/nl3Dq/nT+r0N2dteYlurOcv/8Amldzgg==</latexit>

Y⇡/2

<latexit sha1_base64="rXDuqm9YYO48J6VQpy4xtoUUAvM=">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</latexit>

X⇡/2<latexit sha1_base64="f27ou54EqXVt4vL2eLteJ9eQ4SA=">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</latexit>

X�⇡/2

<latexit sha1_base64="Y4VYHsgk7T2+lNCbRJwMFgEyHZs=">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</latexit>

Y�⇡/2

<latexit sha1_base64="u7QmLmuOjXGPAlhvAFrxqkdDJ5o=">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</latexit>

X�J

<latexit sha1_base64="Jt50Io8gSuXJ8jeO0uDt3jSwfX8=">AAAELXicbZPNbtQwEMfdho+yfLXlyCVihbR7aLQpCPaAqkpcEKcisW1REq0cZ7Jr1bEj28l2ZeUZuMJT8DQckBBXXgNns4u2SS3FGc1v/prxjB3njCo9Gv3a2XXu3L13f+9B7+Gjx0+e7h8cnitRSAITIpiQlzFWwCiHiaaawWUuAWcxg4v46n3NL0qQigr+WS9ziDI84zSlBGvrmnyZmo/VdL8/8kar5XYNf2300XqdTQ+cwzARpMiAa8KwUoE/ynVksNSUMKh6YaEgx+QKzyCwJscZqMisqq3cl9aTuKmQ9uPaXXm3FQZnSi2z2EZmWM9Vm9XO21hQ6HQcGcrzQgMnTaK0YK4Wbn10N6ESiGZLa2Aiqa3VJXMsMdG2Qb1eyGFBRJZhnpjwXWVChvlsdZptABaUWEKuKBO8BYu8qjcspVi0UCIWvGp+GyxhO+DEUrnJeBMN6mIg1YMOGdYiOpvrYQcFa1HQIdFGFHVQWQV+ZMK6t3Fq+n5lu+iGpXt04pa2d3Zmgzgdtk6m7BWYVyZxw7myswBzNPLGcF2FsmAQ+N4xXEemcRnPt3urNbFqssaCJfXgBVtlbteW2LLz+oph1u4tMMs4jhnuqmi5YSFJhG5JSSH/a0NN7T1tzztX1Wq/Zdx5UxFmzBZgX5HffjNd4/zY8994rz697p+O1+9pDz1HL9AA+egtOkUf0BmaIIIo+oq+oe/OD+en89v504Tu7qw1z9CN5fz9Bw/3cg4=</latexit>

YJ

<latexit sha1_base64="Br0Lj82GZzrUesPofFuATx4Oad4=">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</latexit>=<latexit sha1_base64="tNg9J1SVtgpIr0cRqB/y7rTAbOo=">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</latexit>

XYJ

<latexit sha1_base64="QyUb2AXK6pQ7PAU672Wm/gDGSMI=">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</latexit>

(a)

<latexit sha1_base64="CSugIEtCYQmSGVy86ZsseSYL0sc=">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</latexit>

(b)

FIG. 3. Circuit Compilation of two-qubit operations intohardware-native gates for (a) the FSPT model (Fig. 2), and(b) the EDSPT model (Fig. 1). XXθ gates are obtained froma single-qubit basis rotation of ZZθ gates. Here, a line withtwo dots indicates a controlled-Z gate with unitary: uCZ =

e−iπ/4(1+σz)⊗(1+σz), and single qubit gates are listed as Pauli

rotations with unitary uX,Yθ = e−iθ2σx,y .

in either single-qubit (1q) Yb-Ba or two-qubit (2q) Yb-Ba-Ba-Yb linear “crystal” configurations, which can betransported, orientation-swapped, split (2q → {1q, 1q}),or combined ({1q, 1q} → 2q) using an array of elec-trodes to achieve arbitrary pairings of qubit ions. Dur-ing transport, the qubit logical states are essentially per-fectly decoupled from their motion. Laser-based logical1q and 2q gates are performed in parallel across threegate zones with typical infidelities of: p1q ≈ 10−4 andp2q ≈ 3−5×10−3 determined by randomized benchmark-ing (RBN). 1q gates implement arbitrary amplitude rota-tions about arbitrary axis in the σxy-plane, while σz rota-tions are implemented virtually by updating laser phasesof future 1q gates. The native entangling 2q gate is aMølmer-Sørensen (MS) gate wrapped with single-qubitdressing pulses to achieve a phase-insensitive operationuMS = exp

[−iπ4σz ⊗ σz

].

State Preparation and Measurement – In eachexperimental implementation, we perform circuit-basedtime evolution up to time t with unitary U(t) (definedin Figs. 1,2) acting on L = 10 qubits, and measure spinauto-correlations:

Cα(r, t) = 〈σα(r, t)σα(r, 0)〉, (2)

where σα(r) (α ∈ {x, y, z}) are standard Pauli matriceson site r and σα(r, t) = U†(t)σα(r, 0)U (t) denotes thecorresponding (Heisenberg picture) time evolved opera-

tor. Here, (. . .) denotes averaging over Ns = 100 “shots”with a different random initial state in each shot. Onlythe sample error from finite Ns is included in the 1σ errorbars shown in the figures. Due to the low-clock rate ofthe QCCD architecture (each experiment takes between1−2.5 seconds per shot), we employ two tricks to obtainadequate statistical accuracy. First, we prepare the ini-tial states by randomly initializing product states withσz ∈ {±1} on even sites and σx ∈ {±1} on odd sites,so that measurements of Cx and Cz can be conducted inparallel (since there is statistically no difference betweeneven and odd sites). Second, in each model, we focus

6

on a single disorder realization, but verify with extensiveclassical simulations that our results are indicative of thegeneric behavior of the disorder ensemble.

Circuit Compilation and simulation – The compi-lation of the two-qubit circuit elements for implementingthe FSPT and EDSPT into native gates are shown inFig. 3. Each requires a pair of native 2q gates (shownhere as control-Z gates, which differ from the MS gate

only by single qubit dressing).Noisy circuit simulations are performed with

Qiskit [23] using a featureless depolarizing noisechannel with depolarizing probabilities p1q = 5 × 10−4,p2q = 8× 10−3. Though this simple depolarizing noise isnot hardware realistic (e.g., ignores coherent errors), itprovides a useful point of comparison, since departuresof experimental and simulated data signal the presenceof structured noise.

[1] Jiehang Zhang, PW Hess, A Kyprianidis, P Becker,A Lee, J Smith, G Pagano, I-D Potirniche, Andrew CPotter, A Vishwanath, et al., “Observation of a discretetime crystal,” Nature 543, 217–220 (2017).

[2] Soonwon Choi, Joonhee Choi, Renate Landig, GeorgKucsko, Hengyun Zhou, Junichi Isoya, Fedor Jelezko,Shinobu Onoda, Hitoshi Sumiya, Vedika Khemani, et al.,“Observation of discrete time-crystalline order in a disor-dered dipolar many-body system,” Nature 543, 221–225(2017).

[3] Antonis Kyprianidis, Francisco Machado, William Mo-rong, Patrick Becker, Kate S Collins, Dominic V Else,Lei Feng, Paul W Hess, Chetan Nayak, Guido Pagano,et al., “Observation of a prethermal discrete time crys-tal,” Science 372, 1192–1196 (2021).

[4] Aaron J Friedman, Brayden Ware, Romain Vasseur, andAndrew C Potter, “Topological edge modes without sym-metry in quasiperiodically driven spin chains,” arXivpreprint arXiv:2009.03314 (2020).

[5] JM Pino, JM Dreiling, C Figgatt, JP Gaebler, SA Moses,MS Allman, CH Baldwin, M Foss-Feig, D Hayes,K Mayer, et al., “Demonstration of the trapped-ion quan-tum ccd computer architecture,” Nature 592, 209–213(2021).

[6] Fenner Harper, Rahul Roy, Mark S Rudner, andSL Sondhi, “Topology and broken symmetry in floquetsystems,” Annual Review of Condensed Matter Physics11, 345–368 (2020).

[7] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, andMaksym Serbyn, “Colloquium: Many-body localization,thermalization, and entanglement,” Rev. Mod. Phys. 91,021001 (2019).

[8] Yuval R Sanders, Joel J Wallman, and Barry C Sanders,“Bounding quantum gate error rate based on reported av-erage fidelity,” New Journal of Physics 18, 012002 (2015).

[9] Ajesh Kumar, Philipp T Dumitrescu, and Andrew CPotter, “String order parameters for one-dimensional flo-quet symmetry protected topological phases,” PhysicalReview B 97, 224302 (2018).

[10] Curt W von Keyserlingk, Vedika Khemani, and Shiv-aji L Sondhi, “Absolute stability and spatiotemporallong-range order in floquet systems,” Physical Review B94, 085112 (2016).

[11] Philipp T Dumitrescu, Romain Vasseur, and Andrew CPotter, “Logarithmically slow relaxation in quasiperiodi-cally driven random spin chains,” Physical review letters120, 070602 (2018).

[12] Dominic V Else, Wen Wei Ho, and Philipp TDumitrescu, “Long-lived interacting phases of matterprotected by multiple time-translation symmetries in

quasiperiodically driven systems,” Physical Review X 10,021032 (2020).

[13] Ian Affleck, Tom Kennedy, Elliott H Lieb, and HalTasaki, “Rigorous results on valence-bond ground statesin antiferromagnets,” in Condensed Matter Physics andExactly Soluble Models (Springer, 2004) pp. 249–252.

[14] F Duncan M Haldane, “Nonlinear field theory of large-spin heisenberg antiferromagnets: semiclassically quan-tized solitons of the one-dimensional easy-axis neelstate,” Physical Review Letters 50, 1153 (1983).

[15] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen, “Symmetry-protected topological orders ininteracting bosonic systems,” Science 338, 1604–1606(2012).

[16] Frank Pollmann, Erez Berg, Ari M Turner, and MasakiOshikawa, “Symmetry protection of topological phases inone-dimensional quantum spin systems,” Physical reviewb 85, 075125 (2012).

[17] Curt W von Keyserlingk and Shivaji L Sondhi, “Phasestructure of one-dimensional interacting floquet systems.i. abelian symmetry-protected topological phases,” Phys-ical Review B 93, 245145 (2016).

[18] Dominic V Else and Chetan Nayak, “Classification oftopological phases in periodically driven interacting sys-tems,” Physical Review B 93, 201103 (2016).

[19] Andrew C Potter, Takahiro Morimoto, and Ashvin Vish-wanath, “Classification of interacting topological floquetphases in one dimension,” Physical Review X 6, 041001(2016).

[20] Rahul Roy and Fenner Harper, “Abelian floquetsymmetry-protected topological phases in one dimen-sion,” Physical Review B 94, 125105 (2016).

[21] Hoi Chun Po, Lukasz Fidkowski, Takahiro Morimoto,Andrew C Potter, and Ashvin Vishwanath, “Chiral flo-quet phases of many-body localized bosons,” PhysicalReview X 6, 041070 (2016).

[22] Hoi Chun Po, Lukasz Fidkowski, Ashvin Vishwanath,and Andrew C Potter, “Radical chiral floquet phases in aperiodically driven kitaev model and beyond,” PhysicalReview B 96, 245116 (2017).

[23] Hector Abraham, Ismail Yunus Akhalwaya, Gadi Alek-sandrowicz, T Alexander, G Alexandrowics, E Arbel,A Asfaw, C Azaustre, P Barkoutsos, G Barron, et al.,“Qiskit: An open-source framework for quantum com-puting, 2019,” URL https://qiskit. org (2019).

[24] Maksym Serbyn, Zlatko Papic, and Dmitry A Abanin,“Criterion for many-body localization-delocalizationphase transition,” Physical Review X 5, 041047 (2015).

7

A. Diagnosing coherent errors

The System Model H1 QCCD architecture has a vari-ety of different coherent error sources arising from driftsor miscalibrations in gate-laser amplitude or phase orspatial variations in magnetic field. We expect that thetwo-qubit (2q) gate error sources are predominately in-coherent, and focus our attention on possible sources ofcoherent errors in single qubit (1q) gate operations andidle/memory errors.

The FSPT model provides multiple characteristicfinger-prints that enable us to narrow down the phys-ical mechanism for the observed coherent errors. Thecircumstantial evidence is summarized as follows: first,the coherent error in question has a strong anisotropy af-fecting σx correlators but not σz correlators. Second, theamplitude of the error must be such that it can accumu-late ∼ π phase within ≈ 10− 15 Floquet periods. Third,the coherent error effects are much more pronounced forthe periodic-boundary condition (PBC) circuits studiedin Appendix B, where they cause strong deviations fromweakly-open MBL behavior in ≈ 5 Floquet periods, com-pared to the ≈ 10−15 Floquet periods for open boundarycondition (OBC) circuits presented in the main text.

The last point is particularly telling since the PBC andOBC circuits have essentially the same number of one andtwo qubit gates, and differ mainly in the physical pathsof the ions through the processor. Namely, whereas OBCcircuit structures match the physical 1d layout of the H1chip, and require much less transport of ions betweendifferent gate zones to execute, the PBC circuits requireat least a pair of ions to traverse back and forth acrossthe trap. The pronounced ∼ 2− 3× increase in coherenterror effects strongly suggests that the dominant errorsource arises from different accumulation of qubit phasesin different locations of the trap.

From the variance of corrections during (∼ hourly) re-calibrations, we estimate that spatial variations of thelaser phase or amplitude across gate zones contribute≈ 30 − 50mrad deviations in the axis and rotation an-gle of each 1q gate. Since there are ≈ 6 1q-gates perFloquet period, and conservatively assuming that subse-quent errors add linearly, this error-source would require≈ 102 Floquet periods to accumulate a π-error: an orderof magnitude too large to account for the observed data.

A more serious source of coherent errors are deviationsin the magnetic field, B used to split off the F = 1,mF =±1 hyperfine states from the F = 1,mF = 0 qubit state.From Ramsey spectroscopy, we estimate that the the |0〉and |1〉 qubit states energies differ among gate zones withRMS variation of δfRMS ≈ 0.5Hz (see Fig. 4). Thesefrequency offsets drift slowly over the course of manyshots, but are stable over the time scale of a single circuit.Together, these effects can be modeled by including aHamiltonian:

H∆B =∑i

2π δf [xi(t)]σzi (A1)

`

0 0.5 1timestamp (106s)

°2

0

2

freq

uen

cyer

ror

(Hz)

`

°3 °2 °1 0 1 2 3frequency error (Hz)

0

10

20

30

40

occ

urr

ence

freq

uen

cy

outlier threshold

gate zone G2

gate zone G3

gate zone G4


(a)


(b)

FIG. 4. Global qubit frequency calibration error (a)Histogram of the observed difference between the calibratedqubit frequency and the last known good calibration for eachof the three ion trap gate zones: G2,G3, and G4. (b) Time-series of same data. These errors arise due to combinationof background drifts in the magnetic field environment andstatistical error in the actual qubit frequency calibration. Forthe data detailed in this paper, we observe a global RMS fre-quency error of ±0.5(1) Hz. Outliers are filtered (dashed line)for points greater than 4x the standard deviation to bettercapture the average behavior of this error source.

where xi(t) is the position of the ith ion along the lineartrap axis, which has a periodic time dependence inher-ited from the periodicity fo the Floquet circuit. Sinceeach circuit layer takes on the order of tlayer ≈ 50ms, andgiven that there are two circuit layers per Floquet layer,we estimate these B-field induced memory errors can pro-duce a π shift in Nperiods ≈ π/ (2πδfRMStlayer) & 10 Flo-quet periods, roughly consistent with the time scale atwhich the FSPT implementation deviates from the sim-ulations with purely-incoherent depolarizing noise. Theobservation that coherent error effects are exacerbated inPBC circuit compared to the OBC circuit is consistentwith our observations that the frequency variations arelarger between distant gate zones, since ions in the PBCcircuits are transported for longer distances through thetrap during each Floquet cycle.

8

B. Detection of FSPT order through many-bodyinterferometry

In this section we theoretically introduce a nonlocal“bulk order parameter” for the FSPT phase, and exploitthe flexible qubit connectivity of the QCCD architectureto measure it through a many-body interferometry. Aswith the local spin correlations presented in the maintext, these demonstrations are limited to transient shorttime regimes due to the accumulation of coherent errors.

The premise of the non-local order parameter, whichwe dub the “Loschmidt flux echo” is to consider a ringwith closed, periodic boundary conditions (PBC), suchthat there are no edge states and any topology is evidentonly in global features. Next, we effectively “gauge” theprotecting symmetry, i.e. promote the global Z2 symme-try generated by g =

∏i σ

xi to a local gauge-redundancy,

and explore the affect of a dynamically inserting a classi-cal (non-fluctuating) background gauge-flux by measur-ing the trace-overlap of the time-evolution unitaries with-and without- a flux inserted:

Z(t) =tr U†F (t)U(t)

tr1(B1)

where UF (t) denotes the unitary for time-evolution witha symmetry-flux inserted (defined in detail below), and tare assumed to be integers (i.e. multiples of the Floquetperiod).

Based on general properties of MBL and SPT systems,we argue that the long-time behavior of Z(t) can sharplydistinguish trivial MBL, FSPT, symmetry-breaking, andthermal behaviors. In particular, we will argue belowthat:

limt→∞

limL→∞

Z(t) =

1 trivial-MBL

(−1)t Z2-FSPT

0 symmetry-breaking MBL

0 thermal

(B2)

where (. . . ) denotes disorder averaging. Finite size cor-rections to these formulas are O(e−L/ξ) where ξ is thelocalization length. We also note that thermal andsymmetry-breaking MBL can be distinguished by thelatter having non-vanishing |Z(t)| . This predicted be-havior is confirmed by numerically exact diagonalization(ED) simulations of the FSPT model (see Fig. 5).

Intuitive picture – Intuitively, we can understand theFSPT result as follows: the Z2-FSPT is characterizedby a quantized pumping of symmetry parity (henceforthcalled “charge”) [9, 19, 20]. Namely, during each period,for a system with closed, periodic boundary conditionsan odd number of symmetry charges encircle the sys-tem. In the presence of a symmetry flux, the Z2 analogof the Aharonov-Bohm effect dictates that a unit chargeencircling a π-flux acquires phase (−1). Otherwise, thelocal LIOM dynamics is insensitive to the global flux sec-tor (to see this note that for any spatially-well localized

0 5 10 15 20 25 30 35Time t

-1.0

0.0

1.0Z(t)

FSPT MBL Thermal

FIG. 5. Loschmidt flux echo (ED Simulations) Ex-act diagonalization (ED) simulations of the Loschmidt fluxecho, Z(t), for the Floquet model with L = 9 for J = 0.9πin the FSPT phase (blue dots) where it exhibits saturatingperiod-two oscillations, J = 0.5 in the thermal phase (purplediamonds) where Z(t) immediately averages to zero, and forJ = 0.1π in a trivial MBL phase (orange, triangles) whereZ(t) saturates to a non-oscillating value. Each point is aver-aged over all initial states and 100 disorder realizations.

operator, one can always choose a gauge such that thelocal action of flux insertion is equivalent to a physically-inconsequential gauge transformation). Hence, in t = nperiods, the evolution with- and without- the flux willthen differ by (−1)n.

Locally-implementable approximation – We thenconstruct a strictly-local approximate flux-insertion op-erator, F , which has the property that, in MBL phases,F |ψ〉 has finite overlap with the exact flux-inserted state:

|ψF 〉. Denoting by Z(t) the local-approximation ob-tained by evaluating Z(t) with the exact flux insertionoperator replaced by its local approximation: UF →F †UF , we argue that Z(t) ∼ c ·Z(t) where 0 < c < 1 ismodel-dependent constant of order log c ∼ −ξ, i.e. thatthe local approximation to the Loschmidt flux echo tracksthe true value with finite fidelity. We further design andimplement a circuit to measure Z for the FSPT modelutilizing an ancillary qubit. Of course, the experimen-tal implementation suffers from the same limitations dueto coherent errors described in the main text, and yieldsgood agreement with simulations only up to ∼ 5 Floquetperiods.

1. Formalism

a. Gauging the symmetry

Following a standard “gauging” procedure, we define aformal symmetry-gauging procedure in two steps. First,one expands the Hilbert space to include gauge-link vari-

9

ables τ i,i+1 on each nearest neighbor bond where τz rep-resents a Z2-gauge connection (the discrete analog of the

Wilson line segment ei∫A·dr of electromagnetism), and

τx represents the Z2 gauge-electric field. Second, oneprojects into the subspace of this enlarged Hilbert spacethat obeys the Gauss’ law constraint: Gi = 1 ∀i with:

Gi = τxi−1,iσxi τ

xi,i+1, (B3)

(the discrete and lattice analog of Gauss’ law ∇·E(r) =ρ(r) for electromagnetism). Crucially, we consider τ -variables to be completely non-dynamical “background”gauge variables, i.e. which have Hamiltonian = 0.

Any Z2-symmetric local Hamiltonian can be similarlygauged by adding a connected string of τz’s connectingbetween every pair of symmetry-charged single site oper-ators (σy,z) in each term of the Hamiltonian 2.

For example, the Z2 FSPT model in the main textcan be generated by a local time-dependent Hamiltonianwhich changes as follows under the gauging procedure:

H =∑i

[Ji(t) (XiXi+1 + YiYi+1) + hi(t)Xi]

gaugey

HG =∑i

[Ji(t)

(XiXi+1 + Yiτ

zi,i+1Yi+1

)+ hi(t)Xi

],

(B4)

where Ji(t), hi(t) the (piecewise constant) time-dependent coefficients that reproduce the unitary circuitdynamics shown in Fig. 2a.

b. Formal Flux Insertion

After projection, the system retains 2L (gauge-invariant) degrees of freedom consisting of: 2L−1 spinconfigurations that satisfy the charge-neutrality con-straint implied by Gauss’ law:

∏i σ

xi =

∏i τxi−1,iτ

xi,i+1 =

1, and a global Z2 “magnetic” flux: Φ =∏i τzi,i+1 = ±1.

This foreshadows that we will be able to effectively simu-late the gauged system without introducing extra qubitsto represent the τ variables.

We can formally define the action of a flux-insertion op-erator, F , on any local gauge-invariant operator Oi withbounded support, centered at site i as follows: Choosea reference bond, (ia, ia + 1) at which to insert the flux,and divide the system (with periodic boundary condi-tions) into an interval A of length L/2 centered at iaand its complement, B centered at the antipodal point

2 Note that there are guaranteed to be an even number of thesefactors in any term of a symmetric Hamiltonian. In 1d, imposingstrictly locality in all terms removes any ambiguity for differentchoices of pairings.

ib = (L/2 − ia) mod L. Then define the “flux-inserted”version of O as:

O(x)→ OF (x) =

{f†+OF,if+ x ∈ AO(x) x ∈ B ,

f+ =

( ∏xb<x<xa

σxi

)τxia,ia+1. (B5)

For any operator in A away from site ia, f+ acts like apure gauge transformation. Furthermore, this definitionis actually independent of the reference points ia,b (upto a gauge transformation). While the above definitionholds sharply for strictly local operators (with finitely-bounded support), one can extend it to exponentially-well localized operators with the minor caveat that thedefinitions depend on choice of ia,b with sensitivity ∼O(e−L/2ξ).

As an application, we can use this definition to identifypairs of symmetry-preserving gauged-MBL eigenstates indifferent flux sectors, and show that each element of thepair has nearly identical energy to accuracy O(e−L/ξ).Following standard practice, we define a Floquet systemas being MBL if its Floquet operator (time evolution fora single time-step/circuit layer) can be written as:

U = W †Ξe−iE[σxi ]W (B6)

where E[σxi ] is an exponentially-well localized functionof σxi ’s, and W is a (symmetric) finite-depth local uni-tary that implements the weak-local dressing of single-site spins σxi into conserved local integrals of motion (LI-OMs):

`i = W †σxiW. (B7)

In the gauged system (with non-dynamical backgroundgauge fields), the gauged analog of W (defined by ap-plying the gauging procedure to the local generator ofW : −i logW , and which, in a slight notational abuse,we will denote by the same symbol as the ungauged ver-sion) depends on the gauge-variables only through τz ,i.e. W commutes with the gauge-flux operators. Here, Ξis a symmetric operator satisfying [Ξ, E] = 0. For all thephases we consider here, Ξ2 = 1. For example, in thegauged Z2 FSPT with periodic boundary conditions:

ΞFSPT = Φ =∏i

τzi,i+1, (B8)

is the Z2 gauge flux operator (with open boundary condi-tions, Ξ would capture the quantized spin echo dynamicsof the topological edge states).

The key property of symmetric MBL is that the(quasi)-energy eigenstates can be uniquely specified bylisting the eigenvalues of the LIOM operators. In agauged system, for each state in the even flux sector:|{si},Φ = +1〉 defined by `i|{si},Φ = +1〉 = si|{si},Φ =+1〉, there is a partner state in the odd gauge-flux sec-tor: |{si,Φ = −1〉 = F |{si},Φ = +1〉 defined by

10`

<latexit sha1_base64="O0HPVswHp2BkI2ds3SmbT0LxYG0=">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</latexit>a

<latexit sha1_base64="jdLMW9wTHLa2Fsth7aa89A69vO0=">AAAEK3icbZPNbtQwEMfdho+yfLXlyMVihbQ9NNpUFfSAqgouHItg20pJtHKcya5Vx45sJ9tVlEfgCk/B03ACceU9cJJdtE1qKc5ofvPXjGfsKONMm/H419a2c+/+g4c7jwaPnzx99nx3b/9Cy1xRmFDJpbqKiAbOBEwMMxyuMgUkjThcRtcfan5ZgNJMii9mmUGYkplgCaPEWNdnNiXT3eHYHTcL9w1vZQzRap1P95z9IJY0T0EYyonWvjfOTFgSZRjlUA2CXENG6DWZgW9NQVLQYdnUWuHX1hPjRCr7CYMb76aiJKnWyzSykSkxc91ltfMu5ucmOQlLJrLcgKBtoiTn2EhcHxzHTAE1fGkNQhWztWI6J4pQY9szGAQCFlSmKRFxGbyryoATMWtOswnAgoIoyDTjUnRgnlX1RpSSiw6K5UJU7W+NFWwGnFqq1hlvo1FdDCRm1CMHtYjN5uagh/yVyO+RcC0Ke6iofC8sg7q3UVIOvcp2EQcFPjzFhe2dndkoSg46J9P2CsyrMsbBXNtZQHk4dk/gpgpUzsH33CO4CcvWVbqe3TutiXSbNZI8rgcveZO5W1tsy87qK0Z4t7fALRMk4qSvYsWaBTSWpiOlufqvDQyz97Q770xXzX7HuLO2IsK5LcC+Iq/7ZvrGxZHrvXG9T8fDs/er97SDXqJXaIQ89BadoY/oHE0QRTP0FX1D350fzk/nt/OnDd3eWmleoFvL+fsPrWBxMQ==</latexit>

ia<latexit sha1_base64="IEG/INQhDwCujtgKgxbBDTCotAM=">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</latexit>

ia + 1

<latexit sha1_base64="Xqe4AtAJSvhGdMdi2DUArjkH1UY=">AAAEM3icbZPNbtQwEMfdho+yfLXlyCVihbR7aJRUFfSAqgouHIvEthVJVDnOZNeqY0e2s9uVlcfgCk/BwyBuiCvvgJPsom1SS3FG8/NfM54ZJwWjSvv+z61t5979Bw93Hg0eP3n67Pnu3v65EqUkMCGCCXmZYAWMcphoqhlcFhJwnjC4SK4/1PxiDlJRwT/rZQFxjqecZpRgbV1hlGM9I5iZL9XV7tD3/Ga5fSNYGUO0WmdXe85+lApS5sA1YVipMPALHRssNSUMqkFUKigwucZTCK3JcQ4qNk3OlfvaelI3E9J+XLuNd1NhcK7UMk/syTpH1WW18y4Wljo7jg3lRamBkzZQVjJXC7cugJtSCUSzpTUwkdTm6pIZlphoW6bBIOKwICLPMU9N9K4yEcN82txmE4AFcyyhUJQJ3oFlUdUbllIsOigVC161vzWWsHngxFK5jngbjepkINOjHhnXIjqd6XEPhStR2CPxWhT30LwKg9g0s5FkZhhUtopuNHcPTty5rZ3t2SjJxp2bKTsCs8qkbjRTthdgDnzvGG6qSJYMwsA7hJvYtC7jBXbvlCZRbdREsLRuvGBN5G5uqU27qEcMs25tgVnGccJwX0XnaxaRVOiOlJTyvzbS1M5pt9+Fqpr9jnYXbUaYMZuAfUVB9830jfNDL3jjBZ+OhqfvV+9pB71Er9AIBegtOkUf0RmaIIIE+oq+oe/OD+eX89v50x7d3lppXqBby/n7D7G0dOA=</latexit>Z<latexit sha1_base64="6S0pxYOqFWFc2IoKBRh1edNcUx0=">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</latexit>

t


ia


ia

<latexit sha1_base64="O0HPVswHp2BkI2ds3SmbT0LxYG0=">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</latexit>a<latexit sha1_base64="IEG/INQhDwCujtgKgxbBDTCotAM=">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</latexit>

ia + 1


(a)


(b)

<latexit sha1_base64="8UcMii51JU/KKvCrLrMjlXD8UHk=">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</latexit>

(c)

0 5 10 15 20 25 30 35Time t

-1.0

0.0

1.0h�z(t)�z(0)iaux calc. (noisy) calc. (ideal)

0 10 20 30Time t

-1.0

0.0

1.0h�z(t)�z(0)i

0 5 10Time t

h�z(t)�z(0)i

<latexit sha1_base64="xQDPnBpyktC1ahqFOE5p/bWOx6g=">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</latexit>Z(t)

<latexit sha1_base64="xQDPnBpyktC1ahqFOE5p/bWOx6g=">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</latexit>Z(t)

<latexit sha1_base64="HpFgXRlWqRRZbaSDltTk3rA9Cr0=">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</latexit>

Thermal<latexit sha1_base64="ewIcjrmxwwdy3nlF8ZazE+Xq/rU=">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</latexit>

Trivial MBL

<latexit sha1_base64="IHn+thhuM32oLh3RnWWLoPCI6Og=">AAAEM3icbZPNatwwEMeVuB/p9itJj72YLoXNIcs6LW0OIQQKpcctzSYB2wRZHu+KyJKRZO8uwo/Ra/sUfZjSW+m171DZ3i0bOwLLw/zmz4xmpChjVOnR6OfWtnPv/oOHO496j588ffZ8d2//QolcEpgQwYS8irACRjlMNNUMrjIJOI0YXEY3Hyp+WYBUVPBzvcwgTPGU04QSrK3LDzQstPn4ZXxeXu/2R8NRvdyu4a2MPlqt8fWesx/EguQpcE0YVsr3RpkODZaaEgZlL8gVZJjc4Cn41uQ4BRWauubSfW09sZsIaT+u3dq7qTA4VWqZRjYyxXqm2qxy3sX8XCfHoaE8yzVw0iRKcuZq4VYNcGMqgWi2tAYmktpaXTLDEhNt29TrBRzmRKQp5rEJTkoTMMyn9Wk2AVhQYAmZokzwFsyzstqwlGLeQrGY87L5rbGEzYBTS+U64200qIqBRA865KAS0elMH3SQvxL5HRKuRWEHFaXvhSaoehslpu+VtotuULiHp25he2dnNoiSg9bJlL0Cs9LEbjBTdhZgDkfDY1iUgcwZ+N7wCBahaVxm6Nm91ZpINVkjweJq8ILVmdu1xbbsrLpimLV7C8wyjiOGuyparFlAYqFbUpLL/9pAU3tP2/POVFnvd4w7ayrCjNkC7Cvy2m+ma1wcDb13wzef3/bPTlbvaQe9RK/QAHnoPTpDn9AYTRBBAn1F39B354fzy/nt/GlCt7dWmhfo1nL+/gP0KXSq</latexit>

FSPT

FIG. 6. Loschmidt flux echo (a) Interferometric protocol for measuring the Loschmidt flux-echo, Z(t): controlled evolutionand measurement of an ancilla , a, shown here in red enables measurement of the overlap between time evolved states withand without a symmetry flux inserted. (b) Circuit for implementing the ancilla-controlled flux insertion. (c) Simulated andexperimental data for the Loschmidt flux echo in the Floquet model with J = 0.9π (top panel) in the FSPT phase, J = 0.1π(bottom left panel) in the trivial MBL phase, and J = 0.5π (bottom right panel) in the thermalizing phase. For ideal noiselesssimulations (pale line, simulation), symmetry-preserving MBL Z(t) exhibits persistent period-two oscillations in the FSPTphase, saturates to a constant value in the trivial MBL phase, and rapidly decays in the thermalizing phase. For weakly-openbut symmetry-preserving MBL systems (dashed line, simulation) the oscillation or saturation amplitude slowly decays due toincoherent errors. By contrast, the experimental data strongly deviates from the symmetric simulations after t & 5 Floquetperiods due to coherent errors. Experimental data shown for trivial-MBL and thermal include only ten shots, and were takenfor a different disorder realization than the FSPT data.

`F,i|{si,Φ = −1〉 = si|{si,Φ = −1〉 (i.e. labeled bythe same LIOM eigenvalues but with flux-inserted LIOMoperators `F,i). Crucially, since the action of F is lo-cally equivalent to a gauge transformation everywhere,the partner states have the same (quasi)-energy, EF , (upto O(e−L/ξ) corrections).

From these definitions and observations, one can read-ily verify Eq. B2. In particular, for the FSPT:

Z(t = n) =tr U†F (n)U(n)

tr1(B9)

=tr W †F †ΦnFe+inEFWW †Φe−inEW

tr1

=1

tr1

(F †ΦnFΦn

)︸︷︷︸(−1)n

(W †e+i(EF−E)W

)︸︷︷︸

1+O(e−L/ξ)

= (−1)n +O(e−L/ξ

)(B10)

where we have used that [W,Φ] = 0 since W is diagonalin the τz basis.

Behavior of flux-echo in other phases – Similar ar-guments show that a trivial MBL phase (Ξ = 1) would

have Z(t)trivial-MBL = 1. By contrast, in a thermal phasethe localization length diverges, and the system becomessensitive to the global flux (e.g. the ratio of the Thou-less energy to level spacing diverges [24]), and to exhibitchaotic response, such that the evolutions with and with-out flux insertions deviate exponentially with time, re-sulting in exponential decay of Z.

Finally, we note that in MBL systems with sponta-neous symmetry breaking (SSB), the above argumentsfail. For example, there is not quite a complete set ofsymmetry-preserving LIOMs. For example, if we con-sider a ferromagnetic spin glass, a natural set of LIOMswould be a locally dressed version of {σzi σzi+1}, whichcount the parity of domain walls in the magnetic order,but require one additional integral of motion for com-pleteness (e.g. either a local asymmetric operator like σzior a symmetric, but non-local operator such as

∏i σ

xi ).

In the gauged system, the total parity of domain walls islocked to the gauge flux in the system, so that there is alocal energy cost to inserting flux, and the quasi-energieswith and without flux differ by an O(1) constant. As a re-sult, for SSB-MBL, Z(t) ∼ e−iεt where ε depends on thelocation of flux insertion and disorder configuration. In

11

particular, the disorder average Z(t) = 0 at long times,

but the average modulus remains |Z(t)| = 1. The latterproperty distinguishes the SSB-MBL behavior from thatof thermal systems for which |Z(t)| = 0.

Together, these behaviors show that the flux-echo pro-vides a complete non-local “order parameter” diagnos-ing all possible thermal and MBL phases with symmet-ric drives. This property makes the flux-echo a poten-tially useful diagnostic. For example, a previously in-troduced non-local string-order parameter of the FSPTphase [9] could not distinguish between FSPT and SSB-MBL requiring measurement of multiple order parame-ters to uniquely diagnose the phase.

Local approximation to flux-echo – While the abovearguments demonstrate the existence of a formally exactflux-insertion procedure, in practice, this procedure re-quires explicit knowledge of LIOMs. However, we nowargue that it is sufficient to locally approximate the ac-tion of flux insertion without knowledge of the LIOMs,which enables a practical measurement scheme for theflux-echo.

Since the LIOMs, `i of a symmetric MBL system arerelated by local dressing, W to single-site spin operatorsσxi , one can write the exact flux insertion operator F =W †τxiaW (where, recall that ia is an arbitrarily chosenlocation of flux-insertion). While the exact flux insertionoperator F cannot be implemented without knowledge ofW , one can approximate the action of F by simply actingwith τxia which differs only by quasi-local, and symmetricdressing from F , and its correlation function:

Z(t) =tr τxiaU†(t)τxiaU(t)

tr 1(B11)

would hence have finite overlap with the exact fluxLoschmidt echo, Z(t). Approximating the action of Was randomly scrambling operators within a localizationlength ξ, generically, we expect the disorder averaged

magnitude of this quantity |Z(t)| to be smaller than the

|Z(t)| by a finite constant factor of order 0 < c ∼ e−ξ <1.

These arguments are supported by numerical simula-tions of the FSPT model (see Fig. 5).

2. Quantum Circuit Implementation

The flux-echo witness Z can be measured interfero-metrically (see Fig. 6) using an ancilla qubit initializedin an equal superposition of 1√

2(|0〉+ |1〉), to control the

time-evolution, such that the system evolves under H(t)or τxiaH(t)τxia when the ancilla is in the |0〉 or |1〉 staterespectively. For the FSPT model, this amounts to sim-ply allowing the ancilla qubit to flip the sign of a single

e−iJθσyi σ

yi+1 → e+iJθσyi σ

yi+1 , which can be accomplished

with only two extra CNOT gates per Floquet period asshown in Fig. 6b. Then, measuring 〈σx,y〉 for the ancilla

reveals the real or imaginary parts of Z respectively.

We have implemented this protocol both in classicalsimulations and experimentally, with results shown inFig. 6. The experimentally measured flux-echo in theFSPT phase initially shows oscillations that survive upto ≈ 5 Floquet periods, but then are dephased due to co-herent errors (as indicated by the oscillations with largeamplitude, but wrong phase compared to numerical sim-ulations). We note that this dephasing occurs ≈ 2-3×more rapidly than that for the OBC circuits, suggestingthat the more complicated ion transport paths requiredto implement periodic (ring) boundary conditions exac-erbate the effects of errors, consistent with the expectedresults of spatial inhomogeneities in the magnetic-field asdiscussed above. In contrast, in the thermal phase theflux-echo immediately dies, and in the trivial MBL phasethe data is consistent with a saturation to a constantvalue up to an overall slow decay due to weakly-openMBL effects.

C. Long-time (meta)stability ofrecursively-generated quasiperiodic drives

This section addresses the long-time fate of the ED-SPT model, in an idealized perfectly isolated system.We will see that the EDSPT behavior is not a rigidlystable phase, but rather a very-long-lived but ultimatelymetastable behavior. However, the lifetime for the edgemodes can be made exponentially long in a certain pa-rameter regime, and hence can easily be made muchlonger than any experimental lifetime.

Topologically-trivial paramagnetic and time-quasicrystalline behavior arising in such recursiveFibonacci drives were previously studied theoreticallyand numerically in [11], leveraging the exponentialgrowth of tn with number of recursions n to efficientlyreach exponential long times in simulations3. There, itwas observed that, for strong-disorder, MBL only occursas a meta-stable phenomena in these recursively drivensystems: rather than saturating to finite constant,spin-autocorrelators decayed logarithmically slowly,with MBL dynamics eventually melting away intoinfinite temperature incoherent at ultra-long ‘heatingtime-scales th ∼ e1/δ where δ represents the drivestrength (more precisely, deviation from an idealizedperfectly-commuting drive). Numerical simulations(Fig. 7) indicate that this long-lived MBL behavior alsoarises in the EDSPT model.

Additionally, in this appendix we extend a recursiveadaptation of high-frequency (Magnus-type) expansionto show that the dynamics are governed by an effectivetime-independent Hamiltonian with emergent dynamicalsymmetries, for times up to t∗ ∼ δ−p with p = 3, 5for topologically non-trivial (see below) and trivial [11]

3 Asympotically tn ∼ ϕn where ϕ = 1+√5

2is the golden ratio.

12

phases respectively. The long-time dynamics between t∗and th appears to be beyond the purview of any effectiveHamiltonian description, and a controlled theoretical de-scription of this regime remains elusive. However, therecursive nature of the drive permits efficient numericalaccess to long-time dynamics, and our simulations indi-cate that the topological edge spins and emergent dy-namical symmetries persist well beyond t∗ up to th.

1. Recursive Magnus expansion

In Ref. [11], we previously developed a recursive high-frequency (“Magnus”) expansion technique to reduce theevolution under a weak quasiperiodic drive generated bya Fibonacci sequence of two circuit layers to an effectivetime-independent Hamiltonian evolution. This techniquewas accurate when the unitary of the generating circuitlayers was close to the identity by an amount ∼ δ. Theresulting effective Hamiltonian accurately captured theevolution up to time t∗ ∼ δ−5. Comparison to numericalsimulations showed that, beyond t∗, the driven systemexhibited logarithmically slow heating causing completethermalization in timescale th ∼ e1/δ.

Here, we generalize this technique to the case relevantfor the idealized EDSPT model, i.e. when generating uni-taries Ux,z are near-perfect π-pulses about the x, z axesrespectively. I.e. U2

x,z ≈ 1 + O(δ) but Ux,z. We willshow that, in this parameter regime, one can again ob-tain an approximate time-independent Hamiltonian de-scription up to times t∗ ∼ δ−3. Furthermore, the re-sulting effective Hamiltonian has an emergent pair of Z2

(Ising) symmetries whose generates gα with α ∈ {x, z}are related to

∏i σ

α2iσ

α2i+1 by a finite-depth local unitary

transformation whose precise form depends on the de-tails of the drive. Our theoretical picture of the EDSPTphase, is that these emergent dynamical symmetries pro-tect the SPT edge modes of an effective AKLT-chain,“encrypted” in a quasiperiodically rotating frame of thedrive.

While this technique provides our best known analytichandle on recursively generated quasiperiodic drives, nu-merical simulations and experimental results suggest thatthe recursive Magnus expansion dramatically underesti-mates the stability and survival time scales for the emer-gent dynamical symmetries. First, we note that themodel implemented in the main text, which has maxi-mal disorder strength 4π for the K-couplings is actuallynot in the small δ-regime, yet it still exhibits long-livedsignatures of topological edge states. Moreover, basedon the numerical simulations that access long times onmodest system sizes observe that these oscillations decay

exponentially-slowly with respect to 1/δ, suggesting thatthe emergent dynamical symmetries survive up to thismuch longer th timescale.

As a preview, the main result of this section is that, upto third order in a small parameter, δ that quantifies thedeviation from some exactly-solvable ideal drive, we canapproximately reduce the unitary evolution to the form:

U6n = V †e−iϕ6n(D+... )V (C1)

where, D is an effective time-independent Hamiltonianthat obeys an emergent Z2 ×Z2 symmetry generated bygx,z = V †

∏i σ

x,zi V respectively. Here, V is a finite depth

local unitary that we explicitly construct, and (. . . ) in-cludes i) terms with subleading (and generally oscilla-tory) n-dependence (e.g. with coefficients decreasing as∼ ϕ−6n or faster), and ii) O(δ3) terms beyond the va-lidity of the expansion. We note that, for reasons thatwill become apparently shortly, it is convenient to expressthe evolution at Fibonacci-indices that are multiples of6, and note that similar expressions can be obtained forU6n+k with k ∈ {1, . . . 5}.

As with the closely-related expansion for topologicallytrivial Fibonacci drives previously derived in [11], thisexpansion has the peculiar property that it breaks downat finite order independent of expansion parameter δ.Namely, at O(δ3), the expansion results in terms in Dthat grow faster that ϕ6n regardless of δ and finite sys-tem size, whereas on general grounds for sufficiently small

δ one can must always be able to reduce U6n = e−iϕ6nD′n

for some bounded, Hermitian D′n (though not necessarilyone that is time- i.e. n-independent) since the circuit con-tains only ∼ ϕ6n layers. Hence, the generation of termswith coefficients growing faster than ϕ6n signal a break-down in the recursive Magnus expansion, and hint at apossible obstruction to describing the dynamics beyondt∗ by any effective time-independent Hamiltonian.

2. Inflation rule

To derive the recursive Magnus expansion, we exploita self-similar fractal structure of Fibonacci sequencesunder “inflating” the generating unitaries Ux,z. Letus consider a Fibonacci drive generated by two pulses(Ux = XeA,Uz = ZeB) where X =

∏i σ

x2iσ

z2i+1, Z =∏

i σz2iσ

z2i+1 commute and square to one, and A,B are

anti-Hermitian operators with small norm δ ∼ ‖A,B‖ �1 that represent perturbations to the ideal drive. Forconvenience we also define Y = XZ (note that in ournotation X,Y, Z are not single spin Pauli operators, butrather strings of even numbers of σx,y,z products).

The Fibonacci drive is generated by the inflation rule:(XeA, ZeB

)→(ZeB , XeAZeB

)=(ZeB , Y eAxeB

). (C2)

where we have defined Ax = XAX, and similarly define Ay,z, Bx,y,z. It is convenient to recurse three times with the

13

inflation rule, and pull all X,Z’s to the left (conjugating A,B’s along the way) until the X,Z prefactors return totheir original form: (

XeA, ZeB) (1)→

(ZeB , XeAZeB

)=(ZeB , XZeA

z

eB)

(2)→(XZeAzeB , ZeBXZeAzeB

)=(XZeAzeB , XeByeAzeB

)(3)→(XeByeAzeB , ZeAyeBxeByeAzeB

). (C3)

This “3x-inflated” inflation rule yields an recursion relation for A,B alone:

(eA, eB)(3x)−→ (eByeAzeB , eAyeBxeByeAzeB), (C4)

where the arrow now represents this “3x” recursion.At this point, it turns out to be easier to think of the unitaries: Un as being functions of an “alphabet” of 8 “letters”:

(A,B,Ax, Bx, Ay, By, Az, Bz). To avoid a proliferation of subscripts, in what follows we change our notation to labelFibonacci times with indices that are multiples of 3, by defining:

Un ≡ U3n, (C5)

where Un corresponds to t3n = F3n ∼ ϕ3n circuit layers. The recursion rules for Ax, Ay, Az and Bx, By, Bz follow fromthose of A,B simply by conjugating by X,Y, Z. We have the following recursion relation for the unitary operator:

Un+1(eA, eB , eAx , eBx , eAy , eBy , eAz , eBz ) = Un(eByeAzeB , eAyeBxeByeAzeB , eBzeAyeBx , . . . ), (C6)

where we emphasize that n now labels the number of “3x” recursions, and U1 = eA.

3. Effective Hamiltonian and generalized Magnus expansion

Our goal is to check the emergence of an effective Hamiltonian for the dynamics by computing order by orderHn = logUn (defined here to be anti-Hermitian), which obeys

Hn+1(A,B,Ax, Bx, Ay, By, Az, Bz) = Hn(By ? Az ? B,Ay ? Bx ? By ? Az ? B,Bz ? Ay ? Bx, . . . ), (C7)

where A ? B ? C ? · · · = log(eAeBeC . . . ).

To proceed, we relabel those letters(K1,K2, . . . ,K8) = (A,B,Ax, Bx, Ay, By, Az, Bz),and expand

Hn =

8∑α=1

cα(n)Kα +∑αβ

Cαβ(n) [Kα,Kβ ] + . . . . (C8)

where (. . . ) indicate nested commutators with largernumber of K ′s that contribute at O(δ3).

Plugging this expression in the recursion relation (C7),we find that the leading order coefficients are given by:

~cα(n+ 1) = MT~cα(n), (C9)

with the recursion matrix

M =

0 1 0 0 0 1 1 00 1 0 1 1 1 1 00 0 0 1 1 0 0 10 1 0 1 1 0 1 10 1 1 0 0 1 0 01 1 1 0 0 1 0 11 0 0 1 0 0 0 11 0 1 1 0 1 0 1

. (C10)

This recursion relation can be solved straightforwardlyby diagonalizing M . The largest eigenvalue is ϕ3, withan eigenvector (1/ϕ, 1, 1/ϕ, 1, 1/ϕ, 1, 1/ϕ, 1). This im-plies an emergent symmetry between A,Ax, Ay, Az (andsame for B,Bx, By, Bz) at long times. Namely, fromthis dominant eigenvector coefficients we can see thatc1 ∼ c3 ∼ c5 ∼ c7 at large n, and c2 ∼ c4 ∼ c6 ∼ c8. Inother words, at this order Hn commutes with the pulesX and Z at large n (long times).

Let us now consider the next order terms Cα,β . Thereare two different types of contributions: one coming from

14

applying the substitution rule Kα →∑βMαβKβ+. . . to

leading order to the term∑αβ Cα,β(n)

[Kα, Kβ

]on the

right hand side, and the other one coming from the Baker,Campbell, Hausdorff (BCH) formula from

∑α cα(n)Kα .

The first type is readily taken into account, and we find:

C(n+ 1) = MT C(n)M + rn, (C11)

where rn is a skew-symmetric matrix that originates fromthe second type of terms mentioned above.

Keeping track of the contributions to rn is straightfor-ward albeit cumbersome. For example, we have a termc1(n)K1 = c1A with A → By ? Az ? B. This will gen-

erate terms c1(n)2 [By, Az] + c1(n)

2 [By, B] + c1(n)2 [Az, B]

that should be included in rn. The contributions fromc3(n), c5(n), c7(n) can be obtained by conjugating byX,Y, Z, and the other contributions can be dealt within a similar way (except there are now five exponentialsto expand using the BCH formula). With this expressionfor rn, eq. (C11) can be solved by going to the eigenba-

sis of rn and then rotating back. The explicit expres-sion for the matrix C(n) is not particularly illuminat-ing, but we have checked that it grows with n as ϕ3n,corresponding to time, which means that one can writeHn = −iϕ3n(D + . . . ), where D can be interpreted asan effective Hamiltonian for the dynamics. We will showthat Hn has an emergent Z2 × Z2 symmetry at large n(long times), which protects the topological edge modes.

This expansion breaks down at the next order: includ-ing nest commutators in the expression of Hn, we findthat the coefficients of such terms grow with n exponen-tially faster than ϕ3n. Not only does the Hamiltonianinterpretation therefore break down at this order, buthigher orders become more important earlier in time: thisappears to be a general property of recursive drives asnoted in Ref. [11]. This means that this high-frequency,Magnus-type expansion is only strictly valid for times upto t? ∼ δ−3, with δ ∼ ‖A,B‖ � 1.

However, as previously commented, we find numeri-cally that even away from this high-frequency regime,strong disorder and MBL protect this dynamical phaseup to much longer, exponential time scales (see Fig. 7).

4. Emergent symmetry

Let us now analyze the symmetries of Hn including second order terms. Let

gx =

0 0 1 0 0 0 0 00 0 0 1 0 0 0 01 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 0

, gz =

0 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 01 0 0 0 0 0 0 00 1 0 0 0 0 0 0

. (C12)

This is a representation of Z2 ×Z2 on the 8-dimensional “alphabet”, whose action corresponds to conjugating lettersby X and Z, and g2

x = g2z = 1 and [gz, gz] = 0. Note that the recursion matrix is compatible with this symmetry:

[M, gz] = [M, gx] = 0. (C13)

The leading order coefficients cα(n) are symmetric only at long times, as the initial condition cα(n = 0) = δα,1 breaksthe symmetry. Because of this, the matrix rn is also “almost” symmetric up to O(1) errors: [rn, gx,z] = O(1). However,

these non-symmetric terms get amplified in the recursion relation (C11), so C(n) contains some exponentially largenon-symmetric terms of order O(ϕ3n). We will write

C(n) = Cs(n) + δC(n), (C14)

with the symmetrized matrix Cs = (C + gxCgx + gzCgz + gxgzCgxgz)/4, and ||δC(n)|| ∼ O(ϕ3n).Let us now try to cancel out those symmetric term by a (finite-depth unitary) change of frame:

V = e12

∑γ vγKγ , (C15)

with U ′n = V UnV†.

We find

H ′2n = logU ′2n =1

2

∑γα

cα(2n)vγ [Kγ ,Kα] +∑α

cα(2n)Kα +∑αβ

Cαβ(2n) [Kα,Kβ ] + . . . , (C16)

15

where we restricted ourselves to even times to avoid odd/even effects. Our goal is to choose vγ to cancel out the

non-symmetric terms δC(n), in (C14), or more precisely, cancel out the leading non-symmetric terms δC(2n) =

ϕ6n(δC0 + . . .

), where the dots represent exponentially decaying terms. We find that this can be achieved by

choosing vα =(− 2+

√5

4 − 1−√

52 c, c,−

√5

4 − 1−√

52 c, c, 1

4 − 1−√

52 c, c, 3

4 − 1−√

52 c, c− 1

)with c a constant. Note that this

is only possible since the leading order terms∑8α=1 cα(n)Kα in Hn are symmetric at large n.

We conclude that for this choice of dressing unitary,we can write

U2n = V †e−iϕ6n(D+... )V, (C17)

with an effective Hamiltonian D that has a Z2×Z2 sym-metry generated by the pulses, and where the dots de-note exponentially small in n non-symmetric terms. Inother words, the unitary evolution operator U2n com-mutes with the “dressed” symmetry generators

gx = V †XV, gz = V †ZV, (C18)

at long times. This symmetry is manifestly emergent, asV depends on A and B. The effective Hamiltonian hasthe following expression

−iD =1√5

(Bs + ϕ−1As

)+

3√

5− 5

40[A,B]s +

+

√5− 5

40[A,Bx]s +

1

4√

5[A,Bz]s . (C19)

with As = (A+Ax+Ay+Az)/4, and similarly for all sym-metrized quantities, e.g. [A,B]s = ([A,B] + [Ax, Bx] +[Ay, By] + [Az, Bz])/4.

5. Numerical (ED) Simulations

To investigate the long-time dynamics beyond theregime of validity of the recursive Magnus expansion,we have performed numerically exact simulations of theEDSPT model for various J . The recursive structure en-ables access to exponentially long times tn ∼ Fn with nrecursions. Fig. 7 shows the resulting edge-spin correla-tors, averaged over disorder, for various J values. ForJ close to π, we observe that the quasiperiodic oscilla-tions of the edge spins survive up to very long times. Weestimate the heating time th as the time at which theedge-oscillation drops below an arbitrary small thresh-old: ε = 0.02. As shown in the inset of Fig. 7, th showssuperpolynomial growth 1/δ = π/|J − π| for small δ, in-dicating that the edge spin oscillations survive far beyondthe time-scale at which recursive Magnus expansion fails.This behavior is similar to that numerically observed intopologically-trivial recursive Fibonacci drives [11], andsuggests that ultra-long-lived but ultimately metastableMBL-like dynamics are generic features for recursivedrives with strong disorder. However, a detailed theo-retical picture of the logarithmically slow entanglementgrowth and heating in these models remains elusive atthis time.

100 102 104 106

Time t

0.0

0.5

1.0

|〈σz(t)σz(0)〉|1.00

0.98

0.96

0.94

0.92

0.90

0.88

0.86

0.84

0.82

0.80

0.78

0.76

0.74

0.72

0.70

0.7 0.8 0.9J/π

103

104

105 Heating Time

FIG. 7. Slow heating (numerical simulations) Numericalsimulations of the long-time behavior of |Cz| for the edge spinof an L = 10 spin-chain with the Fibonacci drive, for various J(other parameters chosen as in Fig. 1). Each point is averagedover 960 disorder realizations. The inset shows estimate ofheating time th empirically defined as the time where |Cz|drops below a small threshold ε = 0.02 (dashed line), anddemonstrates that the three-fold periodic oscillations persistto time scale exponentially long in the inverse pulse detuning,|J − π|−1.

D. Additional data

In this appendix, we include additional simulation andexperimental data for the FSPT and EDSPT models.

1. Short time FSPT dynamics

Figure 8 shows a close-up of the early time dynamics ofthe FSPT model for nominally ideal J = π parameters,which, absent errors, would be at the fixed point of theFSPT phase. During the time interval shown the systemis not yet effected by coherent errors and exhibits the ex-pected weakly-open MBL FSPT dynamics, characterizedby period two edge oscillations and random/dephasingbulk oscillations.

16

-1.0

0.0

1.0⟨σx(t)σx(0)⟩ ⟨σz(t)σz(0)⟩

edge calc. (noisy) calc. (ideal)

0 5 10 15Time t

-1.0

0.0

1.0

0 5 10 15Time t

bulk calc. (noisy) calc. (ideal)

FIG. 8. Ideal FSPT implementation (short times)FSPT model simulation (noiseless: pale solid lines, noisy:dashed lines) and experimental data (solid points with errorbars) at the ideal J = π “fixed-point” for short times t ≤ 15for which coherent errors have not yet affected the topologicaledge spins. The edge shows the expected period-two oscilla-tions, whereas bulk spins show slowly decaying, weakly-openMBL behavior along the symmetry axis ∼ σx, and rapidlydephase due to random fields and spin-spin interactions per-pendicular to the symmetry axis, ∼ σz.

a. EDSPT with uniaxial disorder and nominal Z2

symmetry

This section shows results for the EDSPT model withrandom fields purely along the y-axis, which, like theFSPT model, nominally has a microscopic Z2 symme-try generated by

∏i σ

yi , but which is broken in imple-

mentation by the same coherent errors that decohere theFSPT edge states. The plots shown in Fig. 9 below pro-vide further evidence that the EDSPT edge states arenot harmed by these coherent errors and do not rely onthis fine-tuned symmetry, a fact that is accentuated bythe data shown in the main text in which this symmetryis manifestly broken by the vector B-disorder. Addition-ally, we show the raw, non-averaged bulk-data in Fig. 10.Note that the bulk spins exhibit random, but slowly de-caying oscillations characteristic of the slow dephasingdynamics of MBL. Upon disorder- and/or site- averagingthese random oscillations wash out, as seen in the otherfigures presented throughout the main text.

17

-1.0

0.0

1.0 ⟨σx(t)σx(0)⟩J = 1.00 · π


-1.0

0.0

1.0 ⟨σz(t)σz(0)⟩ edge

bulk

⟨σx(t)σx(0)⟩J = 0.95 · π


⟨σz(t)σz(0)⟩ edge

bulk

⟨σx(t)σx(0)⟩J = 0.90 · π



bulk

⟨σx(t)σx(0)⟩J = 0.85 · π



bulk

⟨σx(t)σx(0)⟩J = 0.80 · π



bulk


-1.0

0.0

1.0⟨σx(t)σx(0)⟩

1.00 0.95 0.90 0.85 0.80


⟨σz(t)σz(0)⟩

(a)

(b)

FIG. 9. EDSPT with uniaxial disorder. (a) Edge and (site averaged) bulk correlators for various values of J for theEDSPT model with Z2 symmetry (Bi ‖ y), which show that the EDSPT behavior persists over a range of |J − π|/π . 0.25.Pale lines show idealized (noiseless) simulations, dashed lines show simulations with depolarizing noise, and solid dots with1σ-error bars are experimental data. (b) The same edge data for various J replotted on a single plot to facilitate comparisonof curves with different pulse-weights.

18

-1.0

0.0

1.0 i = 1

J = 1.00 · π

-1.0

0.0

1.0 i = 2

-1.0

0.0

1.0 i = 3

-1.0

0.0

1.0 i = 4

-1.0

0.0

1.0 i = 5

-1.0

0.0

1.0 i = 6

-1.0

0.0

1.0 i = 7

-1.0

0.0

1.0 i = 8

-1.0

0.0

1.0 i = 9


-1.0

0.0

1.0 i = 10

i = 1

J = 0.95 · π

i = 2

i = 3

i = 4

i = 5

i = 6

i = 7

i = 8

i = 9


i = 10

i = 1

J = 0.90 · π

i = 2

i = 3

i = 4

i = 5

i = 6

i = 7

i = 8

i = 9


i = 10

i = 1

J = 0.85 · π

i = 2

i = 3

i = 4

i = 5

i = 6

i = 7

i = 8

i = 9


i = 10

i = 1

J = 0.80 · π

i = 2

i = 3

i = 4

i = 5

i = 6

i = 7

i = 8

i = 9


i = 10

FIG. 10. EDSPT Site-resolved Correlators (with uniaxial disorder) shown for each site i ∈ {1, 2, . . . 10} in the chain.Pale lines show idealized (noiseless) simulations, dashed lines show simulations with depolarizing noise, and solid dots with1σ-error bars are experimental data. As described in the Methods section, even sites are prepared and measured in the z basisand odd sites in the x basis. Whereas edge sites (i = 1, 10) exhibit period-three oscillations in Fibonacci time, the bulk sites(2 ≤ i ≤ 9) undergo random oscillations that wash out upon site- or disorder- configuration averaging.

arxiv:2107.09676v1 [quant-ph] 20 jul 2021

Documents