{"id":"aabdfe19-2796-4f61-803e-3096c5b1adb8","arxiv_id":"2411.14359","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"ETH-violating mechanisms such as quantum many-body scars and Hilbert space fragmentation turn complete Hilbert space ergodicity into complete ergodicity within each dynamically decoupled subspace.","lead":"The authors show that quantum circuits with many-body scars or fragmented Hilbert spaces can still be ergodic, but only inside the restricted subspace the dynamics can reach. They introduce the concept of Hilbert subspace ergodicity and use it to design protocols for constructing random state ensembles, called t-designs, in subspaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"CHSSE is defined via convergence for all moments k, but the evidence covers only k=1,2 on N=4, leaving the t-design claim and the central definition unsupported for k>=3.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the dynamics restricted to a Krylov subspace is assumed to be generic enough to converge to the subspace Haar ensemble, but this is argued heuristically in Appendix A and tested only for k=1,2 on small systems. My independent reading confirms that this is the right point to stress. The paper's definition of CHSSE in Sec. III requires convergence for all k and all initial states in the subspace, and the abstract further promises a protocol for constructing t-designs in subspaces. A t-design requires equality of moments up to order t, so the absence of any k>=3 check directly undermines both the definitional claim and the advertised application. The heuristic in Appendix A concerns universality of the unconstrained gates and does not transfer automatically to the restricted dynamics inside a Krylov subspace; the projector and pair-flip constructions could in principle have hidden conserved quantities that leave k=1,2 moments unchanged while distorting higher moments. The DEE results are consistent with subspace ergodicity but are coarse-grained and cannot certify high-order Haar convergence. I do not think this warrants rejection: the paper is careful about what it computes, the low-moment numerics are clean, and the authors explicitly note that higher moments and longer times would be needed for conclusive evidence. The appropriate disposition remains conditional, matching the reader's verdict. The proposed check is concrete, computationally plausible for these system sizes, and would directly settle whether the missing high-moment behavior is a real breakdown or merely a deferred numerical exercise.","tokens_in":23755,"tokens_out":4898,"duration_ms":57726,"concrete_test":"Compute Delta_T^(3) and, if feasible, Delta_T^(4) (Hilbert-Schmidt distance to subspace Haar moments) for (i) the scarred model with N=4, d=2 in the nonscar subspace and (ii) the pair-flip model with N=4, d=3 in the largest 15-dimensional subspace, using all computational basis initial states in each subspace, times up to T=10^5 or until convergence, and averaging over 100 circuit instances. Compare the convergence to the same moments for the generic unconstrained brickwork model at the same N and T. If Delta^(3) and Delta^(4) decay to zero with a similar power law as the generic model, the concern is refuted; if they plateau above the finite-T noise floor while the generic model continues to decay, hidden constraints break CHSSE and the t-design claim must be restricted to k=2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the Fibonacci-driven brickwork circuits with scars or fragmentation exhibit complete Hilbert subspace ergodicity (CHSSE), defined in Sec. III as convergence of temporal ensemble moments to subspace Haar moments for all k and all initial states in K_alpha. The numerical support is limited to k=1,2, mostly N=4 (Secs. V-VIII), and the paper explicitly defers higher moments. Appendix A's universality argument shows only that the unconstrained {U_A,U_B} generate a dense subgroup of U(D); it does not establish that the restriction of the dynamics to a Krylov subspace K_alpha is generic enough to produce all Haar moments. This matters because the local gates in the scarred and pair-flip models have strong block structure (projector embedding, pair-flip constraints), and such constraints can admit additional conserved quantities or algebraic restrictions that leave low moments intact while spoiling k>=3. Since CHSSE is defined for all k and the paper uses it to claim construction of t-designs in subspaces, the unverified higher-moment behavior is the load-bearing gap. The discretized ensemble entropy is a useful coarse diagnostic, but with M~1000 reference bins it cannot resolve high-order moments and therefore does not fill this gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'complete Hilbert subspace ergodicity' (CHSSE), a generalization of complete Hilbert space ergodicity (CHSE) to systems whose Hilbert space decomposes into dynamically decoupled Krylov subspaces due to quantum many-body scars, Hilbert space fragmentation, or conventional symmetries. The authors define CHSSE as equality, at long times, of the temporal ensemble's k-th moments with the Haar moments restricted to each Krylov subspace, and they derive analytic formulas for the distances between full-space and subspace Haar moments. They present numerical evidence from Fibonacci-driven brickwork circuits: generic circuits show CHSE, while circuits with an embedded scar, with pair-flip gates (fragmentation), and with a U(1) symmetry show saturation to the predicted subspace bounds for the first and second moments. They also use a coarse-grained discretized ensemble entropy as an additional diagnostic, and they conclude that such systems form approximate t-designs in the corresponding subspaces.","tokens_in":24007,"tokens_out":6196,"duration_ms":60924,"significance":"If the central claim is fully established, the paper gives a coherent dynamical notion of ergodicity in symmetry- or constraint-restricted Hilbert spaces and identifies a concrete circuit protocol for constructing approximate unitary designs within decoherence-free or fragmented subspaces. The analytic formulas in Eqs. (20)-(21) and Appendix D are parameter-free and appear correct, and the numerical checks for k=1,2 are consistent with those formulas. The systematic test of all computational-basis initial states in the pair-flip model (Sec. VII) is a strength. However, the advertised conclusion that the systems are CHSSE in the sense of all moments rests on evidence limited to low moments and small system sizes, so the significance is conditional on closing that gap.","major_comments":[{"comment":"The definition of CHSSE applies to any initial state in K_alpha, but the numerical tests use only computational-basis product states (e.g., |0>^N, |1>^N, and all product states in Sec. VII). No superpositions within a Krylov subspace are tested. Since the whole point of the definition is that subspace Haar-uniformity should be independent of the initial state in the subspace, the evidence is incomplete even for k=1,2. This is not a criticism of the numerics actually performed, but it is a further reason why the claim should be stated conditionally until either more initial states or a theory argument is supplied.","section":"Sec. III and Secs. VI–VIII"}],"minor_comments":[{"comment":"The statement that the k=1 bound 'reduces to 2/D_alpha' in the large-N limit is inconsistent with Eq. (20), which gives (D-D_alpha)/(D_alpha D) ~ 1/D_alpha; the accompanying k=2 statement, 2/(D_alpha(D_alpha+1)), is correct.","section":"Sec. VII (paragraph after Fig. 8)"},{"comment":"There is a typo in the legend: 'blue]' should be 'blue'.","section":"Fig. 7 caption"},{"comment":"'birckwork' should be 'brickwork'.","section":"Appendix I"},{"comment":"The short example sequence in Eq. (15) could be easier to follow if the first few Fibonacci words were written out explicitly, since the recursive definition W_{j+1}=W_j W_{j-1} and the mapping to U^(A)/U^(B) are otherwise easy to misread.","section":"Sec. IV (Fibonacci word)"},{"comment":"The filtering criterion 'remove states if their overlap is large with any other state, i.e. if |<Phi_j|Phi_k>| > 1 - epsilon' is described but the paper does not state the value of epsilon used in the numerics; reporting it would improve reproducibility.","section":"Appendix C"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest and clearly written, and the analytic subspace-moment formulas are a useful contribution. The main issue is a mismatch between the all-k definition of CHSSE and the low-moment numerical evidence, compounded by the fact that Appendix A does not establish subspace-level Haar uniformity. The correct scope of the central claim should be narrowed, or the missing higher-moment evidence/proof should be added before publication. No concerns about novelty or citation practice."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the definition of CHSSE: complete Hilbert space ergodicity restricted to dynamically decoupled Krylov subspaces. That is a natural but non-obvious extension of the Pilatowsky-Cameo framework, and it is the first attempt I know to treat scars, fragmentation, and ordinary symmetries on the same footing in the CHSE setting. The analytic bounds in Eqs. (20)-(21), giving the distance between full-space and subspace Haar moments, are clean and correct as far as I can tell. The numerics are consistent with those bounds for k=1,2 on N=4, with some k=1 checks at N=5,6,8, and there are no fitted parameters anywhere. Credit is also due for candor: the authors explicitly say that conclusive numerical evidence would require higher moments and longer times, and they leave convergence rates as an open problem.\n\nThe soft spot is real and load-bearing, though not hidden. CHSSE is defined for all moments k and all initial states in the subspace, but the evidence stops at k=2, mostly on a 16-dimensional Hilbert space. The t-design claim in the abstract inherits that gap directly. Appendix A argues that the unconstrained Fibonacci drive generates a dense subgroup of U(D), but it does not establish that the restriction of the projected or pair-flip gates to a Krylov subspace is generic enough to produce all Haar moments. Those gates have strong algebraic block structure, so in principle extra conserved quantities could survive inside a subspace and only spoil moments with k>=3. The discretized ensemble entropy is a useful coarse check, but with about a thousand bins it cannot resolve high-order moments.\n\nI would send this to a serious referee. The definition is worth having, the bounds are useful, and the numerics are consistent as far as they go. But the referee should push for either k=3 calculations on N=4 (which should be feasible) or a softened abstract that presents CHSSE as a low-moment phenomenon and leaves the all-moments statement as an explicit conjecture. As written, \"a protocol for constructing t-designs in subspaces\" overstates what the evidence supports.","headline":"Defines a sensible generalization of CHSE to decoupled subspaces, with clean analytic bounds and consistent low-moment numerics; the all-moments claim outruns the evidence, so treat it as a well-posed conjecture rather than an established theorem.","tokens_in":24503,"tokens_out":1682,"would_cite":true,"duration_ms":18682,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"When scars or fragmentation split Hilbert space, generic aperiodic drives still become ergodic inside each decoupled subspace, a property the authors call Hilbert subspace ergodicity.","keywords":["Hilbert subspace ergodicity","complete Hilbert space ergodicity","quantum many-body scars","Hilbert space fragmentation","Haar-random states","t-designs","aperiodic quantum circuits","Fibonacci drive"],"falsifier":"Take the N=4 scarred brickwork circuit of Sec. VI or the N=5 qutrit pair-flip circuit of Sec. VII, run the Fibonacci drive to T ~ $10^{4}$, and compute the Hilbert-Schmidt distance between the $k=3$ (or $k=4$) temporal moment and the corresponding subspace-Haar moment. If that distance fails to decay toward zero and instead saturates at a positive value, the claimed convergence to subspace Haar randomness for all moments is wrong; if it decays, the heuristic universality argument gains support beyond the tested $k=1,2$ cases.","tokens_in":23554,"feed_emoji":"🎲","tokens_out":9500,"duration_ms":86478,"temperature":0.7,"pith_summary":"Complete Hilbert space ergodicity (CHSE) is the quantum analogue of a system visiting every corner of its state space: under a generic aperiodic drive, the long-time average of any observable matches the Haar average over the full Hilbert space. This paper asks what happens to CHSE when conserved quantities such as quantum many-body scars or Hilbert space fragmentation decouple the dynamics into separate blocks. Its central claim is that such systems, though never globally ergodic, can be ergodic within each dynamically decoupled Krylov subspace: the temporal ensemble generated from any initial state in a subspace converges to the Haar ensemble of that subspace, not of the full space. The authors call this Hilbert subspace ergodicity (CHSSE) and support it with numerical evidence for the first and second moments of the state ensemble in small brickwork circuits with embedded scars, pair-flip fragmentation, and U(1) symmetry sectors. If the claim holds, subspace-restricted randomness—and therefore t-designs in subspaces—can be produced by the same aperiodic driving that gives full CHSE in unconstrained systems.","feed_headline":"Ergodicity survives scars and fragmentation inside subspaces","feed_subtitle":"Aperiodic quantum circuits with many-body scars or fragmented Hilbert spaces still explore each decoupled subspace uniformly.","key_machinery":"The load-bearing objects are the Krylov subspaces $K_\\alpha$ (the dynamically decoupled blocks generated by evolving simple initial states), the subspace Haar measure and its moments, and the Fibonacci-word brickwork circuit built from two generic local unitaries. Projectors of the form $\\hat P = \\hat I - |00\\rangle\\langle 00|$ in the scarred construction force target scar states to remain invariant while leaving the complementary subspace generic; pair-flip gates, which let neighboring qudits change only when they already occupy the same state, produce exponentially many fragments for local dimension $d\\ge 3$. The commutant algebra of each model organizes the decomposition of the Hilbert space into blocks and explains why the moments cannot reach the full Haar ensemble, while the analytic distances between full-space and subspace Haar moments give the saturation levels seen in the numerics. Together these pieces replace one global Haar average by a Haar average per block, with the aperiodic drive supplying enough randomness to fill each block uniformly.","core_discovery":"The central claim is that complete Hilbert space ergodicity does not simply disappear when scars or fragmentation are introduced; it is inherited by each decoupled block. In a brickwork circuit driven by a Fibonacci word of two generic unitaries, the authors embed a projector that leaves designated scar states invariant, and in the pair-flip model with qutrits they realize strong fragmentation. In both settings the Hilbert space splits into Krylov subspaces, so the dynamics cannot converge to the full-space Haar ensemble. Yet the $k$-th moment of the temporal ensemble converges to the subspace-restricted Haar moment for $k=1,2$, while the distance to the full-Haar moment saturates at the analytic bounds set by the difference between the full and subspace Haar ensembles, given in Eqs. (20) and (21). The discretized ensemble entropy also saturates at values consistent with sampling only the available subspace. The authors introduce the term complete Hilbert subspace ergodicity for this behavior and argue, through a universality and Trotterization heuristic in Appendix A and the commutant-algebra structure in Appendix B, that it is the generic fate of aperiodic many-body circuits with nonlocal conserved quantities, so that long-time dynamics forms approximate t-designs within each subspace.","pith_inferences":["Editorial inference: if CHSSE holds for all moments, then any subspace selected by a commutant-algebra block—not just the scar and pair-flip examples—can serve as a platform for approximate Haar-random state generation, suggesting a general design-by-commutant prescription.","Editorial inference: the scarred and fragmented models saturate their full-space moment distances at different rates as system size grows, so the size dependence of these bounds offers a finite-size probe that distinguishes weak from strong ergodicity breaking without directly counting subspaces.","Editorial inference: a numerical check of third- and fourth-order moments or frame potentials on the small circuits used here would test whether the heuristic universality argument survives beyond low moments; if it does, CHSSE becomes a practical tool for randomized benchmarking and state-design construction within symmetry sectors."],"forward_implications":["In a scarred or fragmented aperiodic circuit, initial states in the same Krylov subspace generate long-time temporal ensembles indistinguishable from Haar-random states within that subspace, so each subspace becomes a source of approximate t-designs.","Conventional symmetries are not obstacles: every symmetry sector displays the same subspace ergodicity as a scar or fragment block, so CHSSE holds across scarred, fragmented, and symmetric models alike.","The saturation levels of the moment distances are computable from subspace dimensions, giving a quantitative diagnostic for detecting scars and fragmentation in aperiodic quantum circuits.","The discretized ensemble entropy acts as a witness: it tends to zero for full CHSE, saturates at an intermediate value fixed by the subspace fraction for CHSSE, and stays near its minimum for frozen scar or fragmented states.","The authors note that this subspace t-design protocol could be adapted to decoherence-free subspaces, pointing toward applications in fault-tolerant quantum information processing."],"supporting_citations":[{"why":"Defines complete Hilbert space ergodicity and the generalized Fibonacci drive, supplying the Haar-moment formalism and the lower bound used throughout.","marker":"[29]"},{"why":"Introduces the projector-embedding construction of ETH-violating scar models that the paper adapts to circuits.","marker":"[43]"},{"why":"Adapts projector embedding to quantum circuits, giving the circuit-level scar embedding used for the brickwork model.","marker":"[20]"},{"why":"Introduces the pair-flip model whose local gates and fragment structure are used in the fragmentation section.","marker":"[50]"},{"why":"Provides the commutant-algebra and Krylov-subspace framework that organizes the decoupled subspaces.","marker":"[10]"},{"why":"Supplies the exponential counting and robustness results for fragmentation subspaces used in the pair-flip analysis.","marker":"[12]"},{"why":"Underlies the Appendix A universality argument that generic local bricks generate uniform dynamics.","marker":"[52]"},{"why":"Links approximate Haar behaviour to approximate t-designs, the application the paper draws from CHSSE.","marker":"[49]"},{"why":"Defines the ensemble entropy used as an independent diagnostic of subspace ergodicity.","marker":"[33]"}],"fun_headline_variants":["Scars and fragmentation still yield ergodicity in subspaces","Hilbert subspace ergodicity: a new thermalization mechanism","Decoupled subspaces keep ergodicity alive despite scars","Aperiodic circuits with scars reach subspace t-designs","Complete Hilbert ergodicity splits into subspace-restricted phases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that inside each Krylov subspace the aperiodic drive is generic enough that the long-time temporal ensemble converges to the subspace Haar ensemble for every moment; the authors support this only heuristically in Appendix A and check only the first two moments on systems as small as N=4.","fun_headline_variants_meta":{"raw":{"variants":["Scars and fragmentation still yield ergodicity in subspaces","Hilbert subspace ergodicity: a new thermalization mechanism","Decoupled subspaces keep ergodicity alive despite scars","Aperiodic circuits with scars reach subspace t-designs","Complete Hilbert ergodicity splits into subspace-restricted phases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1535,"prompt_tokens":931,"completion_tokens":604,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":520}},"tokens_in":547,"tokens_out":604,"duration_ms":5698,"temperature":1.0,"reasoning_tokens":520,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:15:32.585821+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the N=4 scarred brickwork circuit of Sec. VI or the N=5 qutrit pair-flip circuit of Sec. VII, run the Fibonacci drive to T ~ $10^{4}$, and compute the Hilbert-Schmidt distance between the $k=3$ (or $k=4$) temporal moment and the corresponding subspace-Haar moment. If that distance fails to decay toward zero and instead saturates at a positive value, the claimed convergence to subspace Haar randomness for all moments is wrong; if it decays, the heuristic universality argument gains support beyond the tested $k=1,2$ cases.","supporting_citations":[{"cited_title":"Obser- vation of many-body scarring in a bose-hubbard quan- tum simulator,","cited_arxiv_id":null,"evidence_quote":"Introduces the projector-embedding construction of ETH-violating scar models that the paper adapts to circuits."},{"cited_title":"Quantum many-body scars in dual-unitary circuits,","cited_arxiv_id":null,"evidence_quote":"Adapts projector embedding to quantum circuits, giving the circuit-level scar embedding used for the brickwork model."},{"cited_title":"Quantum circuits for exact unitary t-designs and applications to higher-order randomized benchmarking,","cited_arxiv_id":null,"evidence_quote":"Introduces the pair-flip model whose local gates and fragment structure are used in the fragmentation section."},{"cited_title":"Embedding quantum many-body scars into decoherence-free sub- spaces,","cited_arxiv_id":null,"evidence_quote":"Underlies the Appendix A universality argument that generic local bricks generate uniform dynamics."}],"review_version":1}