How these numbers were read
Where each number below was read, and how it was checked.
-3.1739018(2), VMC (2D RNN)
Checked 2026-09-29 · arXiv:2002.02973v4 PDF (v1 and v3 identical in this table), pypdf layout + plain; VarBench history; VarBench run script and program
Reported as. -3.1739018(2) (E/N, Pauli, J = 1)
Record row. Instance, convention and class agree: 12 x 12 open, h = 3, Pauli TFIM (Eq. 14), single-layer 2D vanilla pRNN with 100 memory units, final estimate from 2e6 exact autoregressive samples, variational, produced by this paper (PixelCNN -3.1739005(5) is the only quoted column). Finer digits, sigma and variance 0.001429 are first author M. Hibat-Allah's upload (75dddcc, 2021-12-13), which replaced his own mislabelled first upload of the 1DRNN cell (5ec0c50, 2021-11-16: -3.1738969 as '2DRNN'). sqrt(0.001429 / 2e6) = 2.67e-5 = stored sigma, so sigma is the SEM of the final 2e6-sample estimate. Variance not printed in the paper (C7). Not a duplicate: the other row of this paper on the instance is the 1DRNN cell -457.0411536 (-3.1738969(5)). No exact reference exists at 144 sites; the lowest variational row stands.
-3.1739009, DMRG
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/square_144_O_3/dmrg.sh -> programs/dmrg_itensors/ising_2d.jl, common.jl, args.jl (ITensors.jl, Manifest pinned)
Reported as. VarBench TFIsing/square_144_O_3.md: Energy -457.0417309784141 | Sigma (empty) | Energy Variance 0.009210997202899307 | DOF 144 | Einf 0 | Method "DMRG (bond dimension = 1024)"
Script builds the instance (Pauli TFIM +sum ZZ + h sum X on the open 12x12 square, h = 3, equivalent to -sum ZZ - h sum X on this bipartite cluster; no symmetry or magnetisation constraint). Bond dimension = max_B = 1024. Energy is the last sweep's, variational, no extrapolation; variance <H^2> - E^2 of the final MPS (ising_*.jl lines 63-65). No output log (config scope). Value since af1d48f (2022-09-01); script link attached 2024-07-19 (f88f9ba, cd362c4). Not the record: the 2DRNN row -457.041866(27) (other group) is 1.35e-4 lower. Snake-ordered open 12x12, 264 bonds.
-3.1738997, DMRG
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -457.0416 (primary: -3.17389966 (E/N, Hibat-Allah et al. Table II, DMRG, h = 3))
Table 4, row DMRG (not exact); sources/2606.25600.txt line 529; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); the paper's own DMRG, labelled 'DMRG (not exact)'; no bond dimension, code or reference given; qmbl-verify 2026-10-19 SP7-heg3d-tfising#59: reference "H. L. Dao, Two-dimensional Hyperbolic RNN Neural Quantum State, arXiv:2606.25600 (v1, 2026-06-24; v2, 2026-10-10)" -> "M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko & J. Carrasquilla, Recurrent neural network wave functions, Phys. Rev. Research 2, 023358 (2020), arXiv:2002.02973, doi:10.1103/PhysRevResearch.2.023358" (Dao quotes this DMRG energy; it is Hibat-Allah et al.'s. Hibat-Allah et al. arXiv:2002.02973 (qmbl sources/2002.02973.txt line 1734-1736), Table II (App. F): 'h E/N ... DMRG' row h = 3: -3.17389966; c); qmbl-verify 2026-10-19 SP7-heg3d-tfising#60: provenance "primary" -> "secondary" (Quoted from arXiv:2002.02973 Table II (see the reference correction on this row).); qmbl-verify 2026-10-19 SP7-heg3d-tfising#61: energy -457.0416 -> -457.04155104 (Dao prints the primary value rounded to 4 decimals of the total (-457.0416); the primary prints 8 decimals per site. Carry the primary's digits (stored -457.0416 differs by -4.90e-5).); qmbl-verify 2026-10-19 SP7-heg3d-tfising#62: method "DMRG (not exact)" -> "DMRG (bond dimension χ = 1024)" (Method string of the primary source; the label 'DMRG (not exact)' is Dao v1's table label only (v2 prints 'DMRG').) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.1738969(5), VMC (RNN)
Checked 2026-09-15 · source text of arXiv:2002.02973 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -3.1738969(5)
Table II, row h=3, column 1DRNN: -3.1738969(5) | Same table's DMRG column at h=3 (-3.17389966) and 2DRNN column (-3.1739018(2), which exactly equals the instance's recorded best_per_site/exact_per_site) confirm the printed E/N values are already in the instance's per-site Pauli convention; no rescale needed.
-3.173799(4), VMC (RBM)
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/square_144_O_3/vmc_rbm.sh -> programs/vmc_netket/vmc.py, args_parser.py (NetKet 3.13.0 per requirements.txt)
Reported as. VarBench TFIsing/square_144_O_3.md: Energy -457.02705 | Sigma 0.00060 | Energy Variance 0.36743321 | DOF 144 | Einf 0 | Method "RBM (alpha = 1)"
Script builds the instance (open square of 144 sites, h = 3, Pauli TFIM) with nk.models.RBM(alpha = 1, log_cosh, float32; vmc.py lines 125-140); above the best bound -457.041866(27). sigma check: sqrt(Var/2^20) = sqrt(0.36743321/1048576) = 5.92e-4 vs stored 0.0006 (vmc.py lines 268-270: final estimate with estim_size = 2^20 samples); variance is Var(H) of the total energy. The linked script was attached on 2024-07-30 (fdb4bfb Jastrow / 5047bfd RBM) to a number already in VarBench (10f5c12 (2022-07-11) as -457.027, digits extended in af1d48f); vmc.py uses nk.operator.IsingJax (vmc.py line 78), so it is a later reimplementation of the baseline that documents the ansatz and protocol, not the output of the run that produced this number (no log in the methods repo). Value unchanged since 10f5c12; no value was changed and restored.
-3.173711(5), VMC (Jastrow)
Checked 2026-09-29 · varbench/methods ed31bb0 scripts/TFIsing/square_144_O_3/vmc_jastrow.sh -> programs/vmc_netket/vmc.py, args_parser.py (NetKet 3.13.0 per requirements.txt)
Reported as. VarBench TFIsing/square_144_O_3.md: Energy -457.01444 | Sigma 0.00073 | Energy Variance 0.54852065 | DOF 144 | Einf 0 | Method "Jastrow baseline"
Script builds the instance (open square of 144 sites, h = 3, Pauli TFIM) with nk.models.Jastrow (vmc.py lines 118-124); above the best bound -457.041866(27). sigma check: sqrt(Var/2^20) = sqrt(0.54852065/1048576) = 7.23e-4 vs stored 0.00073 (vmc.py lines 268-270: final estimate with estim_size = 2^20 samples); variance is Var(H) of the total energy. The linked script was attached on 2024-07-30 (fdb4bfb Jastrow / 5047bfd RBM) to a number already in VarBench (af1d48f (2022-09-01)); vmc.py uses nk.operator.IsingJax (vmc.py line 78), so it is a later reimplementation of the baseline that documents the ansatz and protocol, not the output of the run that produced this number (no log in the methods repo). Value unchanged since af1d48f; no value was changed and restored.
-3.17338(6), VMC (2D RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -456.9670 (0.0088)
Table 4, row Lorentz 2DRNN Lmax=5.0, column Best, upper line (error in lower line); sources/2606.25600.txt line 526; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0088; qmbl-verify 2026-10-19 SP7-heg3d-tfising#58: sigma 0.008784 -> 0.0088 (The printed error bar is 0.0088 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.000061,), so sigma = round(0.0088/144, 2 s.f.) x 144 = 0.008784. One lo) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.17333(6), VMC (2D RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. −456.9595 (0.0084)
Table 4, row Lorentz 2DRNN Lmax=2.0, column Best, upper line (error in lower line); sources/2606.25600.txt line 523; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0084; qmbl-verify 2026-10-19 SP7-heg3d-tfising#57: sigma 0.008352 -> 0.0084 (The printed error bar is 0.0084 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.000058,), so sigma = round(0.0084/144, 2 s.f.) x 144 = 0.008352. One lo) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.17284(8), VMC (2D RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. −456.8887 (0.0122)
Table 4, row Euclidean 2DRNN , column Best, upper line (error in lower line); sources/2606.25600.txt line 520; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0122; qmbl-verify 2026-10-19 SP7-heg3d-tfising#56: sigma 0.01224 -> 0.0122 (The printed error bar is 0.0122 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.000085,), so sigma = round(0.0122/144, 2 s.f.) x 144 = 0.01224. One loa) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.1542(4), VMC (RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -454.2120 (0.0636)
Table 8, row Poincare GRU, column (N,N)=(12,12); error in the line below; sources/2606.25600.txt line 1045; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0636; qmbl-verify 2026-10-19 SP7-heg3d-tfising#67: sigma 0.06336 -> 0.0636 (The printed error bar is 0.0636 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.00044,), so sigma = round(0.0636/144, 2 s.f.) x 144 = 0.06336. One load) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.1494(5), VMC (RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -453.5092 (0.0691)
Table 8, row Lorentz GRU, column (N,N)=(12,12); error in the line below; sources/2606.25600.txt line 1047; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0691; qmbl-verify 2026-10-19 SP7-heg3d-tfising#68: sigma 0.06912 -> 0.0691 (The printed error bar is 0.0691 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.00048,), so sigma = round(0.0691/144, 2 s.f.) x 144 = 0.06912. One load) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.1411(5), VMC (RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -452.3138 (0.0763)
Table 8, row Euclidean GRU, column (N,N)=(12,12); error in the line below; sources/2606.25600.txt line 1043; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0763; qmbl-verify 2026-10-19 SP7-heg3d-tfising#66: sigma 0.07632 -> 0.0763 (The printed error bar is 0.0763 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.00053,), so sigma = round(0.0763/144, 2 s.f.) x 144 = 0.07632. One load) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.1066(6), VMC (RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -447.3467 (0.0852)
Table 8, row Euclidean RNN, column (N,N)=(12,12); error in the line below; sources/2606.25600.txt line 1037; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0852; qmbl-verify 2026-10-19 SP7-heg3d-tfising#63: sigma 0.08496 -> 0.0852 (The printed error bar is 0.0852 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.00059,), so sigma = round(0.0852/144, 2 s.f.) x 144 = 0.08496. One load) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.0951(7), VMC (RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -445.6974 (0.0944)
Table 8, row Poincare RNN, column (N,N)=(12,12); error in the line below; sources/2606.25600.txt line 1039; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0944; qmbl-verify 2026-10-19 SP7-heg3d-tfising#64: sigma 0.09504 -> 0.0944 (The printed error bar is 0.0944 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.00066,), so sigma = round(0.0944/144, 2 s.f.) x 144 = 0.09504. One load) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.
-3.0936(7), VMC (RNN)
Checked 2026-10-16 · source text of arXiv:2606.25600 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -445.4792 (0.0938)
Table 8, row Lorentz RNN, column (N,N)=(12,12); error in the line below; sources/2606.25600.txt line 1041; version read: arXiv v1 text (sources/2606.25600.txt); every energy string checked identical in v2 (2026-10-10); no journal version (Crossref, arXiv abs page); best over runs with distinct initialisations; the line below the energy prints its standard error, 0.0938; qmbl-verify 2026-10-19 SP7-heg3d-tfising#65: sigma 0.0936 -> 0.0938 (The printed error bar is 0.0938 (total). The loader stored sigma_per_site rounded to two significant digits ("sigma_per_site": 0.00065,), so sigma = round(0.0938/144, 2 s.f.) x 144 = 0.0936. One loade) | Eq. (17): H = -J sum_<ij> sigma^z sigma^z - B_x sum_i sigma^x, Pauli matrices, J = 1, open boundaries in both directions; energies are totals (they scale with N: DMRG -202.5077/64, -316.9770/100, -457.0416/144 = -3.164, -3.170, -3.174 per site); QMBL's h is B_x.