Every published energy
Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (6)
Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.
Projected or fixed-node estimates (2)
Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Each depends on its node or constraint, so two of them are not cleanly comparable to each other or to strict bounds.
How these numbers were read
Where each number below was read, and how it was checked.
-4.01700(9), AFQMC
Checked 2026-10-15 · published PDF Science 386, 296 (2024), sources/journal/science386-296.pdf, sha256 80f8b792, and Supplementary Materials, sources/journal/science386-296-sm.pdf, sha256 4539b5df (node fetch: Cloudflare 403; fetched in the automation browser after the article page loaded); arXiv:2302.04919v2 compared; the number itself from the VarBench dataset v1.1.0 = ref. [63] of the paper (Zenodo 10.5281/zenodo.13377360, the GitHub release of tag v1.1.0, commit 0064b70; the Zenodo API timed out from this host, so the tag in varbench-history was read)
Reported as. -257.088 | 0.006 (total energy, units of t)
8 x 8, 28 + 28, U = -8. Science SM S3 E 1-2 (journal version, same text as arXiv v2 Suppl. S3 E): 'For sign-problem-free systems, e.g., the repulsive Hubbard model (U > 0) at half-filling or the spin-balanced attractive Hubbard model (U < 0) [135, 147], numerically exact results are given by AFQMC using a generalized Metropolis algorithm with force bias [30, 148]'; 'For all systems computed in this work, we have either extrapolated Delta tau to zero or set it to a fixed value (typically Delta tau = 0.01) and verified that the Trotter error is well within our statistical error.' Hamiltonian (S5) H = -t sum c+c + U sum n_up n_dn, no shift, t = 1, N_up = N_dn. Neither the Science main text nor the SM prints any of these energies (searched both PDFs for the stored digits); the dataset is the printing source. Exact (stochastic) with sigma = AFQMC error bar; peer-reviewed venue. Not a duplicate: the nearby rows (-257.0816 Pfaffian, -257.0752 ACE+GFMC, -257.0688 Det-PH/SCALE+GFMC) are other papers' own results, all above this exact (stochastic) value.
-4.0169(1), VMC (transformer BF)
Checked 2026-09-15 · source text of arXiv:2604.21978 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. −4.0169(1)
Table 2 | same instance as above; Pfaffian ansatz, no symmetry projection | variance reported as 0.0014, not stored: convention not verified
-4.0168, FN-GFMC (ACE)
Checked 2026-09-15 · source text of arXiv:2604.25775 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. -4.0168
TABLE IV row 'ACE+GFMC', column '8x8,U=-8' (source ~line 763): 'ACE+GFMC -2.0339 -4.0168 -2.0407 -4.0161' | Same as ACE row above (direct per-site, Nf=28 confirmed); closest own value to the quoted exact DQMC energy.
-4.0167, FN-GFMC (SCALE)
Checked 2026-09-15 · source text of arXiv:2604.25775 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. -4.0167
TABLE IV row 'SCALE+GFMC', column '8x8,U=-8' (source ~line 761): 'SCALE+GFMC -2.0338 -4.0167 -2.0406 -4.0158' | Same as ACE row above (direct per-site, Nf=28 confirmed).
-4.0167(1), VMC (transformer BF)
Checked 2026-09-15 · source text of arXiv:2604.21978 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. −4.0167(1)
Table 2 | same instance as above; Det-PH = particle-hole determinant ansatz, no symmetry projection | variance reported as 0.0024, not stored: convention not verified
-4.0165, VMC (ACE)
Checked 2026-09-15 · source text of arXiv:2604.25775 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. -4.0165
TABLE IV row 'ACE', column '8x8,U=-8' (source ~line 762): 'ACE -2.0338 -4.0165 -2.0405 -4.0159' | Direct per-site fermion energy, no rescaling; Nf=28 (1/8 doping of N=64) matches this instance; agrees with quoted DQMC -4.017(1) to within ~0.012%.
-4.0161, VMC (SCALE)
Checked 2026-09-15 · source text of arXiv:2604.25775 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. -4.0161
TABLE IV row 'SCALE', column '8x8,U=-8' (source ~line 760): 'SCALE -2.0337 -4.0161 -2.0404 -4.0141' | Same as ACE row above (direct per-site, Nf=28 confirmed).
-4.0152(1), VMC (transformer BF)
Checked 2026-09-15 · source text of arXiv:2604.21978 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. −4.0152(1)
Table 2 | same instance as above; T = translational symmetry restored | variance reported as 0.017, not stored: convention not verified
-3.9950(1), VMC (transformer BF)
Checked 2026-09-15 · source text of arXiv:2604.21978 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. −3.9950(1)
Table 2 | 8x8=64 sites (P=PBC); t'=0 ('pure Hubbard model'); attractive U=-8.0; hole doping 1/8 => Nf=28 per spin; catalogue exact_per_site -4.017 for this instance matches the paper's own quoted DQMC value in the same table | variance reported as 0.051, not stored: convention not verified