{"id":"6fee0e0b-108f-4bf9-9f99-b15a056c8139","arxiv_id":"2502.06956","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A tensor cross interpolation method computes quantum resource quantifiers directly from their definitions, demonstrated for stabilizer Rényi entropy in 1D and relative entropy of coherence in 2D Ising models.","lead":"Physicists put forward a general numerical recipe that uses tensor cross interpolation to compute quantum resource measures such as magic and coherence without tailoring the algorithm to each measure. They demonstrate it on Ising model ground states, reaching 64 spins for magic in 1D and computing coherence in the 2D transverse-field Ising model.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No convergence tests in TCI bond dimension or input MPS bond dimension are reported, so the 2D REC results and the claimed polynomial scaling are not established.","rationale":"Section 2 states the algorithm requires O(L d ξ^2) calls, so the entire efficiency argument depends on ξ being small. The results sections set a maximum ξ but never show that the final value is stable as ξ increases; similarly, χ is fixed at 50 without checking the ground-state approximation. In 2D, the ground state of a 2D Hamiltonian is represented as an MPS with χ=50, whose accuracy is not quantified, and the REC is a nonlinear function of the amplitudes, so the TCI low-rankness is not automatic. The 1D SRE results are consistent with known values, which is genuine supporting evidence, but it does not justify the 2D REC volume-law claim or the 'no other efficient algorithm' statement. A concrete convergence and small-system comparison would settle whether the reported numbers are physical or truncation artifacts. This matches the reader's identified weakest assumption, so the conditional verdict remains appropriate.","tokens_in":9444,"tokens_out":6481,"duration_ms":59152,"concrete_test":"For the 2D FM Ising model at L=16 (4×4) and L=36 (6×6), compute REC at h=1.0, 2.0, 3.0 while sweeping the TCI bond-dimension cap ξ∈{10,20,40,80,160} and the input MPS bond dimension χ∈{20,50,100}. Compare L=16 against exact diagonalization (or a fully converged MPS) to isolate TCI error from input-state error. If the REC value changes by more than about 5% across the sweep, the reported 2D numbers and their volume-law scaling are not converged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The method's polynomial cost O(L d ξ^2) is only meaningful if the TCI-MPS bond dimension ξ stays small. The paper caps ξ at 80 (1D SRE) and 40 (2D REC) but never reports the actual ξ reached, a ξ-sweep, or the TCI truncation error; without this, the computed values—and the 2D volume-law claim—could be dominated by truncation. The 2D input is itself an MPS approximation to the ground state with χ≤50, but no comparison with exact diagonalization for small L or with larger χ is given, so the input-state error is also uncontrolled. The 1D SRE comparison with refs. [11,50] validates the pipeline in one well-understood case, but it does not transfer to 2D REC, where the tensor F is a nonlinear function of MPS amplitudes and its low-rankness is not demonstrated. The paper acknowledges that a TTN would be more suitable for 2D (§5), which further undercuts the 2D results as evidence of a fully general framework.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a general numerical procedure for computing quantum resource quantifiers of the form M(ρ)=∑_σ F_σ. The method uses Tensor Cross Interpolation (TCI) to build a matrix-product-state (MPS) approximation of the tensor F, starting from a tensor-network representation of the input state, and then contracts this MPS to obtain the desired quantifier. As demonstrations, the authors compute the Stabilizer Rényi-2 entropy (SRE) for the 1D ferromagnetic transverse-field Ising chain up to L=64 and the Relative Entropy of Coherence (REC) for the 2D ferromagnetic transverse-field Ising model, reporting polynomial scaling in the number of function calls and releasing the code in a public repository.","tokens_in":9596,"tokens_out":4486,"duration_ms":39614,"significance":"If the low-rankness assumption behind the method holds, the approach would be valuable because it is measure-agnostic: the same sampling pipeline applies to different quantifiers and models without tailoring algorithms to each case. The manuscript has clear strengths: the algorithm is described in a reproducible way, the code is publicly available, and the 1D SRE results reproduce the known behavior from refs. [11,50]. However, the central efficiency claim rests on the assumption that the sampled tensor F has a low-rank MPS representation with small bond dimension ξ, and the paper provides no convergence analysis in ξ or in the input MPS bond dimension χ, nor an independent benchmark for the 2D REC results. The 2D demonstration is further weakened by the authors' own admission that a tree-tensor-network representation would be more suitable for 2D. The significance is therefore conditional on additional numerical validation.","major_comments":[{"comment":"The efficiency claim rests on the assumption that TCI produces an MPS approximation of F with a small bond dimension ξ, but no convergence study in ξ or in the input-state bond dimension χ is reported. The caps ξ=80 (Fig. 1) and ξ=40 (Fig. 2) are upper bounds; the paper never states the actual ξ reached, nor does it provide a ξ-sweep or the TCI truncation error. Without such information, the computed values—and especially the 2D volume-law claim—could be dominated by truncation, and the central polynomial-scaling statement O(L d ξ^2) is not established by the data.","section":"§2 and Figs. 1–2"},{"comment":"The 2D REC results lack an independent validation. The input state is an MPS with χ≤50, but there is no comparison with exact diagonalization for small L or with larger χ. The agreement of the 1D SRE with refs. [11,50] validates the pipeline in one well-understood case, but it does not establish the low-rankness of F(S)=|⟨S|ψ⟩|^2 log2|⟨S|ψ⟩|^2 for a 2D ground state, where the same MPS ansatz is less natural. The admission in §5 that a TTN 'would be more suitable for the 2D case' further weakens the claim that this example demonstrates a fully general framework.","section":"§4, Fig. 2"},{"comment":"The stated overall complexity O(2L^2 ξ^2 χ^3) in §3 (and O(2L^2 ξ^2 χ^2) in §4) is not derived cleanly. If TCI requires O(Lξ^2) calls and each call costs O(Lχ^3), the product is O(L^2 ξ^2 χ^3); the extra factor 2 and the '2L^2' form appear without justification. Because the complexity scaling is the central quantitative claim, these expressions should be stated consistently and, ideally, verified by reporting the actual number of function evaluations or wall-clock times for the system sizes studied.","section":"§3, complexity paragraph"}],"minor_comments":[{"comment":"The text says 'we assumed periodic boundary conditions, hence Z_{L+1}=Z_L', but the periodic condition should be Z_{L+1}=Z_1; the current statement is a typo.","section":"§3, Eq. (2)"},{"comment":"Reference [9] has an incomplete author list ('and and.'); the names should be completed.","section":"Ref. [9]"},{"comment":"The abbreviation 'QC' is used without being defined; it should be introduced as 'quantum coherence' at first use.","section":"§4"},{"comment":"The statement that the 1D REC follows a volume law is attributed to the unpublished companion paper [48]; this dependence on an unpublished manuscript should be stated explicitly, and ideally the companion results should be made available or summarized.","section":"§4"},{"comment":"The phrase 'the method is robust' is vague; the authors should specify the sense in which the TCI procedure controls the approximation error, e.g., through deterministic pivoting or documented residual errors.","section":"§2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the algorithmic idea is appealing. The main gap is methodological validation: the paper should report convergence tests in the TCI bond dimension and in the input-state bond dimension, and it should benchmark the 2D REC against exact diagonalization for small systems. I would also flag that the interpretation of the 1D REC volume law relies on an unpublished companion paper by the same group; this should be disclosed more prominently. The central claim is defensible, but the missing convergence analysis is load-bearing and needs to be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a decent paper with a genuinely new application of Tensor Cross Interpolation to quantum resource quantifiers. The 1D SRE results reproduce known behavior, the code is promised in a repository, and the measure-agnostic framework has real utility—especially for something like 2D REC, where they correctly note that no dedicated efficient algorithm exists. If you work on resource estimation, this is worth your time.\n\nWhat I like: the idea is simple and honest. Write the quantifier as a tensor F, sample it with TCI to get an MPS, contract. The 1D SRE benchmark against refs. [11,50] gives a sanity check that the pipeline works. The method is not tied to a particular quantifier, so it is a reusable tool rather than a one-off. The authors also acknowledge that an MPS is not ideal for 2D and that a TTN would be better, which is the right kind of self-criticism.\n\nThe soft spots are real but not fatal. The biggest issue: no convergence tests in the TCI bond dimension ξ or the input MPS bond dimension χ. They cap ξ at 80 and 40 respectively but never report actual ξ reached, truncation errors, or a ξ-sweep. Without that, the 2D REC numbers—including the volume-law claim—could be under-resolved. The lack of an independent benchmark for 2D is also a genuine hole; a small-L exact diagonalization comparison would settle it. The complexity expression O(2L^2ξ^2χ^3) looks odd; I suspect a missing log or a typo, but it should be cleaned up. The reliance on unpublished companion paper [48] for the 1D volume law is a minor self-citation issue, but it doesn't undermine the core method.\n\nThe stress-test note is largely on target. However, I would not call the missing convergence checks a load-bearing flaw. The method is plausible, the 1D validation works, and the 2D results are presented as an illustration of generality rather than as a final quantitative claim. It is under-supported, not wrong.\n\nWho is this for? Someone doing numerical work on magic or coherence in many-body systems will find the approach useful. It deserves serious peer review, with the request that the authors add convergence tests, error bars, and a small-system check. I would accept it as a method paper after those revisions.\n\nRecommendation: send to peer review. It's a solid contribution that needs tightening, not a desk reject.","headline":"A useful and genuinely new numerical tool for computing resource quantifiers via TCI, but the 2D REC claims need convergence data and an independent benchmark before they can be taken at face value.","tokens_in":10168,"tokens_out":1209,"would_cite":true,"duration_ms":12525,"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 that Tensor Cross Interpolation gives a single sampling-based procedure for computing quantum resource quantifiers, from stabilizer magic to coherence, by approximating the quantifier's defining tensor as a matrix…","keywords":["tensor cross interpolation","matrix product states","stabilizer Rényi entropy","quantum magic","relative entropy of coherence","transverse-field Ising model","quantum resource theories","non-stabilizerness"],"falsifier":"Take the 2D transverse-field Ising ground state at a fixed field inside the ferromagnetic phase, say $h=2$, and compute the REC with TCI bond dimensions $\\xi=20,40,80,160$ for $L=16,32,64$, recording the largest $\\xi$ needed to keep the result within a fixed tolerance; if that required $\\xi$ grows linearly or faster with $L$, or if the computed REC drifts with $\\xi$ at fixed $L$, the claimed polynomial efficiency fails. A cleaner test is the same scan at the critical field $h_c\\approx 3.044$, where the sampled tensor should be least compressible.","tokens_in":9182,"feed_emoji":"⚛️","tokens_out":9593,"duration_ms":77084,"temperature":0.7,"pith_summary":"The paper claims that a single, measure-agnostic procedure can compute quantum resource quantifiers that are defined as sums over exponentially many tensor elements. The idea is to treat the quantifier's defining map, such as the fourth power of a Pauli expectation value for magic or the squared amplitude times its logarithm for coherence, as a tensor to be sampled by Tensor Cross Interpolation (TCI), which builds a compact matrix product state approximation from a limited number of function calls. Contracting that approximation yields the quantifier at a cost that grows polynomially with system size whenever the sampled tensor is compressible. The authors demonstrate the recipe on the stabilizer Rényi entropy of the 1D ferromagnetic Ising chain and on the relative entropy of coherence of the 2D transverse-field Ising model, a computation for which they state no other efficient algorithm exists.","feed_headline":"One sampling pass computes both magic and quantum coherence","feed_subtitle":"The same deterministic sampler handles stabilizer entropy in 1D and coherence in 2D, where no other efficient algorithm exists.","key_machinery":"The central object is the tensor $F$, whose elements are the quantities the quantifier sums over, and the Tensor Cross Interpolation (TCI) algorithm that learns a compressed matrix product state representation of $F$ from a small number of deterministic function evaluations. The version used in the paper is based on an LU decomposition; it probes rows and columns of $F$ and iteratively builds an MPS whose bond dimension $\\xi$ controls both accuracy and the number of calls, which scales as $O(L d \\xi^2)$. The input state is first approximated as an MPS with bond dimension $\\chi$, and each call to $f$ is a tensor network contraction of known cost, $O(L \\chi^3)$ for the SRE Pauli-string expectation and $O(L \\chi^2)$ for reading a single amplitude. This combination converts an exponentially large sum into a polynomial number of tame contractions.","core_discovery":"On the paper's own terms, the central claim is that a whole family of quantum resource quantifiers of the form $M(\\rho)=\\sum_{\\sigma} F_{\\sigma}$ can be computed by sampling the tensor $F$ directly with Tensor Cross Interpolation, rather than by designing a dedicated algorithm for each measure. For the stabilizer Rényi-2 entropy the sampled function is $f(P)=\\langle\\psi|P|\\psi\\rangle^4$ over Pauli strings, and for relative entropy of coherence it is $f(S)=|\\langle S|\\psi\\rangle|^2 \\log_2 |\\langle S|\\psi\\rangle|^2$ over computational basis states. The TCI routine performs roughly $O(L d \\xi^2)$ calls to $f$ and returns an approximate matrix product state, which is then contracted to give the quantifier. With the input state also represented as an MPS, the total complexity is $O(2 L^2 \\xi^2 \\chi^3)$ for the SRE and $O(2 L^2 \\xi^2 \\chi^2)$ for the REC, where $\\chi$ is the input bond dimension and $\\xi$ is the bond dimension chosen for the sampled tensor. The authors use this to reach $L=64$ for the 1D SRE by direct application of the definition and to compute the 2D REC, which they say no other efficient algorithm currently achieves.","pith_inferences":["A decisive test of the method's claimed generality is whether $\\xi$ stays small for the 2D REC near the critical field $h_c\\approx 3.044$, where correlations are long-range and the sampled tensor $F$ is likely least compressible.","The same sampling view suggests that other resource monotones expressible through nonlinear functions of amplitudes, such as entanglement entropies or Rényi variants of coherence, could be handled without new algorithmic ideas as long as their defining tensor has low MPS rank.","If the low-rank assumption holds only away from criticality, the method would still be useful for gapped phases but would need tree tensor network or quantics variants for gapless or higher-dimensional settings; the authors already flag tree tensor networks as future work.","A practical extension would be to make $\\xi$ adaptive, increasing it until the computed quantifier stops changing, which would convert the manual $\\xi$ choices in the paper into a convergence-controlled estimate."],"forward_implications":["Any quantifier that can be written as a sum over a tensor $F$, with the state available in a samplable form, becomes computable with the same TCI pipeline and only a change of the function $f$.","The relative entropy of coherence of the 2D transverse-field Ising ground state becomes accessible for system sizes beyond exact diagonalization, a regime the authors state is out of reach for existing algorithms.","The stabilizer Rényi-2 entropy of the 1D Ising chain can be recovered up to $L=64$ by applying the definition directly, reproducing known results without a measure-specific algorithm.","The cost of computing a quantifier is polynomial in the system size whenever the input MPS bond dimension $\\chi$ and the sampled-tensor bond dimension $\\xi$ stay moderate, so the method's reach is tied to the compressibility of $F$ rather than to the structure of the measure.","The procedure opens the possibility of evaluating other nonlinear functions of tensor network states, such as the non-local magic mentioned in the paper, in the same framework."],"supporting_citations":[{"why":"Supplies the LU-based Tensor Cross Interpolation algorithm that the paper uses to sample $F$ and build the MPS approximation.","marker":"[28]"},{"why":"Introduces the TT-cross approximation, the mathematical basis of the deterministic sampling scheme.","marker":"[25]"},{"why":"Provides the parallel cross-interpolation variant behind the claimed $O(L d \\xi^2)$ sampling complexity.","marker":"[27]"},{"why":"Gives the perfect Pauli sampling of matrix product states whose $O(L\\chi^3)$ per-sample cost the SRE function calls inherit.","marker":"[22]"},{"why":"Defines the stabilizer Rényi entropy that the paper computes in Section 3.","marker":"[33]"},{"why":"Defines the relative entropy of coherence, the measure computed for the 2D model.","marker":"[38]"},{"why":"Demonstrates that sampling can estimate entanglement in arbitrary dimensions, the gap the paper extends to coherence and magic.","marker":"[24]"}],"fun_headline_variants":["One TCI pass computes magic and coherence","General TCI framework quantifies quantum resources","Same sampler for 1D magic, 2D coherence","TCI achieves 2D coherence no other efficient algorithm can","Quantum resource measures via tensor cross interpolation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The efficiency claim rests on the assumption that the tensor being sampled (for example, the fourth-power Pauli expectation map for magic, or the squared-coefficient map for coherence) has a low-rank matrix product approximation whose bond dimension grows only mildly with system size, an assumption the paper supports by fixing that bond dimension by hand rather than by convergence testing.","fun_headline_variants_meta":{"raw":{"variants":["One TCI pass computes magic and coherence","General TCI framework quantifies quantum resources","Same sampler for 1D magic, 2D coherence","TCI achieves 2D coherence no other efficient algorithm can","Quantum resource measures via tensor cross interpolation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000641,"raw_usage":{"total_tokens":2957,"prompt_tokens":956,"completion_tokens":2001,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":1928}},"tokens_in":572,"tokens_out":2001,"duration_ms":16370,"temperature":1.0,"reasoning_tokens":1928,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:12:46.128807+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the 2D transverse-field Ising ground state at a fixed field inside the ferromagnetic phase, say $h=2$, and compute the REC with TCI bond dimensions $\\xi=20,40,80,160$ for $L=16,32,64$, recording the largest $\\xi$ needed to keep the result within a fixed tolerance; if that required $\\xi$ grows linearly or faster with $L$, or if the computed REC drifts with $\\xi$ at fixed $L$, the claimed polynomial efficiency fails. A cleaner test is the same scan at the critical field $h_c\\approx 3.044$, where the sampled tensor should be least compressible.","supporting_citations":[{"cited_title":"Tt- cross approximation for multidimensional ar- rays","cited_arxiv_id":null,"evidence_quote":"Introduces the TT-cross approximation, the mathematical basis of the deterministic sampling scheme."},{"cited_title":"Parallel cross interpolation for high-precision calculation ofhigh-dimensionalintegrals","cited_arxiv_id":null,"evidence_quote":"Provides the parallel cross-interpolation variant behind the claimed $O(L d \\xi^2)$ sampling complexity."},{"cited_title":"Nonstabiliz- erness via perfect pauli sampling of matrix prod- uct states","cited_arxiv_id":null,"evidence_quote":"Gives the perfect Pauli sampling of matrix product states whose $O(L\\chi^3)$ per-sample cost the SRE function calls inherit."},{"cited_title":"Stabilizer rényi entropy","cited_arxiv_id":null,"evidence_quote":"Defines the stabilizer Rényi entropy that the paper computes in Section 3."},{"cited_title":"Quantifying coherence","cited_arxiv_id":null,"evidence_quote":"Defines the relative entropy of coherence, the measure computed for the 2D model."},{"cited_title":"Entanglement es- timation in tensor network states via sampling","cited_arxiv_id":null,"evidence_quote":"Demonstrates that sampling can estimate entanglement in arbitrary dimensions, the gap the paper extends to coherence and magic."}],"review_version":1}