Record
-0.4976939(2)
E/N (S.S)
CNN-MPS (h,D,l)=(32,20,20), Marshall sign transformation
Fan et al. (2026) · lowest eligible strict variational bound
(rules §6)
The Hamiltonian
H = ∑⟨ij⟩ σi·σj + J2 ∑⟨⟨ij⟩⟩ σi·σj
J1 = 1, J2 = 0.5, nearest and next-nearest neighbours, written with Pauli matrices.
- lattice
- square
- sites
- 100
- boundary
- periodic
- J2
- 0.5
- energies
- 15
Stored as a total energy in the convention of
VarBench; quoted here as
E/N (S.S), which is what the papers report
(how the conversion works).
Flagged rows
A flag withholds the record and nothing else: the row stays listed, in rank order.
Rules §10 is how a flag is lifted or upheld.
energy-variance-inconsistent — Holographic Quantum Transformer (HQT), zero-shot 8x8->10x10..., -0.49782(3)
Finding. Claimed 1.3e-4 below the best variational energy (CNN-MPS) while the paper itself calls the number 'statistically consistent with the variational state of the art' - it is lower by ~4x its own stated error bar, so if real it is an unclaimed record. There is NO exact reference at 10x10 (ED reaches ~6x6 for this model), so this is a contested record claim, not a proven error.
Diagnosis. The reported energy is inconsistent with the paper's OWN reported variance. At 8x8 the claimed E/N = -0.5001 beats RBM+PP's -0.4989635 while the reported variance (sigma^2 = 1.4e-3 per site, S.S units) gives a V-score of 5.6e-3 against RBM+PP's 9.81e-4 - 5.7x worse. Energy and variance move together - a state further from an eigenstate cannot be lower in energy - so by the V-score calibration this energy should be ~1.2e-3 higher than claimed.
Ruled out. NOT 'below the exact ground state'. There is no exact reference at 8x8 or 10x10: ED for this model reaches about 6x6, and the paper correctly uses 6x6 ED (-0.5038) as its only exact anchor. Chen & Heyl's -0.497715(9) is a zero-variance extrapolation, not a bound, so a lower variational energy would only mean the extrapolation carries systematic error. The sigma^2 is per site, so the V-score is 5.6e-3 and not a range. Separately, the paper is internally inconsistent about it: Table 1 gives sigma^2 = 1.4e-3 for the 8x8 run while the J2 scan lists 0.0034 at J2 = 0.50, which would make the V-score 1.4e-2 and the contradiction larger. No variance is reported at 10x10 at all, so that row's flag rests only on it being an unclaimed record.
Evidence. arXiv:2607.00398 Tables 1-3; V-scores recomputed from this instance's own rows.
How these numbers were read
Rows added after the VarBench import record where the number came from.
-0.49782(3) — Holographic Quantum Transformer (HQT), zero-shot 8x8->10x10...
Checked 2026-09-13 · arXiv HTML parsed locally, no LLM transcription
Reported as. -0.49782 (+/- 0.00003) per site in S.S units
Abstract: "This zero-shot protocol yields an energy of E/N = -0.49782(3), statistically consistent with the variational state of the art". It is not consistent: it is 1.3e-4 BELOW the best variational energy (CNN-MPS -0.4976939(2)) and 1.05e-4 below the zero-variance extrapolated ground state -0.497715(9), i.e. below the ground state itself, which no variational energy can be. See the defect flag.
-0.497715(9) — ResNet2 MinSR, zero-variance extrapolation
Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription
Reported as. -0.497715 (+/- 0.000009) per site in S.S units
PDF text: "estimate the ground-state energy E_GS/N = -0.497715(9) by zero-variance extrapolation". NOT a variational bound.
-0.4976939(2) — CNN-MPS (h,D,l)=(32,20,20), Marshall sign transformation
Checked 2026-09-13 · arXiv HTML parsed locally, no LLM transcription
Reported as. -0.4976939 (+/- 2e-7) per site in S.S units
Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML. Claim: "the best energy obtained by CNN-MPS is -0.4976939(2) ... which is lower than the best previously reported result". Supersedes Chen & Heyl -0.4976921(4) as the record for this instance.
-0.4976923(2) — T-MPS
Checked 2026-09-13 · arXiv HTML parsed locally, no LLM transcription
Reported as. -0.4976923 (+/- 2e-7) per site in S.S units
Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML.
-0.4976921(4) — ResNet2 (64 conv layers, >1e6 params), MinSR
Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription
Reported as. -0.4976921 (+/- 4e-7) per site in S.S units
PDF text: "to attain the best variational energy E/N = -0.4976921(4)"
-0.4976764(7) — Convolutional transformer wave function (CTWF)
Checked 2026-09-13 · arXiv HTML parsed locally, no LLM transcription
Reported as. -0.4976764 (+/- 7e-7) per site in S.S units
Read from Table 1 of arXiv:2603.14425 (square-lattice J1-J2 at J2/J1=0.5, PBC, E per site in S.S units), parsed from the arXiv HTML. Cross-checked against the CTWF paper's own text.
-0.497634(1) — fViT (vision transformer, 2.7e5 params)
Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription
Reported as. -0.497634 (+/- 0.000001) per site in S.S units
Read from Table 1 of the arXiv HTML of 2505.03466; primary source not checked.
-0.497583(6) — ConvNext (6,3,3)[2,2], 2.6e5 params
Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription
Reported as. -0.497583 (+/- 0.000006) per site in S.S units
Table 1 of arXiv:2505.03466, their own result. Paper states L=10, PBC, S.S per site.