{"id":"935d8950-419b-43ba-8b78-c2bd645e4130","arxiv_id":"2501.01107","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Experiments on a 5,627-spin D-Wave annealer suggest 3D spin glass ground states can be found probabilistically with complexity 2^{N/1000}, making annealing the most efficient known method for N below about 10^9.","lead":"Researchers used a D-Wave quantum annealer to find very low energy states, which they argue are the true lowest-energy states, of three-dimensional spin glasses with up to 5,627 spins. The measured time to do so grows as two to the power N divided by 1,000, which for systems up to about a billion spins beats the known worst-case bound for any algorithm.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Circular exactness certification: E0 is produced by the same digital-cooling pipeline used to fit Eq. (6) and to measure β≈10^3; for N=5627 the ancestors do not converge, so an independent exact-solver benchmark is required.","rationale":"The paper's most solid empirical contribution is that cyclic annealing combined with digital cooling drives sampled states to very low energies, with average excess energy scaling approximately linearly in N. Had the paper reported only this, it would be a useful heuristic study. The overclaim enters when digital-cooling output is asserted to be the exact ground state and the efficiency is quoted as β≈10^3. That assertion rests on Eq. (6), which was fit using the same candidate E0, and on the assumption that the sampled ensemble covers all low-energy basins. Neither is independently checked. The N=958 convergence across groups is a consistency check, not a certificate: a reproducible wrong local minimum could survive if the cyclic protocol biases sampling toward one basin. The N=5627 case is more direct: after two generations two near-degenerate distinct ancestors remain, and the authors choose one on belief. Because δ=E-E0 and δ0=-E0/N enter the exponents that define β, a systematic upward bias in the candidate E0 would directly inflate βeff. An independent exact-solver benchmark is therefore the minimal check required to support the abstract's central claim. Credit is due for making data available on Zenodo, so the proposed test is immediately runnable. The reader's rejection is well-founded; this stress test sharpens rather than changes it.","tokens_in":13488,"tokens_out":7280,"duration_ms":71356,"concrete_test":"On the Zenodo dataset (record 14578166), run an independent exact ground-state solver (e.g., branch-and-cut via CPLEX or SpinGlassPEPS) for the N=958 instances and compare each certified E0 to the digital-cooling ancestor; then recompute δ, δ0, βeff, and the fit in Eq. (4) using only certified E0 values. If any ancestor is above E0_exact, or if βeff computed with E0_exact is substantially below 10^3 for the same annealing settings, the exactness/efficiency claim is refuted; if all tested instances pass and βeff remains ≈10^3, the circularity concern is substantially weakened for the sizes that admit exact verification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that exact ground states are found with β≈10^3 rests on identifying the digital-cooling common ancestor as the true ground-state energy E0. This identification is not independently verified. In Sec. IV, for N=5627 the second-generation ancestors differ by a 67-spin cluster, with energies -8961.40 and -8961.11, and the paper states that the first is believed to be the true ground state, with confidence much lower than for N=958. No exact-solver benchmark is reported for any instance. The circularity is quantitative: δ=E-E0 and δ0=-E0/N enter Eqs. (3) and (6), and E0 is the same candidate whose exactness is being argued. If the true E0 is lower, δ is underestimated and βeff (hence β≈2.2βeff) is overestimated. The basin-counting formula itself was fit using that candidate, so it cannot certify E0 without independent ground truth. The N=958 agreement across 25 groups is suggestive but is a self-consistency check, not a certificate; a deterministic bias in the cyclic protocol could reproduce a wrong common ancestor across groups. The largest-N experiment, the headline case, fails to converge to a single ancestor. Thus the abstract's 'exact ground states ... with β≈10^3' is unsupported as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports experiments on the D-Wave Advantage 3D annealer for Edwards-Anderson spin glasses up to N=5627. The authors use a cyclic annealing protocol to generate low-energy states and a 'digital cooling' ancestry search to identify a common ancestor state, which they claim is the exact ground state (in a probabilistic sense) for typical realizations. They fit an empirical basin-count formula m(δ,d)=C0 exp(δ/(2δ0)) (N/2d)^α, observe that the average residual energy scales as δ≈N/βeff with βeff in excess of 10^3, and conclude that the computational complexity of finding exact ground states on this hardware scales as 2^{N/β} with β≈10^3. The paper also sketches a subexponential 2^{O(N^{2/3})} divide-and-conquer algorithm for comparison and argues that annealing devices are the most efficient known method for N<β^3.","tokens_in":13744,"tokens_out":5176,"duration_ms":51363,"significance":"If the central claim were established, this would be a striking result: an analog annealer finding exact ground states of 3D spin glasses at sizes up to 5,627 spins with an effective exponent β≈10^3, far exceeding the reported β≈10^2 for exact branch-and-cut algorithms. The empirical observation of exponential basin proliferation is interesting in its own right, and the authors are transparent about their protocol and data availability. However, the exactness certification is not independent: the ground-state energy E0 used in the analysis is the output of the very same digital-cooling pipeline that the paper seeks to validate, and for the largest system the pipeline does not converge to a unique ancestor. The headline claim is therefore unsupported in its current form.","major_comments":[{"comment":"For N=5627 the digital-cooling ancestry search produces two non-identical common ancestors, with energies −8961.40 and −8961.11, differing by a 67-spin cluster; the authors state, 'We believe the first one is the true ground state, though the confidence level of this assertion is much less than for N=958.' The abstract nevertheless claims exact ground states (in a probabilistic sense) for N≤5627. Since the energy assigned as E0 for this largest size is one of these unverified candidates, and that E0 enters Eq. (6) and the linear fit in Fig. 6(c), the reported βeff and hence β≈10^3 are not certified for the headline system size. An independent exact or rigorous solver (e.g., branch-and-cut) is needed for at least some instances, or the claim of exactness for N=5627 must be withdrawn.","section":"Sec. IV, Fig. 6(b), and Abstract"},{"comment":"The basin-count formula m(δ,d)=C0 exp(δ/(2δ0)) (N/2d)^α is fitted using δ=E−E0 and δ0=|E0|/N, where E0 is the candidate ground state produced by the same digital-cooling algorithm whose exactness the paper argues. This is a quantitative circularity: if the true ground state is lower than the candidate, then δ is underestimated and δ0 is overestimated, so the fitted exponential slope is biased. The data collapse in Fig. 2(c) and the 'single basin at low energy' conclusion of Sec. II are thus not independent evidence that the common ancestor is the true ground state; they are rearrangements of the assumed E0. The paper should validate Eq. (6) on small instances where E0 is independently known from exact algorithms.","section":"Eq. (6) and Secs. II–III"},{"comment":"The statement 'Given a sufficiently large initial set, the common ancestor state must be the ground state' rests on the empirical basin-count formula (6) and on the assumption that the annealer samples all low-energy basins at the sampled energy. The N=958 evidence that 25 independent groups converge to the same ancestor is suggestive, but it is a self-consistency check rather than a certificate: a deterministic bias in the cyclic protocol, which repeatedly biases toward a reference state, could in principle drive all groups to the same non-ground-state ancestor. The authors should either provide a rigorous argument for convergence or benchmark the ancestor against an external exact solver.","section":"Sec. III, paragraph on common-ancestor convergence"}],"minor_comments":[{"comment":"The statement that NP-hardness means 'no known algorithm (classical or quantum) can find or verify an answer in a polynomial time' is imprecise; NP-hardness alone does not imply that verification is hard for every instance class, and the verification complexity of optimality for spin glasses is a separate question.","section":"Sec. I, paragraph on NP-hardness"},{"comment":"There are several typographical errors, including 'D-Wave' vs 'D-wave' in the title/abstract, 'Y et' in Sec. II, and the phrase 'to compare the initial (blue) states were centered' in Sec. IV, which should be corrected.","section":"Throughout"},{"comment":"The recursive algorithm is described with notation '24N^{2/3}' that is easy to misread; the intended meaning is 2^{4N^{2/3}}, and the text should be typeset to avoid this ambiguity.","section":"Sec. V"},{"comment":"The linear fit δ/(2δ0)=0.00105N−0.733 is reported without error bars or a goodness-of-fit measure; given the central role of this fit in determining βeff, the authors should provide at least standard errors and the number of instances used.","section":"Sec. IV, Fig. 6(c)"}],"recommendation":"reject","confidential_remarks":"The core problem is not a disagreement with the community's consensus about annealer performance; it is that the paper's central claim is internally under-supported. The exactness of E0 is the load-bearing assumption for β≈10^3, and the paper states that for the largest N the ancestry search does not converge to a unique state. Adding an independent benchmark would be a substantial new study rather than a local revision, which is why I recommend rejection despite the paper's useful empirical observations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First impression: this paper has a genuinely interesting empirical result buried under an over-claimed headline. The data on low-energy basins in the 3D EA model, the basin-count formula Eq. (6), and the measured beta_eff ~ 10^3 on a D-Wave Advantage (with data on Zenodo) are new and worth knowing about. The digital cooling / common-ancestor method is a natural extension of cluster-flipping postprocessing, but the systematic scaling with N and the fit to Eq. (6) are a useful addition. The paper also deserves credit for stating clearly that for N=5627 the second-generation ancestors do not converge, and for calling out their own confidence level.\n\nThe soft spot is the exactness claim. The abstract says \"exact ground states (in a probabilistic sense)\" with beta ~ 10^3, but the ground-state energy E0 used to define delta is the output of the very same digital cooling pipeline whose reliability is then justified using the fitted basin-count formula. That's circular in a quantitative way: if the true E0 is lower, delta is underestimated and beta_eff is overestimated. The N=958 case looks great—25 groups all collapse to the same ancestor—but that is a self-consistency check, not a certificate; a deterministic bias in the annealer or the postprocessing could produce a common wrong ancestor. And the largest case, N=5627, is exactly where the convergence fails. The authors say they \"believe\" the first ancestor is the true ground state; that is not the same as having found it.\n\nSo the paper's central claim, as stated in the abstract, is not well-supported. That said, the more modest claim—that their protocol produces very low residual energies scaling as N/beta_eff, and that the basin count follows Eq. (6)—does not depend on E0 being exactly right, as long as the candidate E0 is close. The practical value of the annealer for finding near-ground states stands on its own.\n\nMy recommendation for peer review: send it out. The empirical work is substantial, the methods are described in enough detail to reproduce, and the scaling question is important enough to deserve careful referee time. But the authors should be pushed to either benchmark against an independent exact solver (e.g., branch-and-cut on smaller sizes) or to soften the language from \"exact ground states\" to \"very low-energy states\" and reframe the complexity estimate as conditional on the candidate E0. Rejecting the paper entirely would throw out the useful empirical core; accepting the headline as-is would enshrine an unverified claim. A revision with a more honest framing and an explicit discussion of the circularity would be publishable.","headline":"A useful empirical scaling result is over-sold as certified ground-state finding; the exactness claim is circular and the largest-N case does not converge, but the basin-count data and 'very low energies' scaling are worth publishing with more modest language.","tokens_in":14344,"tokens_out":2645,"would_cite":true,"duration_ms":24308,"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 quantum annealer finds exact 3D spin-glass ground states at efficiency $2^{N/\\beta}$, with $\\beta\\approx 10^3$.","keywords":["Ising spin glass","3D Edwards-Anderson model","quantum annealing","NP-hard optimization","ground-state search","basin counting","cyclic annealing","digital cooling"],"falsifier":"Run the full cyclic-annealing plus digital-cooling pipeline on a 3D spin-glass instance whose exact ground state is independently known, for example a small instance solved by an exact branch-and-cut code, and check whether the common ancestor equals that known state; a mismatch, or a second non-collapsing basin, would falsify the claim.","tokens_in":13208,"feed_emoji":"⚛️","tokens_out":15064,"duration_ms":118062,"temperature":0.7,"pith_summary":"This paper reports experiments on a quantum annealing device for three-dimensional Edwards-Anderson spin glasses with up to 5,627 spins. The authors' claim is that, for typical random instances and in a probabilistic sense, they find the true ground state, and that the computational effort scales as $2^{N/\\beta}$ with $\\beta\\approx 10^3$, where $N$ is the number of spins. The efficiency comes from a fitted law for the number of low-energy basins, together with a classical digital-cooling step that drives a large ensemble of annealer outputs down to a common ancestor state. If the claim holds, annealing hardware is the most efficient known way to solve this NP-hard problem for all sizes below roughly $\\beta^3\\approx 10^9$ spins, and the authors argue that $\\beta$ can be pushed still higher with better protocols and quieter devices.","feed_headline":"Annealer finds exact 3D spin-glass ground states up to 5,627 spins","feed_subtitle":"Quantum annealer plus classical cooling reaches efficiency 2^{N/β} with β≈1000, beating known solvers up to ~1e9 spins.","key_machinery":"The load-bearing object is the fitted basin-counting formula, Eq. (6): $m(\\delta,d)=C_0\\,e^{\\delta/(2\\delta_0)}(N/2d)^{\\alpha}$, with $\\alpha\\approx 2.6$, $\\delta_0\\approx 1.6$, and $C_0\\approx 0.08$. It says that the number of basins seen at excess energy $\\delta$ and Hamming-distance threshold $d$ is exponential in $\\delta$ and power-law in $N/(2d)$, and that at $\\delta\\sim O(1)$ the formula yields roughly one basin, the one containing the ground state. The second mechanism is digital cooling: pairwise comparison of low-energy states, decomposition of their differing spins into connected clusters via nonzero couplings, and iterative flipping of the clusters that produce the largest energy decrease, until a common ancestor appears. Together these two pieces convert a sample of annealer outputs into a claimed exact ground state and turn the exponential basin count into the complexity estimate $2^{N/\\beta}$.","core_discovery":"The central discovery is that the low-energy landscape of the 3D Edwards-Anderson model is organized so that the number of distant basins grows exponentially with the excess energy $\\delta$ as $m\\propto\\exp(\\delta/2\\delta_0)$, with $\\delta_0\\approx 1.6$, while at the very lowest energies only a single distant basin remains. On this basis, the authors claim that a sufficiently large ensemble of low-energy states, generated by cyclic annealing at an effective inverse temperature above $10^3$, can be digitally cooled to the true ground state: one identifies connected clusters of flipped spins between pairs of states, flips the clusters that lower the energy, and iterates until independent groups of states collapse to the same common ancestor. They take that ancestor to be the exact ground state, with the error probability decreasing exponentially in the number of independent groups. The measured complexity is $m\\sim 2^{N/\\beta}$ with $\\beta\\approx 2.2\\,\\beta_{\\mathrm{eff}}\\approx 10^3$, which they present as an order-of-magnitude improvement over exact branch-and-cut solvers and as more efficient than the proven subexponential $2^{N^{2/3}}$ algorithm for every $N<\\beta^3$. They conjecture that no fundamental limit prevents a further increase of $\\beta$.","pith_inferences":["A direct falsification test would compare the $N=5{,}627$ common ancestor against an independent exact solver on the same instance; the paper itself notes that for this size the two surviving ancestor candidates are very close and the confidence is much lower.","If the basin-counting law is generic, the same digital-cooling postprocessor should convert any low-energy sampler with comparably low effective temperature, classical or quantum, into a ground-state finder; the authors hint at this but do not test it.","The crossover $N\\approx\\beta^3$ means efficiency gains compound: a tenfold increase in $\\beta$ enlarges the solvable size by a factor of one thousand, so reducing the effective temperature matters more than linear speedups.","The claim is stated for typical random couplings; planted or adversarial instances would clarify whether the efficiency extends beyond the typical-case setting that the paper studies."],"forward_implications":["For $N$ below about $\\beta^3\\approx 10^9$, the annealing-plus-cooling pipeline would be the fastest known way to find exact ground states of typical 3D Edwards-Anderson instances.","The empirical relation $\\beta_{\\mathrm{eff}}\\approx 560(\\tau/20\\,\\mu\\mathrm{s})^{0.16}$ implies that longer annealing cycles raise efficiency, so improving hardware and protocols could enlarge the tractable size range.","The basin-counting formula predicts a definite number of basins for any given $\\delta$ and $d$, which can be checked on other annealing architectures and other short-range spin-glass models.","The authors argue that large $\\beta$ is tied to spatial locality: short-range models support independent excitation clusters, while all-to-all models such as Sherrington-Kirkpatrick remain stuck at small $\\beta$."],"supporting_citations":[{"why":"establishes that finding Ising spin-glass ground states is NP-hard, motivating the problem's worst-case difficulty.","marker":"[19]"},{"why":"proves the ETH-based lower bound $2^{N^{2/3}}$ that the paper's exponential scaling is compared against.","marker":"[55]"},{"why":"reports exact branch-and-cut ground-state computations with efficiency near $\\beta\\approx 10^2$, the baseline the paper claims to exceed by an order of magnitude.","marker":"[57]"},{"why":"gives the contrasting small-$\\beta$ example for the Sherrington-Kirkpatrick model, used to argue that large $\\beta$ is specific to short-range models.","marker":"[58]"},{"why":"introduces the many-body-localization-based iterative optimization idea that underlies the cyclic annealing protocol.","marker":"[73]"},{"why":"describes the cyclic quantum annealing method used to generate the low-energy state ensembles on the 5,000-qubit annealer.","marker":"[74]"},{"why":"provides the cluster-flip algorithm that the digital cooling ancestry search builds on.","marker":"[40]"},{"why":"demonstrates programmable 5,000-qubit spin-glass experiments on the annealing hardware class employed here.","marker":"[53]"}],"fun_headline_variants":["D-Wave finds exact 3D spin-glass states up to 5,627 spins","Annealer achieves 2^{N/β} complexity for 3D spin glass","3D spin-glass annealer reaches N<β^3 efficiency","Annealer exact for 3D spin glass when N<β^3","Quantum annealer solves 3D spin glass exactly under β^3 spins"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire certification of the ground state rests on the fitted basin-counting formula holding down to the lowest energies and on the annealer not having missed a whole basin, so that the repeated common ancestor really is the unique ground state.","fun_headline_variants_meta":{"raw":{"variants":["D-Wave finds exact 3D spin-glass states up to 5,627 spins","Annealer achieves 2^{N/β} complexity for 3D spin glass","3D spin-glass annealer reaches N<β^3 efficiency","Annealer exact for 3D spin glass when N<β^3","Quantum annealer solves 3D spin glass exactly under β^3 spins"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00171,"raw_usage":{"total_tokens":6807,"prompt_tokens":1026,"completion_tokens":5781,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":5677}},"tokens_in":642,"tokens_out":5781,"duration_ms":37627,"temperature":1.0,"reasoning_tokens":5677,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:35:33.611752+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the full cyclic-annealing plus digital-cooling pipeline on a 3D spin-glass instance whose exact ground state is independently known, for example a small instance solved by an exact branch-and-cut code, and check whether the common ancestor equals that known state; a mismatch, or a second non-collapsing basin, would falsify the claim.","supporting_citations":[{"cited_title":"Binder and A","cited_arxiv_id":null,"evidence_quote":"establishes that finding Ising spin-glass ground states is NP-hard, motivating the problem's worst-case difficulty."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"proves the ETH-based lower bound $2^{N^{2/3}}$ that the paper's exponential scaling is compared against."},{"cited_title":"Bernaschi, I","cited_arxiv_id":null,"evidence_quote":"reports exact branch-and-cut ground-state computations with efficiency near $\\beta\\approx 10^2$, the baseline the paper claims to exceed by an order of magnitude."},{"cited_title":"Zhang, Computational complexity of spin-glass three- dimensional (3D) Ising model, J","cited_arxiv_id":null,"evidence_quote":"gives the contrasting small-$\\beta$ example for the Sherrington-Kirkpatrick model, used to argue that large $\\beta$ is specific to short-range models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the many-body-localization-based iterative optimization idea that underlies the cyclic annealing protocol."},{"cited_title":"Zhang, K","cited_arxiv_id":null,"evidence_quote":"describes the cyclic quantum annealing method used to generate the low-energy state ensembles on the 5,000-qubit annealer."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"demonstrates programmable 5,000-qubit spin-glass experiments on the annealing hardware class employed here."}],"review_version":1}