Computing Quantum Resources using Tensor Cross Interpolation

Gianpaolo Torre; Sven Benjamin Ko\v{z}i\'c

REVIEW 3 major objections 5 minor 2 cited by

Computing Quantum Resources using Tensor Cross Interpolation

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2502.06956 v2 pith:4ZG4MQBK submitted 2025-02-10 quant-ph physics.comp-ph

Sven Benjamin Kožić , Gianpaolo Torre This is my paper · ORCID

classification quant-phphysics.comp-ph

keywords tensorcrossinterpolationmatrixproductstatesstabilizerRényientropyquantummagicrelativeofcoherencetransverse-fieldIsingmodelresourcetheoriesnon-stabilizerness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Quick 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.

What 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.

The 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.

The 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.

Who 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.

Recommendation: send to peer review. It's a solid contribution that needs tightening, not a desk reject.

Referee Report

3 major / 5 minor

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.

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 (3)

[§2 and Figs. 1–2] 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.
[§4, Fig. 2] 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.
[§3, complexity paragraph] 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.

minor comments (5)

[§3, Eq. (2)] 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.
[Ref. [9]] Reference [9] has an incomplete author list ('and and.'); the names should be completed.
[§4] The abbreviation 'QC' is used without being defined; it should be introduced as 'quantum coherence' at first use.
[§4] 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.
[§2] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the TCI pipeline evaluates the quantifier definitions directly; the sole self-citation to the unpublished companion paper is not load-bearing.

full rationale

The derivation chain is M(rho) = sum_sigma F_sigma (Sec. 2), with F defined directly from the chosen quantifier: f(P) = <psi|P|psi>^4 for SRE and f(S) = |c_S|^2 log2|c_S|^2 for REC (Table 1). TCI approximates F as an MPS and the quantifier is obtained by contraction, which is a numerical evaluation of the defining equations (1) and (3), not a prediction fitted to target data. No parameter of F is adjusted to reproduce known SRE or REC values. The 1D SRE results are checked against refs. [11,50], and ref. [50] is external to the authors, so the validation is independent. The only self-referential element is Sec. 4's attribution of the 1D REC volume law to the authors' own unpublished companion paper [48]. That citation provides physical context for the 2D trend but is not an input to the TCI computation, so the central claim does not reduce to it. The hand-set caps xi=80 and xi=40 and the absence of convergence sweeps are correctness concerns about whether the reported values are converged, but they are not evidence of circularity. No equation or fitted parameter is equivalent by construction to any claimed prediction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim does not rest on fitted parameters, but it does rest on numerical convergence thresholds (χ, ξ) chosen by hand and on the compressibility of the sampled tensor F. No independent evidence is provided for the 2D case, and the paper's own conclusion acknowledges that the MPS output of TCI may be suboptimal for 2D models.

free parameters (2)

Maximum input MPS bond dimension χ = 50
Set by hand for both the 1D SRE and 2D REC calculations. No convergence study in χ is reported, so the accuracy of the ground-state approximation is not quantified.
Maximum TCI output MPS bond dimension ξ = 80 for SRE, 40 for REC
Chosen by hand as truncation thresholds. The paper reports no analysis of how the final SRE and REC values depend on ξ, leaving the approximation error of F~ unquantified.

assumptions (3)

domain assumption The LU-based TCI algorithm produces a quasi-optimal MPS approximation of F with a controlled tolerance after O(L d ξ^2) function evaluations.
Invoked in Section 2 as the algorithmic foundation. The proof is in the cited TCI papers, not reproduced here.
domain assumption DMRG as implemented in ITensor yields accurate ground states of the 1D and 2D Ising models as MPSs with χ ≤ 50.
Sections 3 and 4 rely on ITensor ground states. Accuracy is assumed, with no DMRG convergence data shown.
domain assumption At h → 0+ the ferromagnetic ground state is the symmetric GHZ superposition, giving SRE = 0 and REC = 1.
Used in Sections 3 and 4 to anchor the low-field limits. The degeneracy argument is standard and not at issue.

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Cite this review

Pith. "Pith review of Computing Quantum Resources using Tensor Cross Interpolation." pith.science (2026). https://pith.science/paper/4ZG4MQBK

@misc{pith2026250206956,
  author       = {Pith},
  title        = {Pith review of: Computing Quantum Resources using Tensor Cross Interpolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZG4MQBK}},
  note         = {Machine review of arXiv:2502.06956}
}

read the original abstract

Quantum information quantifiers are indispensable tools for analyzing strongly correlated systems. Consequently, developing efficient and robust numerical methods for their computation is crucial. We propose a general procedure based on the family of Tensor Cross Interpolation (TCI) algorithms to address this challenge in a fully general framework, independent of the system or the quantifier under consideration. To substantiate our approach, we compute the non-stabilizerness R\'{e}nyi entropy (SRE) and Relative Entropy of Coherence (REC) considering the 1D and 2D ferromagnetic Ising models with minimal modifications to the numerical procedure. This method not only demonstrates its versatility, but also provides a generic framework for exploring other quantum information quantifiers in complex systems.

Figures

Figures reproduced from arXiv: 2502.06956 by the authors.

**Figure 1.** The Stabilizer Rényi entropy Eq. (1) for the 1D ferromagnetic (FM) Ising model Eq. (2) is presented as a function of the magnetic field for various system sizes. The maximum bond dimension of the input state was constrained to χ ≤ 50, while the maximum bond dimension of the TCI MPS was set to ξ = 80. definition itself. We can compute the SRE by rewriting Eq. (1) as M2(|ψ⟩) = − log2 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

**Figure 2.** Quantum coherence Eq. (3) for the 2D FM Ising model Eq. (4) as a function of the magnetic field for different system size. The maximum bond dimension of the input state was constrained to χ ≤ 50, while the maximum bond dimension of the TCI MPS was set to ξ = 40. 5 Conclusions and outlooks We proposed a general approach for computing quantum resources, based on the sampling of the corresponding quantifier. This sam… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Exploring the Effect of Basis Rotation on NQS Performance
quant-ph 2025-12 reject novelty 6.0 of 10

Basis rotation of the Ising ground state relocates the target in the NQS parameter space and can cause shallow networks to converge to low-energy but wrong wavefunctions.
Resource complexity of Symmetry Protected Topological phases
quant-ph 2025-09 conditional novelty 6.0 of 10

Quantum magic in 1D symmetry-protected topological phases is identical at dual trivial/topological points, and boundary-induced magic differences are non-topological.

Reference graph

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