{"id":"e9be0d9c-30ff-4ceb-b4c3-cb0bf53c0ea9","arxiv_id":"2505.00116","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"State-dependent, excitation-conserving quantum circuits with adaptively chosen gate arrangements maintain inhomogeneous non-thermalizing qubit dynamics that open-systems tools can distinguish from random thermalizing circuits.","lead":"This paper designs adaptive quantum circuits whose gate pairings are chosen to keep qubits away from thermal equilibrium, yielding long-lived non-thermal steady states in nominally closed systems. It offers a laboratory-friendly recipe for studying memory, correlations, and work extraction in quantum many-body circuits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Noise-reduced map in Sec. IV.B lacks uncertainty quantification; non-CP 'beyond finite-size noise' may be an ensemble-averaging artifact.","rationale":"The central claim of long-lived inhomogeneous dynamics is supported by multiple independent observables (heatmaps of ⟨σz⟩, trace distance, MI complexity, work statistics), many of which do not rely on the noise-reduced construction. However, the paper's framing as an 'open-systems tool' and its stated primary result—the role of non-Markovian subsystem dynamics—depends on the non-CP propagator analysis. The noise-reduced map is an ensemble-averaged object whose interpretation as typical dynamics is not validated. This is the most load-bearing concern because it affects the paper's distinctive methodological contribution, not merely peripheral measures; a concrete bootstrap or distributional test can settle it. The circularity of R2–R4 optimizing measured quantities is real but partial, since MI complexity and state-space volume are not optimized. The finite-size extrapolation is weak but mainly affects large-N claims. Therefore the reader's CONDITIONAL verdict is appropriate; no change in verdict is needed.","tokens_in":29698,"tokens_out":12678,"duration_ms":133871,"concrete_test":"Recompute the noise-reduced τz for each qubit and layer using bootstrap resampling (e.g., 10,000 resamples) over the 100 ensemble members, and flag a layer as non-CP only if the bootstrap 95% confidence interval for the mean τz lies entirely outside [−sin²(π/15), +sin²(π/15)]. Then compare the fraction of individual members with |τz|>sin²(π/15) at those flagged qubits and layers against the same fraction for R1 on the same central state and layer. If the R2–R4 fractions are not significantly higher than the R1 baseline, or if the confidence intervals include the CP boundary, the 'beyond finite-size noise' claim fails and the non-Markovianity conclusion should be downgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV.B constructs a 'noise-reduced' propagator map that takes the ensemble-averaged state ⟨ρ(ℓ−1)⟩ to ⟨ρ(ℓ)⟩, computing τz from ensemble-averaged partner Bloch vectors and correlations via Eq. (43). Because the adaptive rules make the circuit state-dependent, the ensemble mean does not evolve under any fixed CPTP map; the reconstructed map is a statistical object, not the dynamics of a typical member. The paper interprets non-(C)P noise-reduced maps as evidence that non-thermalizing rules have systematic non-Markovian dynamics 'beyond finite-size noise.' This inference requires that the ensemble-mean τz lies outside the CP interval by more than sampling noise and that this reflects typical, not outlier-driven, individual maps. Neither condition is tested: no error bars, confidence intervals, or comparison between the noise-reduced map and the distribution of individual τz at the same qubits and layers is provided. While a mean outside the interval implies some individual maps are non-CP, it does not establish that the effect is beyond finite-size noise—a few large outliers can shift the mean. The paper's primary open-systems result, the role of non-Markovian subsystem dynamics in non-thermalizing networks, rests on this unverified assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs U(1)-symmetric, excitation-conserving quantum circuits on networks of N qubits, with all gates taken from a fixed two-qubit unitary U* (Eq. 10) and the only freedom at each layer being the pairing of qubits into neighborhoods. Five update rules are compared: R1 (random), R2 (maximize the trace distance of the single-qubit states from the reference thermal state), R3 (maximize the total change in extractable work), R4 (greedy per-qubit maximization of extractable work), and R5 (a heuristic \"strategy mimic\" rule). Starting from ensembles of initial states around four central states (CSP, CS1, CS2, CS3), the authors track the single-qubit phase-covariant dynamics, which reduce to a single parameter tau_z per qubit per layer. They characterize the late-time behavior through heatmaps of <sigma_z>, distributions of tau_z, two-qubit correlations, PCA convex-hull volumes, mutual-information graph complexity, persistence of positive extractable-work changes, and the fraction of non-completely-positive propagator maps. The central claim is that the constrained rules R2-R4, and often R5, produce inhomogeneous non-thermalizing steady states that are clearly distinguishable from the approximately thermalizing random rule R1, and that the non-thermalizing states are associated with non-Markovian single-qubit dynamics signaled by non-completely-positive noise-reduced propagator maps.","tokens_in":29958,"tokens_out":5441,"duration_ms":62102,"significance":"If the results hold, the paper provides a clean, exactly solvable-in-structure family of circuit models in which inhomogeneous subsystem dynamics persist over long times, with the open-systems description reduced to a single parameter tau_z per qubit. The analytical reduction leading to Eqs. (43)-(45) is elegant and correct, and the trace-distance monotonicity bound in Eq. (32) is a solid constraint on what any such rule can achieve. The paper is also commendable for cross-checking the central claim with multiple independent diagnostics (PCA state-space volume, mutual-information graph measures, extractable-work statistics, and non-CP fractions) rather than relying on a single order parameter. However, two load-bearing points need attention: the noise-reduced propagator maps in Section IV.B lack uncertainty quantification and are used to draw the non-Markovianity conclusion, and R2's trace-distance evidence is partly circular because the same quantity is optimized by the update rule. Neither issue appears fatal to the overall picture, but both must be addressed before the non-Markovianity claim can be accepted as stated.","major_comments":[{"comment":"The noise-reduced propagator map is defined by taking element-by-element ensemble averages of the partner Bloch vector z_b and the correlation C^xx_ab and then inserting these averages into Eq. (43) for tau_z. Because the update rules R2-R5 make the circuit state-dependent, the ensemble mean does not evolve under any fixed CPTP map; the reconstructed object is a statistical summary, not the dynamics of a typical member. The conclusion that non-CP dynamics occur 'beyond finite-size noise' requires two things that are not shown: that the ensemble-averaged tau_z lies outside the CP interval by more than sampling uncertainty, and that this reflects typical individual maps rather than a few large outliers. While a mean outside the CP region implies that some individual maps are non-CP, it does not establish that the effect is beyond noise, and the absence of error bars, confidence intervals, or a comparison between the noise-reduced map and the distribution of individual tau_z at the same qubits and layers leaves the primary open-systems result under-supported.","section":"Section IV.B, Eq. (43), Figs. 20-21"},{"comment":"R2 is defined as the rule that, at each layer, maximizes the trace distance of the single-qubit states from the reference thermal state (Eq. (23)). The top panel of Figure 17 then reports the late-time average trace distance as evidence that R2 keeps the network far from the thermal state. This particular diagnostic is therefore not an independent test of non-thermalization for R2; the comparison to R1 is partly by construction. A similar, though weaker, concern applies to R3 and R4, whose objective functions are built from Delta W_ex and whose performance is then evaluated in part by extractable-work measures. The central claim does not collapse, because the paper also reports independent measures such as sigma(tau_z), mutual-information disparity, and non-CP fractions, but the authors should either explicitly label the trace-distance and extractable-work results as consistency checks of the objective functions or provide a control rule that optimizes a quantity not used as a diagnostic.","section":"Section II.D, Eq. (23); Section IV.A, Fig. 17"},{"comment":"The title and abstract describe the systems as \"closed quantum systems,\" but Section II.D states that the dynamics are \"not self-contained, or closed\" because an external controller with exact knowledge of the initial state and a full record of the circuit must compute and apply each layer's extremizing gate arrangement. This is an internal inconsistency in the central framing. The physical results are unaffected, but the scope claim should be qualified: these are adaptively programmed unitary circuits, not autonomous closed Hamiltonian dynamics, and the difference matters for how the \"non-equilibrium steady state\" and the absence of measurements are interpreted.","section":"Title and Abstract vs. Section II.D"}],"minor_comments":[{"comment":"The captions for Figures 3-6 appear as placeholders (\"* CS1\", \"* CS2\", \"* CS3\", \"* CSP\") and the main text refers to them only collectively; the figures need complete captions and in-text references that match the numbering used elsewhere (e.g., Figure 7 for the heatmaps).","section":"Figures 3-6"},{"comment":"The notation for central states in Appendix B is inconsistent with Section II.B: the text lists \"CSP 1, CSP 2, CSP 3, CSP\" where the body uses CSP, CS1, CS2, CS3. The notation should be unified.","section":"Appendix B"},{"comment":"The Bernoulli-process circuits are called \"Markovian circuits,\" but the word \"Markovian\" is used elsewhere in the paper for the quantum open-system divisibility property. Clarify that Eq. (38) defines a classically Markovian sequence of gates, not a quantum-Markovian evolution, to avoid confusion with Section IV.B.","section":"Section III.A, Eq. (38)"},{"comment":"The exponential fit to the variance of <sigma_z> versus N and the inferred large-N limit are load-bearing for the claim that R2 retains non-CP dynamics at large system size, but no fit parameters, confidence intervals, or residuals are reported. Please provide these details or soften the large-N extrapolation.","section":"Appendix A.3, Figs. 31-32"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about its limitations and the derivations are mostly clean, but the non-Markovianity claim in Section IV.B needs substantially stronger statistical support before it can be published as stated. The circularity of R2's trace-distance diagnostic is real but fixable, and the \"closed\" terminology should be reconciled with the external controller. I do not see a fatal error; the central observation of persistent inhomogeneous subsystem dynamics appears to be supported by a range of independent measures."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper actually does something new: it defines adaptive circuit rules (R2-R5) that extremize scalars like trace distance to a thermal reference, extractable work, or a mimic heuristic, and it shows numerically that on small U(1)-symmetric qubit networks these rules create long-lived inhomogeneous steady states that random evolution does not. The phase-covariant map derivation is clean, Eq. (43) is a nice reduction of the dynamics to partner population plus two-qubit correlation, and the multi-measure characterization (trace distance, relative entropy, MI graph complexity, extractable-work persistence) is thorough. I believe the central observation: the non-random rules do stay away from the homogeneous thermal state on multiple independent measures.\n\nThe soft spots are real but not fatal. The biggest is Sec. IV.B's 'noise-reduced' propagator map. The map is built from ensemble-averaged partner Bloch vectors and correlations, and the paper interprets its non-CP violation as evidence of non-Markovian dynamics 'beyond finite-size noise.' But the mean of a distribution can sit outside the CP interval even if the typical member is inside, and the paper gives no error bars, no confidence intervals, and no comparison between the averaged map and the distribution of individual maps at the same qubits and layers. That inference is load-bearing for the strong version of the non-Markovianity claim. The scaling fits in Appendix A.3 help, but they are exponential fits to five system sizes and do not validate the averaging procedure itself. This needs bootstrap or per-trajectory analysis.\n\nSecond, there is partial circularity: R2 maximizes trace distance to the thermal state and then trace distance is reported as evidence of non-thermalization. The same goes for R3/R4 with extractable work. This is real but mitigated because the paper also uses non-optimized measures (MI graphs, tau_z distributions, relative entropy) and the differences persist there. Still, the authors should separate optimized from non-optimized diagnostics. Third, no code or data is released, which makes the numerical claims hard to check; for a computational paper of this type, that should be a condition of acceptance.\n\nWho is this for? People working on quantum thermalization, non-equilibrium steady states, and circuit-based many-body dynamics. It is a useful source of new examples and tools, and the flaws are addressable. I would send it to peer review, with the requests above. It deserves a serious referee, not a desk reject.","headline":"Inventive adaptive circuits that demonstrably avoid thermalization, but the non-Markovianity claim leans on an under-validated ensemble-averaged map.","tokens_in":30503,"tokens_out":5091,"would_cite":true,"duration_ms":50971,"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":"Adaptive, excitation-conserving circuits can keep small qubit networks far from thermal equilibrium.","keywords":["non-equilibrium steady states","closed quantum systems","quantum circuits","phase-covariant maps","non-Markovian dynamics","thermalization","qubit networks","extractable work"],"falsifier":"Examine the distribution of individual (not averaged) two-qubit density matrices in the CS1 ensemble near layer 300: if the fraction of individual circuits whose single-qubit propagator violates the positivity condition is negligible while the ensemble-averaged map violates it, the non-Markovianity claim is an artifact of averaging.","tokens_in":29386,"feed_emoji":"⚛️","tokens_out":5686,"duration_ms":57110,"temperature":0.7,"pith_summary":"This paper tries to show that a closed, unitary quantum circuit can be engineered to resist thermalization and hold a steady non-equilibrium state for long times. The recipe is to conserve excitation number, use a single fixed two-qubit gate, and choose the pairing of qubits at each circuit layer by extremizing a scalar measure of the state. On 12-qubit networks with connectivity 2, these adaptive rules leave single-qubit states visibly inhomogeneous at late times while a random circuit of the same size relaxes to a homogeneous state. The paper argues the two classes are not just different in appearance: they separate on trace distance, relative entropy, mutual-information graph complexity, extractable-work persistence, and the prevalence of non-completely-positive single-qubit propagator maps.","feed_headline":"Adaptive circuits keep small qubit networks out of equilibrium","feed_subtitle":"State-dependent wiring of a fixed gate stops 12-qubit circuits from thermalizing; random circuits do not.","key_machinery":"The load-bearing object is the phase-covariant single-qubit propagator map $\\Lambda_q(\\ell,\\ell-1)$, which because of excitation conservation has the single moving parameter $\\tau_{z,q}(\\ell) = z_{b,\\ell-1}\\sin^2\\theta + C^{xx}_{ab,\\ell-1}\\sin 2\\theta$, with $\\theta = \\pi/15$ fixed by the gate. All variation in single-qubit dynamics therefore lives in one number per qubit per layer, set by the partner's Bloch component and the two-qubit correlation. The update rules R2–R5 choose the next layer's interaction graph by extremizing a scalar function of the state—trace distance from the thermal reference (R2), summed change in extractable work (R3), greedy per-qubit work maximization (R4), or a strategy-mimic pairing rule (R5)—and the paper compares these against random R1 and against Bernoulli circuits with the same emergent edge weights.","core_discovery":"The central discovery is that for qubit networks with conserved excitation number and a single interaction gate, the sequence of two-qubit pairings—not the gate itself—controls whether the closed system thermalizes. Random pairings (R1) drive all single-qubit dynamics toward a common phase-covariant channel whose fixed point is the maximum-entropy state at fixed energy. Adaptive pairings (R2–R4) generate long-lived non-equilibrium steady states in which the single-qubit map parameter $\\tau_z$ has a broad, non-Gaussian distribution, correlations are concentrated in a few qubit pairs, and the noise-reduced propagator maps between layers frequently fail to be completely positive. The failure of complete positivity is the paper's marker that the rest of the network acts as a non-Markovian environment rather than a memoryless bath.","pith_inferences":["A natural extension the paper leaves open: any conserved charge, not just excitation number, may support the same construction, so the mechanism could generalize to fermionic or spin-1 networks by replacing the phase-covariant map with the appropriate covariant class.","One testable prediction is that the non-completely-positive maps should be observable by single-qubit process tomography on the actual circuit: if the positivity violations vanish when tomography is done on individual runs rather than on ensemble-averaged states, the non-Markovianity is an averaging artifact.","The extractable-work interval statistic suggests a practical probe for near-term hardware: circuits run on few qubits could be classified as thermalizing or not by measuring how often a qubit's extractable work increases on consecutive layers.","The strategy-mimic rule's sensitivity to the central state hints that imperfect local information can either preserve or destroy non-thermalizing behavior; mapping when it fails could clarify the role of information access in engineered non-equilibrium steady states."],"forward_implications":["If the claim holds, non-equilibrium steady states can be prepared in closed systems without any engineered dissipation, using only a conserved charge, a fixed gate, and a classical controller that rewires the circuit.","The single-qubit distribution of $\\tau_z$, especially its variance and kurtosis, becomes a cheap diagnostic that separates thermalizing from non-thermalizing circuit dynamics.","Noise-reduced non-completely-positive propagator maps serve as a signature that subsystem memory is retained, tying non-thermalization to quantum non-Markovianity.","The Bernoulli-circuit comparison shows that biased random wiring with the same emergent frequencies does not reproduce the steady state; the state-dependent sequence of layers is essential.","The results give a concrete way to rank non-thermalizing dynamics by thermodynamic utility: extractable-work persistence intervals cleanly separate R1 (exponential decay) from R2–R5 (long intervals)."],"supporting_citations":[{"why":"Establishes the viewpoint of an N-qubit network as an ensemble of N open systems, which the paper's whole analysis uses.","marker":"[26]"},{"why":"Defines the phase-covariant dynamical-map class in which all single-qubit dynamics in this paper live.","marker":"[29–31]"},{"why":"Provides the definition of non-Markovianity through non-completely-positive propagator maps, the key diagnostic.","marker":"[33]"},{"why":"Supplies the mutual-information network and its clustering/disparity complexity measures used to characterize steady states.","marker":"[13]"},{"why":"Earlier work on extractable work in random circuits, which serves as the baseline for the utility measures.","marker":"[45]"},{"why":"Collisional models motivate treating circuit layers as discrete divisibility points for non-Markovianity.","marker":"[22]"}],"fun_headline_variants":["Wiring order, not gate, freezes qubit thermalization","Adaptive pairings halt quantum thermalization","Non-Markovian qubit circuits defy equilibrium","Quantum circuit wiring beats randomness for memory","Ordered qubit pairs lock in non-thermal states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The non-Markovianity result depends on treating the noise-reduced propagator, built from the ensemble-averaged two-qubit state, as representative of what a typical single qubit experiences; if the averaging itself creates the positivity violations, the conclusion fails.","fun_headline_variants_meta":{"raw":{"variants":["Wiring order, not gate, freezes qubit thermalization","Adaptive pairings halt quantum thermalization","Non-Markovian qubit circuits defy equilibrium","Quantum circuit wiring beats randomness for memory","Ordered qubit pairs lock in non-thermal states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":2970,"prompt_tokens":853,"completion_tokens":2117,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":2042}},"tokens_in":469,"tokens_out":2117,"duration_ms":15520,"temperature":1.0,"reasoning_tokens":2042,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:52:30.896726+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Examine the distribution of individual (not averaged) two-qubit density matrices in the CS1 ensemble near layer 300: if the fraction of individual circuits whose single-qubit propagator violates the positivity condition is negligible while the ensemble-averaged map violates it, the non-Markovianity claim is an artifact of averaging.","supporting_citations":[{"cited_title":"Shirai, T","cited_arxiv_id":null,"evidence_quote":"Provides the definition of non-Markovianity through non-completely-positive propagator maps, the key diagnostic."},{"cited_title":"Entangled quantum cellular automata, physical complexity, and Goldilocks rules","cited_arxiv_id":"2005.01763","evidence_quote":"Supplies the mutual-information network and its clustering/disparity complexity measures used to characterize steady states."}],"review_version":1}