{"id":"cf3c68cf-292e-4f8d-ad72-3be1eb40bfe2","arxiv_id":"2606.29133","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Adapts shadow tomography to tensor networks for O(1) scaling in long-range Hamiltonian expectations and stable variational gradients on classical computers.","lead":"This paper adapts shadow tomography from quantum computing to classical tensor network simulations, achieving better sample scaling for observables especially in long-range interacting systems. The approach could make larger-scale classical simulations of quantum chemistry and condensed matter models more practical.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Whether TN contraction costs for adapted shadow estimators remain subdominant for long-range Hamiltonians, preserving the claimed O(1) sample scaling","rationale":"The reader's weakest assumption directly identifies the contraction-cost bottleneck that must hold for the O(1) claim; full-text access does not remove this as the single most load-bearing point, and no machine-checked proof or parameter-free derivation is reported to close it.","tokens_in":1691,"tokens_out":342,"duration_ms":17156,"concrete_test":"For the 2D long-range Heisenberg example, extract the reported bond dimension, contraction cost per shadow sample, and number of samples used; recompute total wall-time scaling versus standard MC on the same observable and check whether the product (samples × contraction cost) is O(1) or better.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline claim of O(1) overall scaling (up to logs) for fixed relative error on long-range Hamiltonians requires that the per-sample cost of evaluating the shadow estimator via TN contraction does not grow with N or interaction range enough to cancel the Monte Carlo variance reduction. Standard TN contractions for long-range operators typically require either high bond dimension or auxiliary indices whose cost scales at least linearly with range; if the shadow protocol introduces additional factors (e.g., via randomized measurements or fermionic parity handling) that interact with this, the net complexity could revert to worse than O(N). The abstract asserts tailored strategies for spin and fermionic cases that achieve the scaling, but the load-bearing step is whether the contraction complexity analysis in the methods section actually bounds the total cost below the sample savings.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript adapts shadow tomography techniques to classical tensor network (TN) simulations of quantum spin and fermionic systems. It claims that tailored shadow estimators yield sample-complexity improvements of O(N) to O(N^3) over standard Monte Carlo sampling, with the key result that expectation values of long-range Hamiltonians can be estimated to fixed relative error with optimal O(1) overall scaling (up to logarithmic factors) in both spin and fermionic cases. The work further asserts that shadow estimators produce more stable gradients than standard Monte Carlo in variational TN optimization and demonstrates the approach on the 2D long-range Heisenberg model and an ab-initio quantum chemistry Hamiltonian.","tokens_in":1866,"tokens_out":576,"duration_ms":22617,"significance":"If the contraction-cost analysis holds, the result would be significant for TN simulations of long-range systems, where standard sampling costs scale poorly with interaction range. Achieving O(1) sample scaling while preserving efficient TN contractions would directly address a practical bottleneck in variational Monte Carlo and related methods. The explicit strategies for both bosonic and fermionic cases, together with the gradient-stability claim and numerical demonstrations, would strengthen the case for broader adoption of randomized measurement ideas in classical TN workflows.","major_comments":[{"comment":"§IV (contraction complexity for long-range operators): The O(1) overall scaling claim for long-range Hamiltonians is load-bearing on the assertion that TN contraction of the adapted shadow estimators remains subdominant and does not grow with N or interaction range. The manuscript must supply explicit bounds (including any auxiliary indices or bond-dimension overhead introduced by the randomized measurement protocol or fermionic parity handling) showing that per-sample cost stays O(poly(log N)) or better; without this, the net complexity could revert to worse than linear in N.","section":"§IV"},{"comment":"§III.B (fermionic adaptation): The fermionic shadow estimator construction relies on a specific mapping to TN contractions; the error analysis and variance bound must be shown to remain independent of the range of the Hamiltonian after this mapping, otherwise the claimed O(1) scaling for fermionic long-range systems is not supported.","section":"§III.B"}],"minor_comments":[{"comment":"Notation for the shadow channel and the TN contraction order should be unified between the spin and fermionic sections to avoid reader confusion when comparing the two cases.","section":null},{"comment":"Figure 3 (gradient stability comparison): axis labels and error-bar definitions are not fully specified in the caption; clarify whether the plotted quantity is the variance of the gradient estimator or its bias.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below and will revise the manuscript to incorporate additional explicit analysis where needed to strengthen the claims.","responses":[{"response":"We agree that more explicit bounds would strengthen the presentation. Section IV already argues that the adapted shadow estimators map to TN contractions whose cost depends only on the bond dimension and local support after randomization, remaining independent of interaction range. To address the concern directly, we will add a new paragraph with rigorous bounds (including overhead from auxiliary indices and fermionic parity) demonstrating per-sample contraction cost O(poly(log N)) independent of N and range. This confirms the overall O(1) scaling claim.","revision_made":"yes","referee_comment":"[§IV] §IV (contraction complexity for long-range operators): The O(1) overall scaling claim for long-range Hamiltonians is load-bearing on the assertion that TN contraction of the adapted shadow estimators remains subdominant and does not grow with N or interaction range. The manuscript must supply explicit bounds (including any auxiliary indices or bond-dimension overhead introduced by the randomized measurement protocol or fermionic parity handling) showing that per-sample cost stays O(poly(log N)) or better; without this, the net complexity could revert to worse than linear in N."},{"response":"We thank the referee for this observation. The fermionic construction in §III.B employs a parity-adapted mapping under which the shadow variance bound depends on the shadow norm of the observable rather than its spatial range; this ensures independence from Hamiltonian range by design. We will revise §III.B to include an expanded derivation (e.g., an additional lemma) explicitly showing that both the error analysis and variance bound remain range-independent after the mapping, thereby supporting the O(1) scaling for fermionic long-range cases.","revision_made":"yes","referee_comment":"[§III.B] §III.B (fermionic adaptation): The fermionic shadow estimator construction relies on a specific mapping to TN contractions; the error analysis and variance bound must be shown to remain independent of the range of the Hamiltonian after this mapping, otherwise the claimed O(1) scaling for fermionic long-range systems is not supported."}],"tokens_in":1422,"tokens_out":450,"duration_ms":29098,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Colleague,\n\nThis paper takes the idea of shadow tomography, which is used on quantum computers to estimate many observables from few measurements, and adapts it for classical tensor network simulations. The main result is improved sample efficiency for long-range interacting Hamiltonians, reaching O(1) scaling up to logs for fixed relative error in both spin and fermionic cases. They also find that the shadow estimators give more stable gradients during variational optimization compared to standard Monte Carlo.\n\nWhat they do well is tailor the approach to the contraction requirements of tensor networks for both bosonic and fermionic systems. This leads to sample reductions ranging from O(N) to O(N^3) depending on the task. The demonstrations on the 2D long-range Heisenberg model and an ab-initio quantum chemistry Hamiltonian provide concrete evidence that the method works in practice for relevant systems.\n\nThe adaptation itself is the novel element here, as it applies an existing quantum technique to the classical setting with system-specific modifications.\n\nOn the downside, the overall O(1) scaling claim rests on the assumption that the computational cost of performing the tensor network contractions for each shadow estimator remains manageable and does not grow significantly with system size or interaction range. Long-range operators can already be challenging for TN methods, and the shadow protocol might add extra factors through randomization or parity handling. The paper needs to demonstrate that the total cost, including contractions, preserves the sample complexity gains. If the contraction overhead is properly bounded, the result stands; if not, the practical benefit could be smaller than advertised.\n\nThis work is relevant for anyone using tensor networks to simulate quantum many-body systems, especially those with long-range interactions in condensed matter or quantum chemistry. The gradient stability finding could be of interest for optimization tasks.\n\nIt deserves a serious referee to verify the cost analysis and the numerical results.\n\nI would recommend sending it to peer review.","headline":"Shadow tomography adapted for tensor networks claims O(1) scaling on long-range Hamiltonians, with the contraction overhead as the main open question.","tokens_in":2360,"tokens_out":457,"would_cite":false,"duration_ms":31819,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Shadow tomography adapted to tensor networks reduces sample needs for long-range observables to constant scaling.","keywords":["shadow tomography","tensor network states","sample complexity","long-range interactions","expectation value estimation","variational optimization","spin systems","fermionic systems"],"falsifier":"A numerical test on increasing system sizes where the number of samples required to hold relative Monte Carlo error fixed grows with N, or where the contraction cost for the shadow estimators exceeds the sample savings.","tokens_in":2588,"feed_emoji":"","tokens_out":594,"duration_ms":23609,"temperature":0.7,"pith_summary":"The paper adapts shadow tomography techniques from quantum computing to estimate observables on classical tensor network states. This yields sample scaling improvements of O(N) to O(N^3) for various tasks in spin and fermionic systems. For expectation values of long-range interacting Hamiltonians, the method reaches optimal O(1) scaling up to logarithmic factors at fixed relative Monte Carlo error. It also produces more stable gradients than standard Monte Carlo during variational optimization and is shown to work on 2D long-range Heisenberg models and ab-initio chemistry Hamiltonians.","feed_headline":"Shadow tomography gives O(1) samples for long-range tensor network Hamiltonians","feed_subtitle":"Adapting the estimators to classical tensor network contractions yields constant sample needs for fixed error on large systems.","key_machinery":"Shadow estimators adapted via tensor network contractions for spin and fermionic systems.","core_discovery":"By tailoring shadow estimators to the contraction requirements of tensor networks, expectation values of observables including long-range Hamiltonians can be estimated with sample complexity independent of system size (up to logs) for fixed relative error, in both bosonic and fermionic cases, while the contractions themselves remain efficient.","pith_inferences":["Tensor network simulations could now tackle interaction ranges that sampling costs previously ruled out.","The same estimator adaptation might transfer to other classical many-body methods that rely on Monte Carlo sampling.","More stable gradients could shorten the number of steps needed to reach convergence in variational tensor network algorithms."],"forward_implications":["Observable estimation gains sample factors of O(N) to O(N^3) depending on task and system type.","Long-range Hamiltonian expectations reach O(1) overall scaling up to logs for fixed relative error.","Variational optimization gradients become more stable than those from standard Monte Carlo estimators.","Practical simulations become feasible for 2D long-range Heisenberg models and ab-initio quantum chemistry Hamiltonians."],"fun_headline_variants":["O(1) samples suffice for long-range Hamiltonians with tensor network shadow tomography","Shadow tomography achieves optimal O(1) scaling in classical tensor network simulations","Constant sample complexity for long-range tensor network observables via shadows","Shadow tomography tailored for tensor networks yields O(1) long-range Hamiltonian samples"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The tensor network contractions needed to apply the shadow estimators remain computationally cheap enough not to cancel out the sample complexity gains for the system sizes and interaction ranges considered.","fun_headline_variants_meta":{"raw":{"variants":["O(1) samples suffice for long-range Hamiltonians with tensor network shadow tomography","Shadow tomography achieves optimal O(1) scaling in classical tensor network simulations","Constant sample complexity for long-range tensor network observables via shadows","Shadow tomography tailored for tensor networks yields O(1) long-range Hamiltonian samples"]},"model":"grok-4.3","cost_usd":0.006847,"raw_usage":{"total_tokens":3154,"prompt_tokens":615,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":68474500,"prompt_tokens_details":{"text_tokens":615,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2462,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":615,"tokens_out":77,"duration_ms":22970,"temperature":1.0,"reasoning_tokens":2462,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T07:57:29.506023+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A numerical test on increasing system sizes where the number of samples required to hold relative Monte Carlo error fixed grows with N, or where the contraction cost for the shadow estimators exceeds the sample savings.","supporting_citations":[],"review_version":1}