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Realization of Two-dimensional Discrete Time Crystals with Anisotropic Heisenberg Coupling

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that a two-dimensional disordered kicked XXZ spin system on a superconducting processor realizes a discrete time crystal with period-doubled oscillations over tens of Floquet cycles.

desk verdict A credible 2D anisotropic Heisenberg DTC phase diagram from tensor networks, but the hardware 'demonstration' rests on a noise-recovery transfer the SI itself flags; the raw data alone show only decaying oscillations. read the letter →

arxiv 2501.18036 v1 pith:UKNCNVBR submitted 2025-01-29 quant-ph

Eric D. Switzer , Niall Robertson , Nathan Keenan , Ángel Rodríguez , Andrea D'Urbano , Bibek Pokharel , Talat S. Rahman , Oles Shtanko
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Sergiy Zhuk Nicolás Lorente
This is my paper · ORCID
classification quant-ph
keywords discretetimecrystalFloquetphasesofmatteranisotropicHeisenbergmodelmany-bodylocalizationquantumsimulationtensornetworknoiserenormalizationtime-translationsymmetrybreaking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a discrete time crystal, a phase of matter that breaks discrete time-translation symmetry by responding at twice the driving period, can exist in a two-dimensional spin system with anisotropic Heisenberg (XXZ) couplings, going beyond the one-dimensional Ising models studied previously. It reports experiments on a 144-qubit superconducting processor arranged in a disordered heavy-hex lattice, at kick angle $\phi = 0.45\pi$ and spin-flip coupling $\epsilon = 0.05$, where the time-correlation function $\Delta(t)$ keeps period-doubled oscillations and nearest-neighbor spin order over 30 to 50 Floquet cycles. Sweeping $\epsilon$ and $\phi$ maps a phase diagram with spin-glass, time-crystalline, and ergodic regions, and a fully polarized initial state shows a particularly stable scar-like subharmonic response. If true, this would establish that Floquet time-crystalline order is realizable in two dimensions with more natural interactions and that current noisy quantum processors can access such phases with tailored noise renormalization and tensor-network cross-checks.

What carries the argument

The argument is carried by a Floquet circuit $U_F = U_{XXZ}^{(3)}U_{XXZ}^{(2)}U_{XXZ}^{(1)}U_X$, where $U_X$ is a global X-rotation by angle $\phi$ and each $U_{XXZ}^{(k)}$ is a layer of non-overlapping two-qubit gates $U_{ij} = \exp[-i J_{ij}(\epsilon X_i X_j + \epsilon Y_i Y_j + Z_i Z_j)]$ with disordered couplings $J_{ij}=1+\delta_{ij}$. The disorder is designed to induce many-body localization, blocking heating from the periodic drive. The second load-bearing piece is a noise-renormalization ansatz, stated in Eq. (S.1) of the supplementary information, that $\langle O\rangle_{\mathrm{noisy}} = f(t)\langle O\rangle + c(t)$ with a period-2 offset $c(t)=c(t+2)$; the four offset parameters are learned on small $2\times2$ or $3\times3$ subsystems by fitting to classical simulation, then exported to the 144-qubit $3\times7$ lattice. The classical reference comes from two-dimensional tensor-network simulations updated and contracted with belief propagation, which places the network in an approximate canonical gauge so that truncation is quasi-optimal, cross-checked against matrix-product-state simulations.

What would settle it

Run the same 144-qubit experiment at $\epsilon=0.05$, $\phi=0.45\pi$ on a device with low enough error rates to execute depth-450 circuits directly, or with an independently validated error-mitigation method, and check whether the period-doubled $\Delta(t)$ and the $\omega=\pi$ peak persist for 30 to 50 Floquet cycles; the claim falls if the raw or independently mitigated signal does not show subharmonic order. A purely classical falsifier is to increase the bond dimension of the two-dimensional tensor-network belief-propagation simulations at late times and check whether the predicted $\Delta(t)$ converges to a period-2 plateau or decays to the ergodic value.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that a kicked, disordered XXZ model on a two-dimensional decorated hexagonal lattice realizes a discrete time crystal. At the phase point ($\epsilon=0.05$, $\phi=0.45\pi$), the spin-memory order parameter $\Delta(t) = \frac{1}{N}\sum_i s_i \langle \psi_t|Z_i|\psi_t\rangle$ oscillates with period 2 for tens of Floquet cycles, and its Fourier transform has a sharp peak at $\omega=\pi$, the signature of period doubling. The nearest-neighbor spin-spin correlation $\chi(t) = \frac{1}{M}\sum_{\langle i,j\rangle} \langle Z_i Z_j\rangle^2$ likewise remains nonzero over the same time window, indicating persistent spatial order. After noise renormalization, the hardware data agree with classical two-dimensional tensor-network simulations. Sweeping the spin-flip coupling $\epsilon$ and the X-gate angle $\phi$ reveals distinct glassy, time-crystalline, and ergodic regions, and the order parameters become more stable as the system grows from a $2\times2$ to a $3\times7$ lattice. A fully polarized initial state, being an eigenstate of the spin-flip term, shows a much more persistent subharmonic response than an alternating-spin state, a behavior the paper links to quantum many-body scars and cat-scar discrete time crystals.

Load-bearing premise

The load-bearing assumption is that the noise-recovery ansatz, $\langle O\rangle_{\mathrm{noisy}} = f(t)\langle O\rangle + c(t)$ with a period-2 offset, remains valid when its four offset parameters, learned on small $2\times2$ or $3\times3$ lattices by matching classical simulation, are applied to the 144-qubit $3\times7$ lattice.

Editorial extensions

If this is right

  • Period-doubled time-crystalline order is not confined to one-dimensional Ising couplings; it persists in a two-dimensional system with anisotropic Heisenberg interactions over tens of Floquet cycles.
  • The $(\epsilon,\phi)$ phase diagram of the kicked disordered XXZ model contains distinct spin-glass, time-crystal, and ergodic regimes, so interaction anisotropy and the driving protocol become tunable controls for these phases.
  • Initialization is decisive: a fully polarized state, being an eigenstate of the spin-flip coupling, shows a much more stable subharmonic response than an alternating-spin state, even at larger $\epsilon$.
  • Many-body localization, despite theoretical arguments questioning its stability in two dimensions, can protect subharmonic response against heating on the timescales probed.
  • The noise-renormalization method extends reliable extraction of collective observables to circuit depths where standard error-mitigation techniques fail, permitting the study of 144-qubit Floquet dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit is that the same Floquet circuit on other two-dimensional lattices, such as square or honeycomb, would separate the time-crystal mechanism from the heavy-hex topology; survival of period doubling there would make the phase generic in two dimensions.
  • A testable extension: on a system small enough for exact classical simulation, one could validate the recovery ansatz by comparing $f(t)/c(t)$-recovered expectation values with exact values at every cycle; systematic growth of the discrepancy with time would mark where transferred offsets cease to be valid.
  • A practical consequence the paper does not develop is that a fully polarized state's strong subharmonic response is a sensitive state-dependent probe of the kick angle and weak perturbations, which could be exploited in Floquet-based sensing.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper reports an experimental study of a 144-qubit two-dimensional Floquet spin system with disordered anisotropic Heisenberg (XXZ) couplings and periodic transverse kicks, performed on the IBM Heron processor 'ibm_fez'. The authors define time-correlation and spatial spin-glass order parameters, present raw and noise-recovered data for up to 50 Floquet cycles, and construct phase diagrams in the spin-flip coupling strength and kick angle. They supplement the hardware data with tensor-network simulations using both MPS and belief-propagation 2dTNS methods. The central claim, stated in the abstract and conclusion, is the demonstration of a discrete time crystal (DTC) in a two-dimensional system with non-Ising couplings.

Significance. If the central claim is sound, this would be a notable extension of DTC experiments beyond one-dimensional Ising models, addressing a gap in the study of Floquet matter in two dimensions. The paper also introduces a noise-recovery scheme and uses state-of-the-art tensor-network methods on a current quantum processor, which adds methodological interest. The raw subharmonic response and the independent classical tensor-network predictions of a DTC phase, if both validated, would provide genuine evidence for a 2D Heisenberg DTC. The manuscript is honest in its supplementary material about the limitations of the recovery method, which is a strength, but those limitations are not adequately reflected in the main-text claims.

major comments (3)
  1. [SI §IV A, Eqs. (S.9)–(S.10); Figs. 2a and 4b] The central 144-qubit DTC evidence rests on the noise-recovered curves, yet the SI explicitly acknowledges that the transfer of offsets learned on a smaller lattice can produce a denominator in Eq. (S.9) that is underestimated or approaches zero, and that this 'explains the ... signal increase observed at late times Figs. 2a and 4b.' A noiseless order parameter defined in Eq. (5) cannot grow under unitary evolution, so the late-time increase is a direct indication that the recovered curve is not a faithful estimate of the noiseless dynamics. The paper must either provide a quantitative validation of the transfer (for example, learning offsets on a 3×3 system and benchmarking against a second 3×3 or an independent classical simulation, with error bars) or substantially weaken the main claim from 'demonstrate the existence of a DTC' to 'observe signals consistent with a DTC in raw data and classical simulation.' As written, the main-text claim is not supported by the presented evidence.
  2. [Main text Eqs. (7a)–(7b); Fig. 3a] The phase diagram and the DTC order parameter are based on a single disorder realization and show no error bars. The sentence following Eq. (4) states that one disorder realization is sufficient for sums of single-qubit observables given the large number of qubits, but this is an assumption rather than a demonstrated fact, and it is especially consequential for a phase diagram where boundaries are claimed. Shot-noise uncertainties are also absent. Without error bars or disorder averaging (or a clear theoretical argument for self-averaging), the location of the ergodic/DTC boundaries in Fig. 3a and the scaling statement in Fig. 3b are not quantitatively supported.
  3. [SI §III B, Eqs. (S.9)–(S.10) and Figs. S9–S10] The 2dTNS belief-propagation simulations used to fit the noise offsets (Eq. S.10) are acknowledged to have uncontrolled truncation errors in the presence of loops, and the comparisons in Figs. S9–S10 show that MPS and BP diverge at late times. Since the offsets are learned from these classical simulations on 2×2 and 3×3 lattices, any late-time error in the classical target propagates directly into the recovered 3×7 curves. The paper should include a convergence analysis of Δ(t) and χ(t) as a function of bond dimension for the specific parameter points used in the offset fitting, with a quantitative estimate of the remaining error and a discussion of how that error affects the recovered hardware curves.
minor comments (5)
  1. [Abstract and Conclusion] The claim 'demonstrate the existence of a DTC' is stronger than the evidence, which spans at most 30–50 Floquet cycles; consider phrasing such as 'observe signatures consistent with a DTC' or include an argument for why 50 cycles is sufficient to establish the phase.
  2. [SI §IV A, Eq. (S.9) and surrounding text] The second offset in the numerator of Eq. (S.9) should be δ(ε,φ,t), not δ(ε,φ₀,t); the same typo appears in the sentence describing the failure mode ('δ_k(0, φ0, t) and δ_k(ε, φ0, t)').
  3. [Reference [3]] The journal name 'Phys. Mod. Phys.' should be 'Rev. Mod. Phys.'.
  4. [Fig. 3a caption] The color scale and the precise definition of the plotted order parameters (raw versus noise-recovered) are not specified; please clarify so the reader can interpret the phase boundaries.
  5. [SI §IV B, Eq. (S.18)] The factor (N−1) in Eq. (S.18) appears inconsistent with the definition of the expectation value over the pair set S in Eqs. (S.11)–(S.12); please check the normalization and clarify the size of the pair set used in the 2×2, 3×3, and 3×7 geometries.

Circularity Check

1 steps flagged · score 4.0 of 10

Hardware-verification loop is partially circular: the noiseless 144-qubit DTC curves are reconstructed with offsets fitted to classical simulations of the same observable on smaller lattices, though the independent 2dTNS phase prediction keeps the central physics claim grounded.

  1. fitted input called prediction [Supplementary Information, Section IV A, Eqs. (S.9)-(S.10) and Fig. S11; used for main-text Figs. 2a, 3a, and 4b]
    "In the present series of experiments, the results obtained from the optimization of the expression in Eq. (S.10) for a classically simulable, smaller qubit subset (e.g., 2 × 2) are utilized to mitigate the results for a larger qubit subset (e.g., 3 × 7), where reliable classical simulation is not feasible. The plots shown in Figs. 2a and 4b are generated using this approach."

    The "noiseless" Δ(t) curves presented as hardware evidence are not independent measurements. Eq. (S.9) reconstructs them using four offsets δ learned by Eq. (S.10), which minimizes the squared mismatch between the recovered curve and a classical simulation of the same collective observable Δ(t) on a 2×2 lattice. Transferring these fitted offsets to the 3×7 system makes the recovered 144-qubit signal an estimator biased toward the small-system classical simulation used in the fit. The SI explicitly concedes the failure mode: if the learned offsets do not transfer, the denominator of Eq. (S.9) can approach zero, "also explains ... signal increase observed at late times Figs.

full rationale

Most of the derivation is self-contained: the Floquet model (Eqs. 1-4), observables (Eqs. 5-6), order parameters (Eqs. 7), and the 2dTNS/MPS classical simulations are defined from first principles with no fitted parameters, and the belief-propagation tensor-network results are benchmarked against MPS and bond-dimension convergence in Figs. S9-S10. No load-bearing claim rests on a self-citation chain: the decorated-lattice and belief-propagation references are external prior work, and the multiproduct-formula references appear only in a forward-looking discussion. The partial circularity is confined to the hardware-verification loop. The noiseless 144-qubit curves and the hardware panels of the phase diagram are not direct measurements: Eq. (S.9) reconstructs Δ(t) using offsets obtained by fitting Eq. (S.10) to a classical simulation of the same observable on 2×2 (or 3×3) lattices. The paper itself flags that if these offsets do not transfer, the denominator in Eq. (S.9) can approach zero, producing the unphysical late-time signal increase in Figs. 2a and 4b. Thus the hardware cannot independently confirm the large-system DTC; the recovered curves are model-assisted estimates. The independent 2dTNS classical simulation does support the DTC phase in the Heisenberg model, which is why the circularity is partial rather than total. Score 4 reflects this mix: a real fitted-input-as-prediction step in the hardware evidence, but an independent tensor-network anchor for the central physics claim.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central physics rests on the model being simulated, the classical tensor-network approximation of that model, and the noise model used to correct hardware. No new particles, forces, or dimensions are introduced. The scar-like DTC behavior is an interpretation of existing dynamics. The noise-recovery offsets are genuine fitted parameters, and their transferability across system sizes is the least externally constrained step.

free parameters (4)
  • Noise offsets c(t) and c'(t), four values in total = not reported numerically
    Fit by convex optimization against classical simulation of the target observable on 2x2 or 3x3 lattices (Eq. S.10), then applied to the 3x7 lattice. If wrong, the recovered large-system data lose meaning.
  • Hamming flip probability p(t) = depth-dependent, shown in Fig. S12a
    Learned from Clifford-point data through Eq. (S.28) and used to invert the independent-spin-flip noise kernel for Hamming distance distributions.
  • Trial distribution shape parameters d0, sigma, k, q = not reported
    Fit to noisy Hamming distributions under the loss in Eq. (S.30); the variance of this distribution is tied to the quantum Fisher information claim.
  • Regularization weights q, lambda1, lambda2 = not reported
    Auxiliary parameters that select unique offsets and constrain the trial distribution; their values are not specified in the manuscript.
assumptions (4)
  • ad hoc to paper Noise on collective observables has the linear form <O>_noisy = f(t)<O> + c(t), with f(t) independent of model parameters and c(t) 2-periodic.
    Invoked in SI Section I A (Eq. S.1). The offsets are fit to classical simulation and transferred to larger lattices, which is the least externally validated step in the hardware analysis.
  • domain assumption The 2D tensor network simulation using belief propagation, with bond dimension up to 128, accurately approximates the noiseless dynamics of the 3x7 lattice for 30 to 50 cycles.
    SI Section III B and Figs. S9 and S10 show bond-dimension convergence for several observables, but truncation is not optimal or rigorously controlled when the network has loops, and MPS and BP diverge at later times.
  • domain assumption A single disorder realization is sufficient to describe collective single-qubit observables on N=144 qubits.
    Main text, Model section. The authors state this without disorder averaging or a quantitative statistical test, so the phase boundaries in Fig. 3 rest on one sample of random couplings.
  • standard math Standard limit theorems justify neglecting the cross term eta(t) in the noise model for large N.
    SI Eq. (S.6) uses independence between attenuation factors and spin values, then law of large numbers and central limit theorem. This is a mild but nonzero assumption about the noise statistics.

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Cite this review

Pith. "Pith review of Realization of Two-dimensional Discrete Time Crystals with Anisotropic Heisenberg Coupling." pith.science (2026). https://pith.science/paper/UKNCNVBR

@misc{pith2026250118036,
  author       = {Pith},
  title        = {Pith review of: Realization of Two-dimensional Discrete Time Crystals with Anisotropic Heisenberg Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UKNCNVBR}},
  note         = {Machine review of arXiv:2501.18036}
}
read the original abstract

A discrete time crystal (DTC) is the paradigmatic example of a phase of matter that occurs exclusively in systems out of equilibrium. This phenomenon is characterized by the spontaneous symmetry breaking of discrete time-translation and provides a rich playground to study a fundamental question in statistical physics: what mechanism allows for driven quantum systems to exhibit emergent behavior that deviates from their counterparts with time-independent evolution? Unlike equilibrium phases, DTCs exhibit macroscopic manifestations of coherent quantum dynamics, challenging the conventional narrative that thermodynamic behavior universally erases quantum signatures. However, due to the difficulty of simulating these systems with either classical or quantum computers, previous studies have been limited to a set of models with Ising-like couplings -- and mostly only in one dimension -- thus precluding our understanding of the existence (or not) of DTCs in models with interactions that closely align with what occurs in nature. In this work, by combining the latest generation of IBM quantum processors with state-of-the-art tensor network methods, we are able to demonstrate the existence of a DTC in a two-dimensional system governed by anisotropic Heisenberg interactions. Our comprehensive analysis reveals a rich phase diagram encompassing spin-glass, ergodic, and time-crystalline phases, highlighting the tunability of these phases through multiple control parameters. Crucially, our results emphasize the interplay of initialization, interaction anisotropy, and driving protocols in stabilizing the DTC phase. By extending the study of Floquet matter beyond simplified models, we lay the groundwork for exploring how driven systems bridge the gap between quantum coherence and emergent non-equilibrium thermodynamics.

Figures

Figures reproduced from arXiv: 2501.18036 by the authors.

Figure 1
Figure 1. Two-dimensional transitions driven by spin-flip coupling. a, Discrete time crystals (DTC) are stabilized through a long-range order induced by disordered Ising couplings. Introducing XY spin-flip couplings acts as an additional mechanism for state-dependent thermalization. However, the stability of time-crystal ordering in two dimensions in the presence of such coupling, along with the precise location of phase boun… view at source ↗
Figure 2
Figure 2. Subharmonic behavior. Performance of the device in the discrete time-crystalline regime. The results are compared to numerical simulations, where we use a two-dimensional tensor network state (2dTNS), see Section III B of the Supplementary Information. Noise recovery is performed using the renormalization methods described in Sections I A and IV of the Supplementary Information. a, The order parameter ∆(t), defined … view at source ↗
Figure 3
Figure 3. Exploring the parameter space. a, Dependence of the MBL order parameter [Eq. (7a)] (left) and the DTC order parameter [Eq. (7b)] (right) on the spin-flip interaction strength ϵ and X-gate rotation angle ϕ for T = 30 Floquet cycles. Regions where the order parameters approach unity correspond to crossovers into the glassy localized regime (left) and the discrete time crystal (DTC) regime (right). Key reference points… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Prethermal scar-like behavior. Data corre￾sponding to the polarized initial state |ψ⟩ = |↑⟩⊗N . a, The MBL order parameter as a function of spin-flip interaction strength ϵ and X-gate angle ϕ. In contrast to Fig. 3a, the or￾der parameter remains substantially large acr…
Figure 31
Figure 31. Figure 31: TN heavy hex 1 λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ [PITH_FULL_IMAGE:figures/full_fig_p016_31.png]
Figure 32
Figure 32. Figure 32: TN heavy hex 7 Figure S4. The two-dimensional tensor network state with 2 × 2 heavy hexagons. known as the canonical form or the “Vidal gauge.” There is no simple way to construct a canonical form that can be truncated optimally for more general tensor network states,…
Figure 32
Figure 32. Figure 32: TN heavy hex 7 Figure S5. The quasi-canonical form of the the two-dimensional tensor network state with 2 × 2 heavy hexagons. Each site has a vertex tensor Γ and each edge has a bond tensor λ. We use a belief propagation algorithm to ensure that the Γ and λ tensors ap…
Figure 1
Figure 1. Figure 1: Vidal gauge 1 Figure S6. In the Vidal gauge, the vertex and bond tensors at each site satisfy the constraint defined by the contraction diagram shown above. This constraint can be stated as follows: at any given site, if one takes the vertex tensor Γ and contracts it w…

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