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 (2)
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 (5)
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.
How these numbers were read
Where each number below was read, and how it was checked.
-0.5298(6), AFQMC
Checked 2026-09-15 · arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription
Reported as. E = -8.476(9) (total), 4 x 4, U = 8, PBC
Table IV, "Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit." Row 4 x 4, column U = 8 PBC. Half filling: N_up = N_dn = 8. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.
-0.5293047, ED
Checked 2026-09-29 · ed_full2.py (code B: plain Lanczos, Hubbard kernel with fixed N_up = N_dn = 8, 165,636,900 states, true residual 5.2e-11 at U = 4 and 3.7e-11 at U = 8) and hub_fact.py (code A: the factorised matvec T x 1 + 1 x T + U D of checks/hubbard-u-labels/ed_check.py with ARPACK, residual 9.8e-12 and 5.7e-12), an AMD EPYC 9654 server, 2026-09-29
Reported as. -8.468875014199401 (ed_full2.py) | -8.468875014196898 (hub_fact.py)
Total energy, t = 1, H = -t sum (c+c + h.c.) + U sum n_up n_dn on the periodic 4x4 lattice (32 bonds); E/N = -0.529304688. Replaces the quoted exact row -8.46896 (-0.52931 x 16), which lay below this ground state: removed under RULES.md 11 (removals.mjs); the producer prints -8.46888 (arXiv:1207.4847 Table IV), this energy to its printed digits. Carried to 10 decimals, the agreement of the two codes.
-0.5292(3), VMC (PITQS)
Checked 2026-09-15 · source text of arXiv:2602.03031 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. -0.52919(35)
Table 7, row 'PITQS', column E(U/t=8): -0.52919(35), N_p=4,752. | Same Table 7, same convention as ED row above, no rescale.
-0.5291, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. -0.5291
Table II, row '4 × 4', column 'U = 8, K = 2' | Consistent with instance exact_per_site -0.52975 (own value lies above it, as expected variationally).
-0.5291(3), VMC (TQS)
Checked 2026-09-15 · source text of arXiv:2602.03031 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. -0.52909(25)
Table 7, row 'TQS', column E(U/t=8): -0.52909(25), N_p=11,424. | Same Table 7, same convention as ED row above, no rescale.
-0.5281, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. -0.5281
Table II, row '4 × 4', column 'U = 8, K = 1' | Consistent with instance exact_per_site -0.52975 (own value lies above it, as expected variationally).
-0.4898, HF
Checked 2026-09-15 · source text of arXiv:2606.00924 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. -0.4898
Table II, row '4 × 4', column 'U = 8, K = 0 (HF)' | Same row's quoted QMC value (-0.5298(1)) matches instance exact_per_site -0.52975 to displayed precision.