Hubbard square 8x8, U = -8, n = 1

Exact energy and best values (rules §6)

exact (stochastic)	-4.52589(9)	AFQMC Wu et al. (2024) · sign-problem-free QMC, exact within its error bar
best variational	-4.525834	VMC (HFPS) Chen et al. (2026) · the lowest unflagged variational energy, a strict variational bound

Energies as E/site. Classes are different claims and are not ranked against each other.

The Hamiltonian

H = −t ∑_⟨ij⟩,σ (c^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = -8, N_↑ = N_↓ = 32 on 64 sites.

lattice: square
sites: 64
boundary: periodic
Nf: 32
U: -8
energies: 2

Stored as a total energy; quoted here as E/site, which is what the papers report (how the conversion works).

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-4.52589exact (stochastic)	0	9.4e-5	n/a	n/a	AFQMC	Wu et al. (2024)	run scripts	2024

Strict variational upper bounds (1)

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.

E/site	distance to class best	σ	Var(E)	V-score	method	source	code	year	comment
-4.525834best variational	0	n/a	1.47e-3	3.6e-6	VMC (HFPS) peer reviewed	Chen et al. (2026)	uses Quantax	2026

How these numbers were read

Where each number below was read, and how it was checked.

-4.52589(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. -289.657 | 0.006 (total energy, units of t)

8 x 8, half filling, 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 (HFPS -289.653376 is Chen et al.'s own).

-4.525834, VMC (HFPS)

Checked 2026-09-15 · source text of arXiv:2507.10705 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. -4.525834

TABLE III, row U=-8, column 'E/N_site', sigma^2/N_site=2.3e-5 | Slightly above exact_per_site -4.525890625, consistent with a variational bound.

Coverage

Newest published number here: 2026. Literature last checked 2026-09-15 (table harvest of cached arXiv sources (match_tables.mjs --all), every matched paper read for this instance): 3 papers read, 1 of them with an energy on this page.

This instance as data

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