{"id":"96512e6c-1021-40aa-b08f-e8915780e460","arxiv_id":"2510.24059","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Strong Ising interactions approximately conserve domain-wall number, splitting Fock space into sparse sectors and yielding a long-lived discrete time crystal in a clean 72-qubit Floquet system.","lead":"A 72-qubit experiment shows that strong interactions can split a driven quantum system's state space into sparse networks, slowing heating and keeping a time-crystalline oscillation alive for 120 drive cycles. The paper proposes 'Fock space prethermalization' as a disorder-free mechanism for preserving nonequilibrium quantum order.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SI §2C's assertion that FSP conserves DW number for exponentially long time is the load-bearing unsupported step; 120-cycle and L≤72 evidence does not establish the thermodynamic/prethermal claim.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: approximate DW conservation and its claimed exponential lifetime are the theoretical foundation for FSP and the thermodynamic extrapolation. The experimental observation of 120-cycle DTC order is real and well supported by the site-resolved correlators, the parameter-free butterfly-velocity prediction, MPS convergence checks, and noisy simulations. Those elements justify a conditional acceptance. However, the unproven 'exponentially long time' assertion in SI §2C is what elevates the observation from a finite-time transient to a distinct prethermal mechanism. Without either a rigorous derivation (e.g., a Schrieffer-Wolff or Lieb-Robinson-type bound on inter-sector leakage) or direct numerical evidence of exponential scaling in 1/λ, the central claim remains conditional. The proposed ED/MPS test directly addresses this gap and would settle whether the concern lands. No additional objections beyond the reader's were identified; I do not see an internal inconsistency or a reason to reject, only the need to verify or soften the exponential-timescale claim.","tokens_in":42983,"tokens_out":6083,"duration_ms":79580,"concrete_test":"Perform exact diagonalization for L=14,16,18,20 (with MPS extrapolation for larger L if needed) at λ1=2λ2=0.05,0.1,0.15,0.2, J=1, averaging over φ2 realizations. Extract the leakage timescale τ defined by the time when D(w0,t) decays below a fixed threshold, or from the saturation of participation entropy S[w,t] in the initial DW sector. Verify whether log τ grows linearly in 1/λ (or at least super-polynomially in J/λ) and whether τ grows with L as predicted by the light-cone picture. If log τ vs 1/λ is not linear, or τ does not scale with L as expected, the SI §2C exponential-lifetime claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central FSP mechanism rests on the claim that for λ1,λ2 ≪ J, inter-sector hybridization is suppressed by the Ising gap 2J|w−w'| so that D(w,t) remains exponentially localized on the initial DW sector (Eq. S8, Fig. S8). SI §2C then asserts that this DW number 'persists for exponentially long time' (first bullet), but no derivation is provided. The available evidence is finite-time and finite-size: experiments run to 120 cycles, ED eigenstructure only for L=10 (main) and L=18 (SI), and MPS simulations are extrapolated to L=72. The finite-size scaling in Fig. 4c/e uses fixed time windows (t∈[2T,20T] and [10T,30T]); if the light cone has not crossed any of the simulated systems, data collapse can occur for a merely algebraic transient and does not distinguish an exponentially long prethermal plateau from a finite-size/finite-time effect. Moreover, Fig. S9c shows DTC lifetime growing linearly with L, not exponentially, so the meaning of 'exponentially long' is ambiguous and unsupported. If the DW conservation is only algebraic in λ or linear in L, the claim that FSP is a robust ergodicity-breaking mechanism distinct from ordinary weak perturbation would be significantly weakened.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes 'Fock space prethermalization' (FSP) as a disorder-free mechanism for suppressing heating in a periodically driven Ising chain with interaction strength J comparable to the drive frequency. The central idea is that strong Ising interactions split the Fock-space graph into sectors labeled by approximate domain-wall (DW) number, with inter-sector hybridization suppressed by the Ising gap and intra-sector connectivity limited by local DW-conserving moves. The authors support this with exact-diagonalization eigenstructure, a perturbative butterfly-velocity formula, and an experiment on up to 72 superconducting qubits showing wave-packet localization in Fock space, subharmonic time-crystalline oscillations persisting for 120 drive cycles, site-resolved light-cone dynamics, and finite-size scaling across L = 24, 40, 56, 72 that identifies a crossover near λ1 ≈ 0.4. The manuscript argues that FSP is distinct from conventional high-frequency Floquet prethermalization, MBL, and quantum scars.","tokens_in":43213,"tokens_out":4668,"duration_ms":57790,"significance":"If the central claim holds, FSP is a potentially important addition to the known mechanisms of ergodicity breaking: it requires no disorder, operates at interaction strengths comparable to the drive frequency, and organizes the entire Fock-space network into sparse sectors rather than a few scarred states. The experimental work is substantial: 72-qubit programmable circuits, 120-cycle dynamics, careful context-aware gate calibration, noisy simulations that reproduce the data, and an analytic butterfly velocity that matches the measured light cone without fitting parameters. The paper also gives explicit credit to prior work on Fock-space dynamics and prethermal DTCs. The main risk is that several statements about 'exponentially long' conservation and thermodynamic-limit robustness go beyond what the finite-time, finite-size evidence establishes.","major_comments":[{"comment":"The manuscript asserts that the DW number in FSP 'persists for exponentially long time' and uses this to distinguish FSP from U(1) prethermal DTCs. No derivation is provided for this exponential timescale. The evidence shown is finite-time and finite-size: experimental data extend to 120 cycles, ED is limited to L ≤ 18, and Fig. S9c gives a DTC lifetime that grows linearly with L (t0 ≈ 2.9 + 4.3L), not exponentially. The exponential localization in Fig. S8a is a statement about a fixed L = 18 system, not about the thermodynamic-limit lifetime. This is load-bearing for the abstract's 'robust mechanism for breaking ergodicity' and for the claimed separation from ordinary prethermalization. Either provide a rigorous perturbative bound with explicit λ- and L-dependence, or soften the claims to 'long-lived' and avoid stating an exponential timescale.","section":"SI §2C, first bullet and Figs. S9, S10"},{"comment":"The finite-size scaling collapses are computed over fixed time windows: t ∈ [2T,20T] for the Fourier amplitude Fw and t ∈ [10T,30T] for ⟨Δx⟩/√L. Given the measured vB ≈ 0.074 sites/cycle and the linear fit t0 ≈ 2.9 + 4.3L in Fig. S9c, the light cone has not crossed even the L = 24 ring until t ≈ 100T, and for L = 72 it requires t ≈ 300T. Thus all plotted data lie in the transient local light-cone regime, where data collapse can occur even if the asymptotic dynamics is ergodic. To support the claim of a size-independent crossover and thermodynamic-limit FSP, the authors should show that the collapses remain stable when the time window is extended past the light-cone traversal time for at least the smaller systems, or provide a scaling argument that separates a genuine prethermal plateau from an algebraic transient.","section":"Main text, 'Scaling of FSP-thermal crossover', Fig. 4c,e"},{"comment":"The wording 'FSP is a robust mechanism for breaking ergodicity' and 'robustly suppressing heating' is stronger than the presented evidence. The 72-qubit experiment demonstrates a long-lived (120-cycle) non-thermal transient for specific initial states and perturbations, and the eigenstructure calculations demonstrate approximate DW sectors at small L. But without either a rigorous theorem or a scaling analysis that extends beyond the light-cone time, the manuscript has not established that the mechanism persists in the thermodynamic limit as true ergodicity breaking. I recommend rewording the central claim to describe a long-lived prethermal regime, or adding the missing theoretical support.","section":"Main text, Abstract and Conclusions"}],"minor_comments":[{"comment":"There are encoding artifacts in the author list: 'Y ang-Ren Liu' and 'Y u Gao' should be 'Yang-Ren Liu' and 'Yu Gao'. Please proofread the PDF generation.","section":"Author list and front matter"},{"comment":"The formula for vB(1) contains an explicit i inside an absolute value. Define the notation as the complex modulus and state that the result is real; otherwise the expression appears to be complex. Also state the units of vB (sites per Floquet cycle).","section":"Eq. (S18), SI §2B"},{"comment":"The captions state that 'across all figures in the text' error bars come from 10 random samples of φ2, but the main text also describes averaging over five globally shifted initial patterns. Please clarify in each caption which averaging is used for the plotted quantity.","section":"Fig. 2f and Fig. 4 captions"},{"comment":"The derivation of Δx ∝ L for the subspace-thermalized case is clear, but the subsequent bullet in the main text says that in the FSP regime Δx/√L 'exceeds 0.5 in late times' yet the idealized prediction is Δx ∝ L. Please explain the role of noise-induced depolarization more explicitly at the point where this comparison is made, since the reader is left to reconcile the two scalings.","section":"SI §2D, Eq. (S37)"}],"recommendation":"major_revision","confidential_remarks":"The experimental core is strong and likely correct; the main issue is that the theoretical framing overclaims an exponential lifetime and a thermodynamic-limit result on the basis of finite-time, finite-size data. This is fixable by either adding a rigorous derivation or carefully rephrasing the claims. I do not see a fundamental flaw that would require rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my honest read of arXiv:2510.24059.\n\nThe headline: this is a serious experimental paper. They demonstrate a DTC on 72 superconducting qubits that persists for 120 drive cycles without disorder, and they explain it with Fock space prethermalization (FSP): strong Ising interactions, with J comparable to the drive frequency, split Fock space into approximately conserved domain-wall sectors, slowing thermalization. The data are convincing on their own terms. The light-cone speed in the site-resolved correlators matches a parameter-free analytic expression, the finite-size scaling across L=24-72 collapses near lambda1≈0.4, and noisy simulations reproduce the crossover. That is a genuinely useful advance, both as a conceptual label and as an experimental benchmark.\n\nNow the soft spots. The most important is the claim in SI §2C that domain-wall number 'persists for exponentially long time.' There is no derivation of that. The eigenstructure for L=10 and L=18 shows w localization, and the participation entropy in Fig. S10d reaches a plateau on the timescale shown, but that doesn't establish exponential scaling. Worse, their own Fig. S9c shows the DTC lifetime scaling linearly with L (t0≈2.9+4.3L), which is exactly what you'd get from a finite light cone crossing the system. So the 'exponentially long' statement is doing a lot of work, and it is unsupported. The paper should either prove a rigorous prethermal bound for this class of Floquet systems, or soften the claim to 'long-lived on the experimentally accessible timescale.' As it stands, the DTC is a robust finite-size transient, not a proven thermodynamic phase. That distinction matters for the novelty claim of FSP as a distinct ergodicity-breaking mechanism.\n\nTwo smaller issues. The butterfly velocity in Eq. S18 is parameter-free and matches the data, but the derivation assumes the very DW conservation under test; that's a mild circularity, not fatal. And the data/code are not publicly available yet—only 'upon publication' and 'upon reasonable request.' For a paper making this kind of claim, the raw data and circuit-level simulation code should be part of the review package.\n\nOverall: this deserves a serious referee. The experiment is strong, and the FSP concept will be useful to the driven-many-body community even if the exponential-timescale claim needs to be walked back. I'd send it to peer review, with a clear request to address the lifetime scaling and release the artifacts.","headline":"A convincing 72-qubit DTC experiment with a useful new label, Fock space prethermalization, but the exponential-lifetime claim is asserted, not proven, and the data/code are not yet public.","tokens_in":43927,"tokens_out":4179,"would_cite":true,"duration_ms":44024,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A disorder-free mechanism, Fock space prethermalization, keeps a 72-qubit driven Ising chain from heating and sustains time-crystalline order for over 120 cycles.","keywords":["Fock space prethermalization","time crystal","discrete time-crystalline order","domain wall conservation","Floquet driving","ergodicity breaking","superconducting qubits","quantum thermalization"],"falsifier":"Measure the late-time leakage out of the initial domain-wall sector, i.e., D(w,t) for w≠w0, over several hundred drive cycles on a 72-qubit processor or in a large-scale numerical simulation; if the leakage grows with system size or reaches the density-of-states value dos(w) on timescales short compared with an exponential in J/λ1, the claimed robust ergodicity breaking would collapse.","tokens_in":42774,"feed_emoji":"⚛️","tokens_out":3457,"duration_ms":37400,"temperature":0.7,"pith_summary":"This paper tries to establish that periodically driven quantum many-body systems can be protected from heating by a mechanism that needs neither disorder nor very high driving frequency. The protection comes from strong Ising interactions, which make the number of domain walls—the boundaries between up and down spins—nearly conserved, splitting the huge space of spin configurations into many small, weakly connected islands. Because a wavefunction stays on its original island for a long time, the system fails to thermalize, and a simple periodic drive produces period-doubled (time-crystalline) oscillations that survive for at least 120 drive cycles on a 72-qubit superconducting processor. If correct, this gives a new, experimentally accessible route to stable non-equilibrium phases in interacting quantum devices.","feed_headline":"Domain-wall conservation gives 120-cycle time crystal on 72 qubits","feed_subtitle":"Strong Ising interactions, not disorder or high-frequency driving, split the quantum state space and slow heating.","key_machinery":"The approximately conserved domain-wall number W(s)=Σ_j [s_j(1−s_{j+1})+(1−s_j)s_{j+1}], enforced by the large Ising energy gap 2J|w−w'| between sectors of different W. Each Floquet eigenstate acquires a nearly good quantum number w at weak perturbation; inside a fixed-w sector, only flips of spins whose two neighbors are antiparallel are allowed, so the dense Fock-space network decomposes into linearly many sparse subnetworks, delaying ergodicity even at high energy density.","core_discovery":"For a kicked Ising model with Ising coupling J comparable to the drive frequency, and with generic single- and two-qubit perturbations λ1, λ2 much weaker than J, the Floquet eigenstates carry an almost well-defined domain-wall number w. The Ising energy gap 2J|w−w'| suppresses hybridization between sectors of different w, and within a sector only spin flips that locally conserve w are allowed. As a result, the Fock-space network of many-body states becomes sparse: the wavefunction D(w,t) stays exponentially localized around its initial domain-wall sector even when the initial state sits at high energy density. Measurements on rings of up to 72 qubits show the wave packet remaining far from t","pith_inferences":["The exponential localization D(w,t)∼e^{−|w−w0|} hints at an emergent sector structure that could be captured by an effective rotor or hard-constraint model; one testable extension is to check whether dynamics inside a single w-sector are integrable or thermal.","Because the mechanism is disorder-free, it could be ported to platforms where disorder is hard to engineer, such as photonic or ultracold-gas simulators; a concrete test would compare the measured butterfly velocity with the analytical expression in Eq. (S18) across platforms.","The claim that domain-wall conservation 'persists for exponentially long time' (SI §2C) is asserted without a rigorous derivation, and the experimental and numerical support covers only 120 cycles up to L=72; whether the thermodynamic-limit leakage time is truly exponential remains open.","One could test the stability of FSP against coupling to an environment, following recent ideas that strong Fock-space bottlenecks protect fragmented Hilbert spaces against thermal avalanches."],"forward_implications":["Discrete time-crystalline order can be stabilized without quenched disorder or high-frequency driving, using interaction strengths comparable to the drive frequency.","The thermalization rate becomes spatially controllable: initializing different spin patterns produces inhomogeneous light cones, enabling local management of heating in many-body devices.","Fock-space wave-packet dynamics are demonstrated as a sharp probe of Floquet eigenstructure, useful for diagnosing prethermal and ergodicity-broken regimes.","The mechanism is compatible with additional structures, opening routes to other non-equilibrium phases (e.g., time quasicrystals) and to protected quantum information processing.","Finite-size scaling in the crossover near λ1≈0.4 suggests the phenomenon is not a finite-size artifact and survives to large systems."],"fun_headline_variants":["Time crystal survives 120 cycles via Fock-space splitting","Disorder-free time crystal on 72 qubits lasts 120 cycles","Fock space prethermalization yields long-lived time crystal","Domain-wall conservation drives 120-cycle time crystal","72-qubit time crystal persists 120 cycles via FSP"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire scheme rests on the assumption that for weak perturbations the domain-wall number of each Floquet eigenstate is exponentially well conserved, so that the wavefunction remains localized on its initial sector for very long times; the paper evidences this only for 120 cycles and up to 72 qubits and supplies no rigorous bound on the leakage time in the thermodynamic limit.","fun_headline_variants_meta":{"raw":{"variants":["Time crystal survives 120 cycles via Fock-space splitting","Disorder-free time crystal on 72 qubits lasts 120 cycles","Fock space prethermalization yields long-lived time crystal","Domain-wall conservation drives 120-cycle time crystal","72-qubit time crystal persists 120 cycles via FSP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1442,"prompt_tokens":753,"completion_tokens":689,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":606}},"tokens_in":497,"tokens_out":689,"duration_ms":6705,"temperature":1.0,"reasoning_tokens":606,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:50:35.242107+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the late-time leakage out of the initial domain-wall sector, i.e., D(w,t) for w≠w0, over several hundred drive cycles on a 72-qubit processor or in a large-scale numerical simulation; if the leakage grows with system size or reaches the density-of-states value dos(w) on timescales short compared with an exponential in J/λ1, the claimed robust ergodicity breaking would collapse.","supporting_citations":[],"review_version":1}