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REVIEW 2 major objections 7 minor 28 references

A classical tensor-network baseline for the Lipkin–Meshkov–Glick model up to 1400 particles shows subspace quantum diagonalization staying accurate far longer than variational eigensolvers on present hardware.

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2026-07-31 03:25 UTC pith:WBPXEOEV

load-bearing objection Solid LMG benchmarking package: large DMRG tables plus a clean hardware VQE-vs-SQD comparison that holds inside the exactly solvable window. the 2 major comments →

arxiv 2607.28570 v1 pith:WBPXEOEV submitted 2026-07-30 quant-ph

Benchmarking Quantum Simulations of the Lipkin-Meshkov-Glick Model Using Large Tensor Networks

Maggie Bao , Rushil Dandamudi , Jerimiah Wright , Joan \'Etude Arrow , Henry Zou , Vardaan Sahgal , Brian J. McDermott This is my paper
classification quant-ph
keywords Lipkin-Meshkov-Glick modelDMRGtensor networksVQEsample-based quantum diagonalizationNISQ benchmarkingDicke statesquantum many-body simulation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper builds a high-accuracy classical reference for ground-state energies of the Lipkin–Meshkov–Glick many-body model by running density-matrix renormalization group calculations out to 1400 particles, plus a phase diagram over particle number and interaction strength. That reference is then used to score two leading near-term quantum algorithms on real hardware: a compressed variational quantum eigensolver and sample-based quantum diagonalization that projects the Hamiltonian into a subspace spanned by measured bitstrings from Dicke-state circuits. Variational optimization stays inside a one-percent error band only near six particles and degrades steadily thereafter, while the subspace method holds sub-percent agreement out to roughly seventeen particles before a fixed shot budget can no longer cover the growing symmetry sector. The result supplies both a public classical benchmark dataset for a nuclear-physics-relevant Hamiltonian and concrete evidence that, on today’s noisy devices, sampling a symmetry-restricted subspace and diagonalizing classically currently balances accuracy, circuit depth, and noise better than pure variational search on this model.

Core claim

Against a DMRG-computed ground-truth energy curve for the Lipkin–Meshkov–Glick Hamiltonian (ε=1, V=1, W=0) that extends to N=1400, variational quantum eigensolver runs on a superconducting processor remain within about one percent error only near six particles, whereas sample-based quantum diagonalization of the same model, using an ensemble of Dicke-state circuits and symmetry-aware post-selection, maintains sub-percent agreement out to roughly seventeen particles before shot-budget limits cause rapid divergence.

What carries the argument

The Density Matrix Renormalization Group (DMRG) representation of the LMG ground state as a matrix-product state, used to produce reference energies that score two NISQ algorithms: compressed VQE and sample-based quantum diagonalization (SQD) that projects the Hamiltonian into the subspace of measured bitstrings from fixed-parity Dicke states.

Load-bearing premise

The claim that the large-N DMRG energies are accurate enough to serve as sole ground truth rests on checks only up to nineteen particles plus smooth scaling and chosen sweep settings, not on an independent certificate once exact diagonalization is impossible.

What would settle it

Recompute selected large-N LMG ground energies (for example N=50, 100, 500 at V=1) with an independent high-precision classical method or substantially tighter DMRG bond-dimension and cutoff schedules; if those energies differ from the published DMRG curve by more than the one-percent success threshold used in the paper, the quantum-method rankings lose their reference.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The released LMG energy tables and phase diagram become a reusable classical yardstick for any future quantum algorithm claiming progress on this Hamiltonian.
  • On present noisy hardware, subspace-projection methods that exploit LMG pair-excitation symmetry can reach larger particle numbers at sub-percent accuracy than compressed variational eigensolvers.
  • SQD accuracy on this model is gated by whether the shot budget can cover the 2^{N-1}-sized symmetry sector; once that coverage fails, error rises sharply rather than gradually.
  • For all-to-all LMG Hamiltonians, classical MPO construction—not the DMRG sweeps—dominates runtime at large N, so classical baselines remain limited by Hamiltonian assembly cost.
  • Improved SQD sampling or shallower Dicke preparations would be required before the quantum subspace approach can push the accurate regime past the present shot-budget wall.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same DMRG-versus-SQD comparison pattern is likely to appear for other collective spin models whose eigenstates live in low-dimensional Dicke or fixed-parity sectors.
  • Once shot budgets or error-mitigated sampling grow enough to cover larger symmetry sectors, SQD could become a practical cross-check against tensor networks precisely in the intermediate-N window where both remain feasible.
  • Hamiltonian libraries aimed at nuclear and condensed-matter applications would benefit from shipping companion large-N tensor-network energy tables as first-class benchmark artifacts alongside the operators themselves.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. The manuscript generates large classical reference datasets for the Lipkin–Meshkov–Glick (LMG) model (W=0) via DMRG on NERSC Perlmutter—ground-state energies for N up to 1400 at (ε,V,W)=(1,1,0) and a phase diagram over N∈[2,100], V∈[0,1]—and uses them to benchmark VQE and Sample-Based Quantum Diagonalization (SQD) on IBM hardware. Against a 1% relative-energy success rule, VQE stays within threshold only near N=6 while SQD reaches roughly N=17–18 before a shot-budget cliff; the authors conclude that subspace-based NISQ methods currently balance accuracy, depth, and noise better than VQE on this model. Supporting material includes exact-diagonalization checks for N≤19, literature-spectrum replication, SPSA/ansatz details, Dicke-state preparation and error mitigation for SQD, and runtime profiling that attributes DMRG wall time mainly to O(N²) MPO construction.

Significance. Application-centered benchmarks that place NISQ algorithms next to strong classical baselines remain scarce; an openly usable LMG energy library at this scale is a concrete contribution for HamLib/HamPerf-style work and for the nuclear many-body community that already uses LMG as a testbed. The side-by-side VQE vs SQD comparison on the same model, with explicit shot-budget and O(N²) circuit-resource analysis for SQD, is timely and gives a clear, falsifiable ranking inside the exactly solvable window. Strengths include public code/dataset pointers, statistical stability checks of DMRG for small N, and transparent reporting that SQD can beat DMRG absolute error for N∈[3,15] when the symmetry sector is fully sampled. If the large-N DMRG energies are adequately certified and hardware naming/figure issues cleaned up, the paper is a useful reference dataset plus a solid NISQ methods comparison rather than a claim of quantum advantage.

major comments (2)
  1. [§4.1.1–4.1.2, Appendix B.2] §4.1.1–4.1.2 and Appendix B.2 certify DMRG only for N≤19 (exact diagonalization, >99.99% agreement, X=100 init stability) plus qualitative literature spectra. The central classical claim—accurate ground states through N=1400 and the phase diagram—rests on a fixed empirical schedule (nsweeps=5, maxdim=[10,20,100,100,200], cutoff=1e-10), smooth thermodynamic curves, and the assumption that modest bond dimension captures the all-to-all LMG ground state. For a dataset marketed as a high-precision benchmark, please add load-bearing convergence evidence at representative large N (e.g. N=100,500,1400): energy vs max bond dimension, discarded weight / truncation error, and if feasible a variance or two-site residual. Without that, large-N energies should be labeled provisional reference values, not unqualified ground truth. Note that the NISQ ranking itself is already supported inside N≤19 (Figs
  2. [§3.2, §4.2, §4.4, Abstract] VQE is run exclusively on the W=0 Gray-encoded compressed Hamiltonian of dimension d=J+1 (§3.2, Eqs. 11–12), whereas SQD samples Dicke sectors of the N-qubit formulation and DMRG builds the full Pauli/MPO form (Eq. 4). The paper states the W=0 restriction once, but the abstract and combined comparison (§4.4) read as a head-to-head on “the LMG model.” Please state explicitly in the abstract, §4.2, and §4.4 that VQE solves the symmetry-reduced problem, report the effective qubit counts, and avoid implying identical resource scalings. The comparative conclusion can stand if framed as method-plus-encoding pairs rather than pure algorithm ranking.
minor comments (7)
  1. [Abstract, §5] Abstract says “IBM Eagle”; Conclusion §5 says “IBM Heron processor.” Align the device name everywhere and specify the backend used for each of VQE and SQD.
  2. [§4.2, Figure 5] Figure 5 caption reads “Iterations to convergence across four problem sizes…” while the surrounding text describes VQE energies vs DMRG. Figure 17 in the appendix appears to be the convergence plot. Fix the Fig. 5 caption and cross-references.
  3. [Abstract] Abstract: “exceedingthatthresholdforallothervalueswhile” — missing spaces (also elsewhere, e.g. “NoisyIntermediate-ScaleQuantum”). Copy-edit for concatenated words.
  4. [§4] Success is defined as “within 1% of the true ground state” (§4) without motivation. A short sentence on why 1% (nuclear phenomenology, prior LMG VQE papers, or hardware noise floor) would help readers interpret Figs. 6, 8, 12.
  5. [§4.3.2, Figure 10] Figure 10 notes unexplained local oscillations in SQD error vs N; either a brief hypothesis (parity sectors, weighted shot allocation, recovery bias) or a clearer “left open” statement would suffice.
  6. [Appendix A, §5] Dataset access is “upon request” in the appendix while GitHub is cited for code [26]. Prefer a stable archival link (Zenodo/Figshare) for the N≤1400 energy tables so the benchmark is fully reproducible without gatekeeping.
  7. [Appendix B.4, Table 1] Table 1: GPU-DMRG sweep times exceed CPU-DMRG (e.g. N=1400: 73s vs 11s). A one-line explanation (transfer overhead, small bond dim, ITensor GPU maturity) would prevent misreading the profiling message.

Circularity Check

0 steps flagged

No significant circularity: DMRG baseline is independent of VQE/SQD; quantum errors are measured against external references.

full rationale

This is a benchmarking paper, not a first-principles derivation of a physical law or a fitted predictive model. The load-bearing classical reference is DMRG/MPS variational minimization of the LMG Hamiltonian (Eqs. 5–9), constructed from the standard quasi-spin form (Eqs. 1–4) and validated against exact sparse diagonalization for N≤19 plus published spectra. VQE and SQD energies are then compared to that external baseline (and to exact diagonalization in the overlapping window N≤19, Figs. 11–12). Nothing in the quantum pipelines defines or fits the DMRG energies, and the comparative claim (SQD reaches ~N=17–18 within 1% while VQE does not) is independently supported inside the exactly solvable regime. Empirical DMRG sweep schedules, Gray encoding, and Dicke post-selection are methodological choices, not predictions that reduce to their inputs by construction. No self-definitional loop, fitted-input-as-prediction, load-bearing self-citation uniqueness theorem, or renaming of a known result appears. Score 0 is the honest finding.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The work sits on standard many-body and NISQ methodology. Load-bearing choices are empirical DMRG truncation schedules, the W=0 compression used for VQE, Dicke-subspace sampling and Hamming-weight recovery for SQD, a fixed shot budget, and a 1% energy-error success threshold. No new physical entities are postulated; free parameters are algorithmic hyperparameters and resource caps, not fitted physical constants.

free parameters (5)
  • DMRG sweep schedule (nsweeps, maxdim ladder, cutoff) = nsweeps=5; maxdim=[10,20,100,100,200]; cutoff=1e-10
    Default nsweeps=5, maxdim=[10,20,100,100,200], cutoff=1e-10 chosen empirically for convergence; large-N energies inherit this truncation policy.
  • Success threshold of 1% relative energy error = 1%
    Binary ‘algorithm solutions are considered successful only if within 1% of the true ground state’ frames headline VQE/SQD particle-number reach.
  • SQD total shot budget and weighted Dicke allocation = ~10^6 total shots; max 8192 shots noted as limiting
    Performance cliff is tied to ~10^6 shots (and 8192-shot mentions elsewhere) versus 2^{N-1} symmetry-sector size; weighted sampling near k≈N/2 is a heuristic.
  • VQE ansatz layer pattern and SPSA settings = SPSA; 8192 shots/eval; selected 3-layer real RY/CNOT-style ansatz
    Three-layer RY/CRY search outcome and SPSA with 8192 shots per expectation fix the VQE baseline quality.
  • Hamiltonian parameters for main dataset (ε,V,W) = ε=1, V=1, W=0 (main); V∈[0,1] step 0.1 for phase diagram
    Primary large-N curve fixes ε=1, V=1, W=0; phase diagram scans V in 0.1 steps with W=0.
axioms (6)
  • domain assumption The LMG Hamiltonian in quasi-spin/Pauli form (Eqs. 1–4) correctly encodes the intended two-level collective model with W=0 for the main study.
    Section 2 adopts the standard Lipkin et al. model and the common W=0 truncation used in prior quantum papers.
  • domain assumption An MPS with the chosen bond-dimension schedule can represent LMG ground states to high accuracy even for all-to-all interactions at large N.
    DMRG is used as ground truth beyond exact-diagonalization reach (N>19); justified by small-N tests and smooth scaling, not a proof.
  • standard math Variational principle: circuit or MPS expectation values upper-bound the true ground energy.
    Stated for VQE (Eq. 10) and implicit in DMRG energy minimization (Eq. 5).
  • domain assumption For W=0, pair excitations yield a block structure (even/odd M) allowing Gray-encoded compression of VQE to dimension ~N/2+1.
    Section 3.2; all VQE results are restricted to this regime by construction.
  • domain assumption LMG eigenstates live in fixed Hamming-weight parity sectors well approximated by superpositions of Dicke states, so ensemble Dicke sampling plus weight recovery yields a faithful SQD subspace.
    Section 3.3 and Appendix D; enables SQD and the post-selection/recovery mitigation.
  • ad hoc to paper NISQ sampling noise can be adequately mitigated for benchmarking by symmetry post-selection and configuration recovery without biasing the energy comparison unfairly.
    Two-layer mitigation in §3.3; authors note recovery can obscure raw device fidelity and report pre-recovery fidelities in Fig. 18.

pith-pipeline@v1.2.0-daily-grok45 · 22126 in / 4289 out tokens · 85224 ms · 2026-07-31T03:25:57.336497+00:00 · methodology

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read the original abstract

As quantum computing matures, it is critical to benchmark its real-world problem solving performance against competitive classical methods, such as tensor networks. In this work, we leverage the Density Matrix Renormalization Group (DMRG) algorithm to compute ground state energies of the Lipkin Meshkov Glick (LMG) model as a comparative benchmark against popular noisy intermediate-scale (NISQ) algorithms like the Variational Quantum Eigensolver (VQE) and Sample-Based Quantum Diagonalization (SQD) method. By running DMRG on the NERSC Perlmutter supercomputer, we provide one of the largest LMG ground state energy datasets in literature, containing accurate ground state energies for systems up to 1400 particles. We compare these results with VQE and SQD implementations on an IBM Eagle quantum computer for comparison. VQE achieved results within 1 percent error for 6 particles, while exceeding that threshold for all other values while SQD extended that range to 17 particles, suggesting that in a noisy intermediate scale quantum era, subspace-based approaches may strike the best balance between accuracy, circuit depth, and noise resilience.

Figures

Figures reproduced from arXiv: 2607.28570 by Brian J. McDermott, Henry Zou, Jerimiah Wright, Joan \'Etude Arrow, Maggie Bao, Rushil Dandamudi, Vardaan Sahgal.

Figure 1
Figure 1. Figure 1: Comparison between DMRG-computed ground-state energies and exact numerical solutions [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relative error of DMRG against exact numer [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ground-state energy of the LMG model com [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: DMRG-computed ground-state energy phase diagram as a function of particle number N and inter￾action strength V . To further examine the behavior of the model across parameter space, we capture a two￾dimensional scan of the ground-state energy over particle number N ∈ [2, 100] and interaction strength V ∈ [0, 1] in step sizes of 0.1 show in Fig￾ure 4. Similar to [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: compares SQD and DMRG ground-state energies at V = 1, W = 0 as a function of sys￾tem size N, with SQD results obtained on the IBM processor using 106 total shots distributed across the Dicke state ensemble. Two distinct regimes are visible. For N < 20, SQD and DMRG agree to within <0.5%, demonstrating near-exact diagonalization when the symmetry sector is sufficiently small. Beyond this thresh￾old, SQD ene… view at source ↗
Figure 6
Figure 6. Figure 6: Relative error of VQE against DMRG baseline [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Relative error in SQD ground-state energy with [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Phase diagram for plotting SQD energy error [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Phase diagram for SQD plotting the ground [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relative error with respect to exact numerical [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Replication of benchmarking plots from Hlatshwayo et al. [ [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Statistical analysis of DMRG ground-state energy estimates across repeated runs ( [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Cumulative mean relative error of DMRG ground-state energies as a function of the number of repeated [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Runtime of the DMRG optimization stage as a function of the maximum bond dimension used in the [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Iterations to convergence across four problem sizes: [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: fidelity between the target Dicke state for given [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Worst-case 2Q gate count of |Dk N ⟩ after transpilation to a IBM heavy-hex device (optimization level 3, best of 20 transpiler seeds), selecting k = ⌊N/2⌋ for each N. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Two-qubit circuit depth of each |Dk N ⟩ after transpilation to a IBM heavy-hex topology (optimization level 3, best of 20 transpiler seeds), with color encoding the optimized 2Q depth per (N, k) pair. Taken together, these results establish that SQD achieves near-exact accuracy for the LMG model in the regime N < 20, where the symmetry sector is fully coverable by available shot budgets and circuit depths… view at source ↗

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