{"id":"a5eedec1-be66-44b2-ba85-16db61a2d95f","arxiv_id":"2607.14223","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Hybrid feedback on a superconducting processor stabilizes regular orbits and maps coexisting regular and chaotic regions—a many-body analog of mixed phase space.","lead":"This experiment uses a 24-qubit superconducting processor with a quantum-classical feedback loop that automatically finds and stabilizes orderly, long-lived motion in a chaotic quantum system. The result is a map of coexisting regular and chaotic regions—a many-body version of classical mixed phase space, previously only a prediction.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim depends on the untested premise that the 3-parameter TDVP flow mirrors full quantum dynamics; the main experimental evidence uses a first-revival observable that the SM shows misclassifies regular orbits.","rationale":"The reader's weakest assumption is exactly the ansatz-faithfulness premise I identify as load-bearing. My reading of the manuscript surfaces two reinforcing pieces of evidence for that concern: (1) the explicit caveat in Methods acknowledging that the ansatz is a minimal description; (2) SM S8, which shows the main-text first-revival observable misclassifies a regular trajectory and requires a refined diagnostic to recover TDVP agreement. Together these indicate that the experimental evidence for the central claim is weaker than the main text alone suggests. However, the paper does include substantial independent support—iMPS entropy-slope separation in the thermodynamic limit, ED revival-map overlays, robustness to boundary conditions and diagonal perturbations—so the concern does not justify rejection. It does justify maintaining the CONDITIONAL verdict until the concrete test—a systematic exact/MPS fidelity map over the full phase space—is performed. I therefore leave the reader's verdict unchanged while agreeing with the identified weakest assumption.","tokens_in":26062,"tokens_out":5863,"duration_ms":70323,"concrete_test":"Perform high-bond-dimension MPS time evolution (χ ≥ 512) of the 24-qubit Hamiltonian (1) for a fine grid of initial states (θ, φ2) covering the experimental map in Fig. 2d, and compute the full Loschmidt fidelity F(t) = |⟨ψ(0)|ψ(t)⟩|² (or the subsystem fidelity) up to t ≈ 300 ns. Overlay regions where F(t) exhibits revivals/slow decay onto the TDVP Poincaré section from Fig. 1d. If the TDVP regular islands coincide with exact-dynamics revival regions, the central claim is supported; if the correspondence is absent or weak, the observed mixed phase space is a variational projection artifact rather than a property of the quantum many-body system.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline claim—'first experimental evidence for a mixed phase space in a quantum many-body system'—rests on identifying the TDVP flow, Eq. (5), with the low-entanglement sector of the 24-qubit Hamiltonian (1). This identification is the load-bearing premise: if the three-parameter manifold Eq. (2) is only a truncation, the mixed phase space is a property of the classical nonlinear equations (5), not of the quantum system. The authors themselves flag this in Methods: 'our variational ansatz is the minimal description that captures the essential dynamical features... With this caveat in mind.'\n\nA sharper, self-identified weakness appears in SM S8: the first-revival map I_max^z used in main-text Fig. 2c,d misclassifies regular trajectories such as θ=0, where the strongest revival occurs at the third peak, not the first. The SM then proposes a refined diagnostic (max_{t>t1} I_z(t)) that improves agreement with TDVP, but the main-text experimental evidence is presented with the unrefined observable. Thus the experimental maps that most directly support the central claim are generated by an observable the authors show is not a faithful regularity probe near the boundaries.\n\nThe iMPS thermodynamic-limit results (SM S9) do provide independent support, but only for two representative families, not a systematic phase-space map, and they rely on bond-dimension truncation (χ=256). They do not rule out that the TDVP islands are variational artifacts. Consequently, the key question—whether the mixed phase space is intrinsic to the quantum dynamics or an artifact of the projection—remains open.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports a hybrid quantum–classical feedback protocol implemented on a 24-qubit superconducting processor to discover and stabilize coherent, low-entanglement trajectories in an interacting SSH ladder. The central theoretical object is a three-parameter shallow-circuit variational manifold (Eq. 2), on which the authors derive TDVP equations of motion (Eq. 5) that exhibit mixed phase space: regular islands and a chaotic sea in Poincaré sections (Fig. 1d). Experimentally, they map a first-revival imbalance observable I_max^z over parameter space (Fig. 2c,d), measure a co-moving imbalance (Fig. 2e,f), and show that an evolution–projection feedback loop converges to stable periodic orbits whose geometry varies with coupling strength (Fig. 3). The paper concludes that this provides the first experimental evidence for a quantum many-body mixed phase space.","tokens_in":26396,"tokens_out":6902,"duration_ms":77938,"significance":"If the central identification between the TDVP flow and the low-entanglement sector of the 24-qubit Hamiltonian is accepted, the result is significant: it extends scar phenomenology from isolated initial states to a structured coexistence of regular and chaotic regions, and it introduces a practical feedback method for distilling coherent dynamics on noisy hardware. The manuscript is strong in several respects: the TDVP equations are derived in full in the Supplemental Material, the numerical maps are cross-checked with exact diagonalization and iTEBD, and the authors are transparent about the variational nature of the ansatz and about the limited diagnostic power of the first-revival observable. The main open question is whether the observed mixed phase space is a faithful property of the full quantum model or an artifact of the three-parameter truncation; this must be settled before the 'first experimental evidence' claim can stand.","major_comments":[{"comment":"The paper's headline claim that the experiment reveals a mixed phase space of the 24-qubit Hamiltonian (1) is not fully established, because Eq. (5) is a 3-parameter TDVP truncation. The authors acknowledge in Methods that the ansatz is 'the minimal description...' and SM S9 only compares two initial-condition families in the thermodynamic limit; the exact-diagonalization check in Fig. 2c is made through the same first-revival diagnostic. A quantitative comparison of exact dynamics and TDVP for representative regular and chaotic initial conditions—e.g., time-dependent fidelity F(t)=|⟨ψ(0)|ψ(t)⟩|^2 and leakage out of the manifold—is needed to rule out that the mixed phase space is a property of the ansatz rather than of the model. Without this, the phrase 'first experimental evidence for a mixed phase space in a quantum many-body system' overstates the result.","section":"Methods, Eq. (5)"},{"comment":"The primary experimental Poincaré map uses the first-revival peak I_max^z. SM S8 shows that this diagnostic misclassifies regular trajectories such as θ=0, ϕ1=ϕ2=0: the strongest revival occurs at the third peak, so the first-peak criterion labels it irregular. The refined observable max_{t>t1} I_z(t) restores agreement with TDVP. Because Fig. 2d is the central experimental evidence for the coexistence of regular and chaotic regions, presenting the unrefined map as the main result is misleading. Either the refined diagnostic should be used in the main text, or the authors should demonstrate that the first-revival map is faithful in the region used to support the claim.","section":"Fig. 2c,d; SM S8"},{"comment":"The hybrid feedback loop is a dissipative map P_M U(Δt), and its convergence to a periodic orbit is governed by the projection onto the variational manifold rather than solely by the original Hamiltonian. The converged orbit satisfying e^{-iHT}|ψ*⟩≈e^{iφ}|ψ*⟩ (SM S6) is a fixed point of the feedback map, not necessarily a periodic orbit of the original Hamiltonian. To support the claim that Fig. 3d probes the stability of the model's mixed phase space, the authors should show that the converged states exhibit low leakage over times comparable to the experiment without feedback, or otherwise clarify that the stabilized orbits are properties of the variational feedback dynamics rather than of H itself.","section":"Eq. (4), Fig. 3"}],"minor_comments":[{"comment":"The text states 'ℏ/J_o ≈ 200 ns' for J_o/2π = 5 MHz. With this value, ħ/J_o ≈ 31.8 ns, while 1/J_o = 200 ns. The numerical factor of 2π should be checked and the notation made consistent with the units used in Fig. 1.","section":"Methods (timescales)"},{"comment":"The refined diagnostic max_{t>t1} I_z(t) is only shown numerically. An experimental refined map, even for a subset of parameters, would substantially strengthen the claim that the improved agreement with TDVP is not a numerical artifact.","section":"SM S8"},{"comment":"The 'first experimental evidence' claim should be more carefully qualified relative to prior scar experiments on similar platforms [22,28] and the theoretical proposal of Ref. [23]. The distinction between isolated scar states and a full mixed phase space is clear, but it should be stated explicitly in the introduction and conclusion to avoid overclaiming novelty.","section":"Conclusions"},{"comment":"The schematic shows the feedback loop with a single 'U(Δt)' block, but the text explains that the reverse circuit U_M†(z') is applied before measurement. Adding the inverse unitary to the schematic would make the protocol easier to follow.","section":"Fig. 3a"}],"recommendation":"major_revision","confidential_remarks":"This is an interesting and potentially important paper, and the Supplemental Material already contains much of what is needed to address the main concerns. The required revision is not prohibitive: use the refined revival diagnostic in the main experimental maps, add a direct TDVP-versus-exact comparison for representative initial conditions, and soften or carefully qualify the 'first experimental evidence' claim. I would not reject the paper, but I would not accept it in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper does something real: it takes the ScarFinder idea and actually runs it on a superconducting processor, with enough experimental care to show the feedback loop converges to a stable orbit that deforms smoothly as J_e is varied. That is a genuine algorithmic result, and the TDVP derivation in the SM is complete and internally consistent. The imbalance-based Poincaré-like maps are striking, and the 16-qubit ED comparison supports the qualitative structure.\n\nThe central claim, though — \"first experimental evidence for a mixed phase space in a quantum many-body system\" — is softer than it looks. The mixed phase space lives in the three-parameter TDVP flow, not directly in the 24-qubit Schrödinger dynamics. The authors know this; in Methods they say the ansatz is the minimal description capturing essential features, \"with this caveat in mind.\" That caveat is load-bearing. The iTEBD thermodynamic-limit results in SM S9 give independent support, but only for two one-parameter families, not a systematic phase-space map, and they rely on bond-dimension truncation. So the gap between \"variational mixed phase space\" and \"intrinsic many-body mixed phase space\" remains open.\n\nThere is also a self-identified diagnostic problem. The main-text experimental maps use the first-revival Imax_z, and SM S8 shows this misclassifies regular trajectories like θ=0, where the strongest revival is the third. The refined max_{t>t1} diagnostic improves agreement, but the main-text evidence is built on the unrefined one. That is not fatal — the refined map exists and points the same way — but it does weaken the experimental support for the headline claim as presented.\n\nWhat is solid: the TDVP equations of motion, the feedback protocol itself, the convergence data, and the robustness of the phase-space structure to boundary conditions and a small diagonal perturbation. The paper is honestly written about its own limitations in the SM, which is a good sign.\n\nWho is this for? Researchers working on quantum many-body scars, TDVP methods, and hybrid quantum-classical algorithms. It will be a useful citation for the feedback-control technique even if the mixed-phase-space claim is softened in review.\n\nRecommendation: send it to peer review. The experimental protocol and the derivation deserve referee time. Referees should push on the interpretation — the burden should be to show that the mixed phase space is a property of the quantum model, not just the projection. I would likely accept after major revision.","headline":"A genuinely useful feedback-control experiment and a clean TDVP derivation, but the headline claim of \"first experimental evidence for quantum many-body mixed phase space\" overreaches: the mixed phase space lives in the variational projection, and the main-text diagnostic is shown in the SM to misclassify some regular orbits.","tokens_in":26927,"tokens_out":1882,"would_cite":true,"duration_ms":19659,"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":"This paper claims to experimentally reveal a many-body mixed phase space—regular and chaotic trajectories coexisting within the same interacting 24-qubit model—and attributes it to nonlinear variational dynamics rather than a classical limi","keywords":["quantum many-body scars","mixed phase space","time-dependent variational principle","quantum feedback control","eigenstate thermalization hypothesis","Poincaré section","superconducting qubits","SSH ladder"],"falsifier":"A decisive test: simulate the identical dynamics with a much larger variational manifold (for example, a matrix-product-state ansatz with bond dimension 100 or more) and re-measure the imbalance revival map and the hybrid-feedback convergence. If the regular islands disappear or shift substantially, or if feedback no longer converges to a stable periodic orbit, the claimed mixed phase space is a truncation artifact. Alternatively, use exact diagonalization of the 24-qubit chain to check whether the specific initial states in the regular island exhibit long-lived revivals under full quantum evo","tokens_in":25951,"feed_emoji":"⚛️","tokens_out":6683,"duration_ms":59213,"temperature":0.7,"pith_summary":"The paper tries to establish that a genuinely quantum many-body system can host a mixed phase space, meaning regular, coherent orbits embedded in a chaotic sea, in direct analogy with KAM theory in classical mechanics. It does so using a three-parameter shallow-circuit ansatz and projecting the exact Schrödinger dynamics onto that manifold via the time-dependent variational principle, yielding nonlinear effective equations for the variational parameters. On a 24-qubit superconducting processor realizing an interacting Su–Schrieffer–Heeger ladder, it measures an experimentally accessible imbalance whose revival peaks reproduce the Poincaré-section structure predicted by the projected dynamics. A hybrid quantum-classical feedback loop that alternates short evolution with projection onto the manifold converges from a chaotic initial state to a stable periodic orbit, and the orbit deforms smoothly as couplings are tuned. If correct, this is the first experimental evidence for a quantum many-body analog of mixed phase space, with consequences for quantum scars, thermalization, and coherent state preparation.","feed_headline":"Feedback control maps order and chaos in a 24-qubit system","feed_subtitle":"A feedback loop stabilizes regular orbits inside a chaotic sea — a many-body analog of KAM phase space.","key_machinery":"The argument rests on three linked objects. (1) A three-parameter shallow circuit ansatz, |ψ(z)> = U2(phi1,phi2)U1(theta)|0101...>, defines a three-dimensional variational manifold M inside the exponentially large Hilbert space. (2) The time-dependent variational principle projects the exact Schrödinger evolution onto the tangent space of M, producing the nonlinear equations of motion (Eq. 5) for z=(theta,phi1,phi2); these are the effective classical-like dynamics in which regular islands and a chaotic sea appear. (3) The imbalance I_z, a locally measurable proxy for fidelity that equals one exactly when the evolved state lies on M, and the feedback map |ψ(n+1)> = P_M U(Δt)|ψ(n)> together pr","core_discovery":"In the interacting SSH ladder of Eq. (1), the low-entanglement sector captured by the shallow circuit |ψ(z)> = U2(phi1,phi2)U1(theta)|0101...> has nonlinear TDVP dynamics (Eq. 5) that exhibit a genuine mixed phase space: invariant tori (regular islands) coexist with a chaotic sea, visible in a many-body analog of a Poincaré section. The revival amplitude of the imbalance I_z, measured on 24 qubits, reproduces the TDVP Poincaré-section features, and the co-moving imbalance confirms that regular trajectories remain confined to the variational manifold while chaotic trajectories leak out. The hybrid feedback iteration |ψ(n+1)> = P_M U(Δt)|ψ(n)>, implemented as an evolution-projection cycle, con","pith_inferences":["If the claimed structure is genuine, quantum many-body scars may be reinterpreted as the most stable periodic orbits of a variational phase space; the feedback protocol would then be an automated way to discover scar-like states without prior model knowledge.","Because the mixed phase space is established on a deliberately three-parameter projection, a natural test is to repeat the analysis with a larger variational manifold: islands that persist under increasing ansatz depth would indicate genuine many-body structure, while islands that dissolve would expose truncation artifacts.","The finite-time feedback step acts as an effective dissipation that selects a particular orbit; this suggests the protocol could be tuned to prepare states with prescribed revival properties, connecting to measurement-and-feedback-driven entanglement transitions.","The exact rainbow-scar point, where the TDVP flow reproduces known oscillatory trajectories, suggests that the mixed-phase-space picture may unify previously separate families of nonthermal states; one could test this by searching for additional island-centered trajectories at special coupling values."],"forward_implications":["Regular and chaotic trajectories can coexist in a strongly interacting many-body system at the same energy density, signaling a weak breakdown of the eigenstate thermalization hypothesis.","A local observable—the imbalance—can serve as a scalable proxy for global fidelity and map a many-body Poincaré section without reconstructing the wave function.","The hybrid feedback protocol can stabilize long-lived coherent dynamics starting from a chaotic initial state, using only short evolutions and local measurements, and can thereby prepare nonthermal states.","The regular islands deform smoothly as the Hamiltonian couplings are varied, matching the KAM picture of structural stability of mixed phase spaces.","The method extends to deeper variational circuits and other Hamiltonians, offering a general framework for discovering and controlling coherent sectors of chaotic quantum systems."],"fun_headline_variants":["Hybrid feedback maps order and chaos on 24 qubits","Feedback control reveals quantum mixed phase space in 24 qubits","Stable orbits in a chaotic sea: quantum feedback on 24 qubits","Hybrid control tames chaos, revealing quantum order on 24 qubits","Feedback finds hidden regular islands in a quantum chaotic sea"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the three-parameter shallow-circuit ansatz, together with the TDVP projection, faithfully represents the relevant low-entanglement dynamics of the full 24-qubit Hamiltonian; if that projected flow is only a truncation artifact, the reported mixed phase space is a property of the ansatz rather than of the quantum system.","fun_headline_variants_meta":{"raw":{"variants":["Hybrid feedback maps order and chaos on 24 qubits","Feedback control reveals quantum mixed phase space in 24 qubits","Stable orbits in a chaotic sea: quantum feedback on 24 qubits","Hybrid control tames chaos, revealing quantum order on 24 qubits","Feedback finds hidden regular islands in a quantum chaotic sea"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000667,"raw_usage":{"total_tokens":2852,"prompt_tokens":686,"completion_tokens":2166,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":2076}},"tokens_in":430,"tokens_out":2166,"duration_ms":15300,"temperature":1.0,"reasoning_tokens":2076,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T02:44:08.545932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test: simulate the identical dynamics with a much larger variational manifold (for example, a matrix-product-state ansatz with bond dimension 100 or more) and re-measure the imbalance revival map and the hybrid-feedback convergence. If the regular islands disappear or shift substantially, or if feedback no longer converges to a stable periodic orbit, the claimed mixed phase space is a truncation artifact. Alternatively, use exact diagonalization of the 24-qubit chain to check whether the specific initial states in the regular island exhibit long-lived revivals under full quantum evo","supporting_citations":[],"review_version":1}