{"id":"ae8dd9bb-e5c0-4aa9-8d94-526f28b646bc","arxiv_id":"2607.10227","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"Engineered dual-rail code-space recovery extends SQD to symmetry-free Hamiltonians and yields lower Ritz energies than unencoded sample support on Ising models up to 36 sites.","lead":"Dual-rail encoding turns noisy quantum samples into recoverable bitstrings for sample-based diagonalization, even when the Hamiltonian has no particle-number or similar symmetry. The method beats unencoded sample-support baselines on Ising models up to 36 spins despite using more qubits and deeper circuits.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the key premise—that pair-violation signals stay informative enough for stochastic repair to offset the encoding overhead—and correctly rates the work CONDITIONAL on limited scope (Ising/SqDRIFT/one Heron device, fixed hyperparameters, residual 36-site errors). After checking the matched circuits, the RR property of the full-support baseline, the recovery map (Eqs. 29–33), and the consistent energy ordering, that premise holds for the data shown and does not collapse the claim. No stronger load-bearing flaw (baseline confound, variational inconsistency, or algebraic error) appears. Therefore the verdict and confidence stay as the reader left them.","tokens_in":18319,"tokens_out":484,"duration_ms":55075,"concrete_test":"Re-run the self-consistent recovery (N_C=2 BMM, ρ=1/2, δ=0.01, N_B=10, ϵ_carry=10^{-3}) on the existing 25-site 1D TFIM encoded sample pool at D_tar=2.5e6 and confirm the final Ritz energy remains strictly below the unencoded full-support value −25.714085; a reversal would indicate an implementation or stochastic-selection artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an empirical comparison: dual-rail code-space recovery produces lower Ritz energies than the full unencoded sample-support baseline at equal or smaller D_proj (Table 1, Figs. 2–4). The design matches logical SqDRIFT sequences, and Rayleigh–Ritz makes the unencoded full-support baseline the strongest possible one-shot competitor from those samples (any subset can only raise the Ritz value). Fully valid encoded strings are rare (0.02–4.8 %), so recovery is necessary; yet the repaired logical bases still win despite higher depth/N2q (Table 2). No internal inconsistency or hidden assumption undermines the reported results on these benchmarks. The residual 36-site gaps and single-device scope are ordinary limitations already reflected in the CONDITIONAL verdict, not a flaw that falsifies the claim as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript proposes code-space recovery for sample-based quantum diagonalization (SQD): dual-rail encoding maps each logical qubit to a physical pair (|0⟩→|01⟩, |1⟩→|10⟩) so that Hamming-weight-one violations in noisy samples become detectable recovery signals. Encoded counterparts of logical sampling operations are implemented so that the pair constraint is preserved in the noiseless limit (Eqs. 1–4, 20–23). After sampling, invalid pairs are repaired by a self-consistent, cluster-adaptive stochastic procedure guided by reference vectors, decoded logical bitstrings span a projected subspace, and the original logical Hamiltonian is diagonalized. Benchmarks on 1D/2D TFIM and MFIM instances (n=25 and 36) use matched SqDRIFT logical sequences on IBM Heron r2. Across all five problems, recovered subspaces at D_tar ≤ 2.5×10^6 yield lower Ritz energies and smaller |E−E_ref| than the full unencoded sample-support baseline (Table 1, Figs. 2–4), despite roughly doubled qubits and higher transpiled depth/N2q (Table 2).","tokens_in":18554,"tokens_out":1016,"duration_ms":9325,"significance":"If the empirical advantage holds more broadly, the work supplies a concrete route to apply SQD-style recovery to eigenvalue problems that lack a native, measurement-visible constraint such as particle number. The matched logical SqDRIFT design, identical shot budgets, and Rayleigh–Ritz projection of the original logical Hamiltonian make the comparison fair and falsifiable; energy-variance intercepts for the 25-site cases agree closely with exact/DMRG references, strengthening the claim that the recovered subspaces track the target low-energy branch. The explicit resource trade-off (Table 2) and the demonstration that recovery can still win when fully valid encoded strings are rare (0.02–4.8 %) are useful contributions to the quantum-centric diagonalization literature.","major_comments":[{"comment":"The central claim rests on a single sampling protocol (SqDRIFT with fixed K, Δt, M_seq) and one IBM Heron device. Section 2.2 and Methods 4.1 do not test whether the subspace-quality advantage survives under alternative samplers (e.g., different Krylov constructions or non-randomized circuits) or under substantially different noise. A short additional experiment or a clear statement that the advantage is protocol- and device-specific would make the scope of the claim precise.","section":null},{"comment":"For the 36-site stress tests (Fig. 4), residual gaps to DMRG remain large (≈0.22 and ≈0.17 at D_tar=2.5×10^6) and the manuscript reports that a controlled zero-variance extrapolation was not possible. The Discussion attributes this to sampling budget and 72-qubit noise exposure, but does not quantify how much of the residual is due to incomplete support versus residual recovery error. Clarifying this distinction (e.g., by comparing recovered support overlap with a classical high-quality basis when available) would strengthen the interpretation of the stress-test results.","section":null}],"minor_comments":[{"comment":"Table 2 reports median (min–max) active qubits, depth, and N2q; a brief note on how routing/layout inflation of active qubits affects the encoded vs unencoded comparison would help readers interpret the overhead.","section":null},{"comment":"The modified-ReLU parameters (ρ,δ)=(1/2,0.01) and N_C=2 are fixed without a sensitivity check (Methods 4.2). A short remark or supplementary scan would reassure that the advantage is not finely tuned to these choices.","section":null},{"comment":"Fig. 1 is dense; labeling the six numbered stages more explicitly in the caption would improve readability for readers new to SQD recovery loops.","section":null},{"comment":"The Discussion correctly notes that engineered and native constraints can be complementary, but does not cite or discuss any prior dual-rail or pair-code use outside photonic QC; a sentence situating the recovery-only use of dual-rail would help.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid empirical methods contribution with a clean matched design. The residual 36-site gaps and single-device/protocol scope are ordinary limitations already reflected in the authors’ own discussion; they do not undermine the reported claim as stated. Fit for a quantum-information or quantum-computing methods venue is good; I would not require multi-device or multi-sampler experiments for acceptance if the scope is stated clearly."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this is a real empirical methods result, not a rebrand. They take dual-rail encoding—old hat as a code—and use it only as a sample-level recovery interface for SQD when the Hamiltonian has no particle-number-style U(1) to lean on. On TFIM/MFIM up to 36 sites, recovered logical subspaces give lower Ritz energies than the full unencoded sample support, often at smaller D_proj, even though the encoded circuits are deeper and use ~2× qubits (Tables 1–2, Figs. 2–4).\n\nWhat is actually new is the packaging: encode the sampling circuit so pair violations are visible, repair with a self-consistent cluster-adaptive loop, decode, then Rayleigh–Ritz the original logical H. The comparison design is careful. Same logical SqDRIFT sequences, same shot budget, same GHZ support augmentation, and the unencoded baseline is the full observed support—so any subset would only raise the Ritz value. Fully valid encoded strings are rare (0.02–4.8%), so recovery is doing real work, not postselecting a free lunch. Energy-variance intercepts on the 25-site cases sit close to exact/DMRG, which is a useful consistency check.\n\nSoft spots are ordinary, not load-bearing. Residual gaps remain on the 36-site runs; they do not claim a clean zero-variance extrapolation there. One device, one sampling family, fixed recovery hyperparameters, and code/data only on request. The paper does not show that engineered recovery beats native recovery when both exist, and it does not claim a general theory of when pair noise stays informative. Those are scope limits, not internal contradictions. The central claim as stated holds on the reported benchmarks.\n\nThis is for people already in the SQD / quantum-centric diagonalization lane who care about problems without native bitstring constraints. Math and citations look solid; dual-rail and SqDRIFT are properly sourced. I would send it to peer review. Worth a reading-group slot if your group is in hybrid sampling methods; otherwise a careful skim of Results + Methods is enough.","headline":"Clear methods result: dual-rail recovery beats a strong unencoded sample-support baseline on symmetry-free Ising models, with fair matched hardware design and ordinary near-term limits.","tokens_in":19176,"tokens_out":549,"would_cite":true,"duration_ms":9041,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Dual-rail code-space recovery lets sample-based quantum diagonalization repair noisy samples without native symmetries, beating raw unencoded supports on Ising models up to 36 spins.","keywords":["sample-based quantum diagonalization","dual-rail encoding","code-space recovery","Ising model","quantum Krylov","projected subspace","self-consistent recovery","SqDRIFT"],"falsifier":"On the same device, SqDRIFT protocol, and shot budget, if the encoded workflow’s lowest Ritz energy at target basis size 2.5 million were higher than the unencoded full-support Ritz energy on the reported 25- or 36-site Ising benchmarks, the central claim would be false.","tokens_in":19187,"feed_emoji":"⚛️","tokens_out":923,"duration_ms":19874,"temperature":0.7,"pith_summary":"Sample-based quantum diagonalization builds a compact classical subspace from quantum measurement samples and diagonalizes the target operator there. Prior recovery of noisy samples leaned on native constraints such as particle-number symmetry, which many eigenvalue problems lack. This paper engineers a recovery constraint instead: each logical qubit is dual-rail encoded so that only weight-one physical pairs are valid, making code-space violations visible in the bitstrings. Invalid pairs are stochastically repaired with self-consistent reference vectors, decoded, and used for projected diagonalization. On transverse- and mixed-field Ising models with up to 36 sites—models with no particle-number-like U(1) symmetry—the recovered subspaces give lower projected Ritz energies than the full unencoded sample support, even at smaller basis sizes and despite roughly doubled qubit count and deeper circuits. The result is that recoverable structure can be designed into sampling rather than required of the target problem.","feed_headline":"Code-space recovery beats raw samples on 36-spin Ising models","feed_subtitle":"Dual-rail encoding builds a repairable constraint so sample-based diagonalization works without native symmetries.","key_machinery":"Dual-rail code-space recovery: map each logical qubit as |0⟩→|01⟩ and |1⟩→|10⟩, implement sampling operations that intertwine with this encoding, then repair invalid pairs (00 or 11) stochastically using cluster-specific, self-consistently updated rail-occupation reference vectors before decoding and classical projected diagonalization.","core_discovery":"For Ising Hamiltonians without a native measurement-visible symmetry, dual-rail code-space recovery produces lower projected Ritz energies than direct diagonalization on the full unencoded sample support, even when the recovered logical basis is smaller than that support and the encoded circuits use more qubits and more two-qubit gates.","pith_inferences":["On noisier devices or encodings, pair statistics may randomize faster than recovery can correct, erasing the subspace-quality advantage.","Other error-detecting codes with local, measurement-visible constraints could replace dual-rail if their encoded operations remain practical.","The residual gap to DMRG on the 36-site tests may close with denser Krylov sampling or larger shot budgets rather than only better recovery.","The work reframes simple encodings as classical post-processing aids for hybrid subspace methods, not only as quantum memories."],"forward_implications":["SQD-style projected diagonalization can be aimed at general Hermitian eigenvalue problems whose low-lying states are sparse in a computational basis, not only chemistry Hamiltonians with particle-number symmetry.","Encoding overhead need not erase sample quality if the engineered pair constraint supplies enough repair information to improve the projected subspace.","Native constraints and engineered code-space constraints can be composed: repair dual-rail pairs first, then enforce a logical sector such as particle number.","Any sampling circuit whose operations admit encoded lifts that preserve the code space in the noiseless limit can feed the same recovery loop."],"fun_headline_variants":["Dual-rail code-space recovery lowers Ritz energies on 36-spin Ising","Encoded recovery beats unencoded SQD samples for symmetry-free Ising","Code-space recovery extends SQD past native constraints on 36 sites","Dual-rail encoding yields smaller better bases for Ising SQD","Self-consistent code recovery tops raw samples on mixed-field Ising"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Noise from the larger, deeper encoded circuits still leaves pair-violation patterns that self-consistent repair can turn into a better logical basis than the raw unencoded samples provide.","fun_headline_variants_meta":{"raw":{"variants":["Dual-rail code-space recovery lowers Ritz energies on 36-spin Ising","Encoded recovery beats unencoded SQD samples for symmetry-free Ising","Code-space recovery extends SQD past native constraints on 36 sites","Dual-rail encoding yields smaller better bases for Ising SQD","Self-consistent code recovery tops raw samples on mixed-field Ising"]},"model":"grok-4.5","effort":"low","cost_usd":0.006392,"raw_usage":{"total_tokens":1616,"prompt_tokens":730,"num_sources_used":0,"completion_tokens":101,"cost_in_usd_ticks":63920000,"prompt_tokens_details":{"text_tokens":730,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":785,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":730,"tokens_out":101,"duration_ms":5745,"temperature":1.0,"reasoning_tokens":785,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T13:21:56.586595+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On the same device, SqDRIFT protocol, and shot budget, if the encoded workflow’s lowest Ritz energy at target basis size 2.5 million were higher than the unencoded full-support Ritz energy on the reported 25- or 36-site Ising benchmarks, the central claim would be false.","supporting_citations":[],"review_version":1}