Rows added after the VarBench import record where the number came from.
-0.5262(5) — AFQMC, sign-problem-free at half filling (tau = 0.01...
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 = -33.68(3) (total), 8 x 8, 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 8 x 8, column U = 8 PBC. Half filling: N_up = N_dn = 32. 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.
-0.5258200 ○ — Transformer backflow + MARCH optimizer
Checked 2026-09-15 · arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription
Reported as. -0.52582 per site, no error bar
Supplementary Table S1 of arXiv:2507.02644, "Benchmark energy in pure Hubbard model at half-filling with PBC", column 8 x 8, row "NQS"; the AFQMC reference in the same table is -0.5262(5). Same method string as this paper's 16 x 4 row. No error bar, so it is eligible for nothing (RULES.md 6). It sits 1.6e-4 per site below the stored AFQMC exact row -0.5256563(78) - see the defect flag. Read from the arXiv supplement; the Nat. Commun. version was not compared.