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All instances / Hubbard

Hubbard square 16x16, U = 8, n = 0.875, t12 (t2 = -0.2)

Record

-0.74411(1) E/site

SBP (symmetric: unspecified)

Rende et al. (2026) · lowest unflagged energy published, a strict variational bound (rules §6)

The Hamiltonian

H = −t1 ∑⟨ij⟩,σ (c†iσcjσ + h.c.) −t2 ∑⟨⟨ij⟩⟩,σ (c†iσcjσ + h.c.) + U ∑i ni↑ni↓

t1 = 1, t2 = -0.2, U = 8, N↑ = N↓ = 112 on 256 sites.

This instance is a t12 variant: a next-nearest-neighbour hopping t2, written as in VarBench's Eq. (S6), so t2 is the t′ of the sources' H = −t ∑⟨ij⟩ − t′ ∑⟨⟨ij⟩⟩ + U ∑ ni↑ni↓. Couplings from the Hamiltonian each source prints (see the rows).

lattice
square
sites
256
boundary
periodic
Nf
112
U
8
t1
1
t2
-0.2
energies
15

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. Where the instance is not solved, the lowest unflagged energy of any group is the record.

Strict variational upper bounds (13)

Where the instance is not solved, the record is the lowest unflagged energy in this group or among the projected and extrapolated ones, with or without an error bar; where it is solved, the exact energy is the record and the lowest of these is the closest challenger.

E/sitedistance to recordσVar(E)V-score methodsourceyear
-0.74411record 0 1.0e-5 n/a n/a SBP (symmetric: unspecified)
preprint
 Rende et al. (2026) 2026
-0.743 1.1e-3 n/a n/a n/a ACE (16 conv layers)
peer reviewed
 Gu et al. (2026) 2026
-0.7408 3.3e-3 1.0e-4 n/a n/a Tensor backflow (pinning field PF3 in the optimisation only, UHF orbitals at U = 4, 9013 steps after its removal, 1 Lanczos step)
preprint
 Liang (2026) 2026
-0.7401 4.0e-3 2.0e-4 n/a n/a Tensor backflow (pinning field PF3 in the optimisation only, UHF orbitals at U = 4, 1925 steps after its removal, 1 Lanczos step)
preprint
 Liang (2026) 2026
-0.7400 4.1e-3 1.0e-4 n/a n/a Tensor backflow (pinning field PF2 in the optimisation only, UHF orbitals at U = 4, 2152 steps after its removal, 1 Lanczos step)
preprint
 Liang (2026) 2026
-0.7396 4.5e-3 1.0e-4 n/a n/a Tensor backflow (pinning field PF1 in the optimisation only, UHF orbitals at U = 4, 2484 steps after its removal, 1 Lanczos step)
preprint
 Liang (2026) 2026
-0.7373 6.8e-3 n/a n/a n/a SCALE (1 conv layer)
peer reviewed
 Gu et al. (2026) 2026
-0.7360 8.1e-3 7.0e-4 n/a n/a Tensor backflow (pinning field PF3 in the optimisation only, UHF orbitals at U = 4, 9013 steps after its removal)
preprint
 Liang (2026) 2026
-0.7360 8.1e-3 1.0e-4 n/a n/a HFPS (half-filled stripe + d-wave superconductivity)
preprint
 Roth et al. (2025) 2025
-0.7351 9.0e-3 2.0e-4 n/a n/a Tensor backflow (pinning field PF2 in the optimisation only, UHF orbitals at U = 4, 2152 steps after its removal)
preprint
 Liang (2026) 2026
-0.7349 9.2e-3 7.0e-4 n/a n/a Tensor backflow (pinning field PF2 in the optimisation only, UHF orbitals at U = 0, 2020 steps after its removal)
preprint
 Liang (2026) 2026
-0.7342 9.9e-3 2.0e-4 n/a n/a Tensor backflow (pinning field PF1 in the optimisation only, UHF orbitals at U = 4, 2484 steps after its removal)
preprint
 Liang (2026) 2026
-0.7261 1.8e-2 1.0e-4 n/a n/a HFPS (filled stripe)
preprint
 Roth et al. (2025) 2025

Projected or fixed-node estimates (2)

Variational only within a constraint: fixed-node, constrained-path, GFMC on a trial state. Node- or constraint-dependent, so not cleanly comparable to each other or to strict bounds. Like any unflagged energy, one holds the record where it is the lowest on an unsolved instance.

E/sitedistance to recordσVar(E)V-score methodsourceyear
-0.744 1.1e-4 n/a n/a n/a Fixed-node GFMC (trial state: ACE)
peer reviewed
 Gu et al. (2026) 2026
-0.7405 3.6e-3 n/a n/a n/a Fixed-node GFMC (trial state: SCALE, 1 conv layer)
peer reviewed
 Gu et al. (2026) 2026

How these numbers were read

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

-0.74411(1) — SBP (symmetric: unspecified)

Checked 2026-10-16 · source text of arXiv:2608.12465 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. -0.74411(1)

TABLE I (Ground-state variational energies and variances per site, PBC, t'=-0.2, U=8, delta=1/8), 16x16 / SBP; sources/2608.12465.txt line 298; version read: arXiv v1 (sources/2608.12465.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t' = -0.2, U = 8, doping 1/8, L x L periodic; Table I: energies and variances per site.

-0.744 — Fixed-node GFMC (trial state: ACE)

Checked 2026-10-16 · 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. -0.7440

Table I (16x16 Hubbard, U=8 and delta=1/8 per Sec. III C text; caption prints neither); columns t'=0 OBC | t'=0 PBC | t'=-0.2 PBC; row ACE+GFMC, column t'=-0.2 PBC; sources/2604.25775.txt line 481; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).

-0.743 — ACE (16 conv layers)

Checked 2026-10-16 · 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. -0.7430

Table I (16x16 Hubbard, U=8 and delta=1/8 per Sec. III C text; caption prints neither); columns t'=0 OBC | t'=0 PBC | t'=-0.2 PBC; row ACE, column t'=-0.2 PBC; sources/2604.25775.txt line 480; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).

-0.7408(1) — Tensor backflow (pinning field PF3 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7408(1)

TABLE III, row E_p=1, column 5 (PF3, U_UHF=4, steps=9013); sources/2609.22618.txt line 407; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7405 — Fixed-node GFMC (trial state: SCALE, 1 conv layer)

Checked 2026-10-16 · 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. -0.7405

Table I (16x16 Hubbard, U=8 and delta=1/8 per Sec. III C text; caption prints neither); columns t'=0 OBC | t'=0 PBC | t'=-0.2 PBC; row SCALE+GFMC, column t'=-0.2 PBC; sources/2604.25775.txt line 479; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).

-0.7401(2) — Tensor backflow (pinning field PF3 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7401(2)

TABLE III, row E_p=1, column 4 (PF3, U_UHF=4, steps=1925); sources/2609.22618.txt line 407; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7400(1) — Tensor backflow (pinning field PF2 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7400(1)

TABLE III, row E_p=1, column 2 (PF2, U_UHF=4, steps=2152); sources/2609.22618.txt line 407; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7396(1) — Tensor backflow (pinning field PF1 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7396(1)

TABLE III, row E_p=1, column 1 (PF1, U_UHF=4, steps=2484); sources/2609.22618.txt line 407; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7373 — SCALE (1 conv layer)

Checked 2026-10-16 · 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. -0.7373

Table I (16x16 Hubbard, U=8 and delta=1/8 per Sec. III C text; caption prints neither); columns t'=0 OBC | t'=0 PBC | t'=-0.2 PBC; row SCALE, column t'=-0.2 PBC; sources/2604.25775.txt line 478; version read: arXiv v1 (sources/2604.25775.txt); APS copy not reached (403, 2026-10-16) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, U = 8, doping 1/8 (Sec. III), periodic unless the column says OBC; energies per site. Pinning fields only in pre-training (App. C).

-0.7360(7) — Tensor backflow (pinning field PF3 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7360(7)

TABLE III, row E_p=0, column 5 (PF3, U_UHF=4, steps=9013); sources/2609.22618.txt line 406; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7360(1) — HFPS (half-filled stripe + d-wave superconductivity)

Checked 2026-10-16 · source text of arXiv:2511.07566 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. -0.7360(1)

Table V (App. D 'Supplementary data': 256-site 16x16 cluster, U=8, delta=1/8; columns Filled Stripe | Half-filled stripe + No SC | Half-filled stripe + SC); row t'=-0.2, column Half-filled stripe + SC; sources/2511.07566.txt line 1196; version read: arXiv v1 (sources/2511.07566.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = U sum n_up n_dn - sum t_xx' c+c, t_xx' = t' on next-nearest bonds, periodic; Table V: 16x16, U = 8, doping 1/8, energy per site. The d-wave field Delta_0 enters only the mean-field Hamiltonian of the initial orbitals (Eq. 5, App. D).

-0.7351(2) — Tensor backflow (pinning field PF2 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7351(2)

TABLE III, row E_p=0, column 2 (PF2, U_UHF=4, steps=2152); sources/2609.22618.txt line 406; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7349(7) — Tensor backflow (pinning field PF2 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7349(7)

TABLE III, row E_p=0, column 3 (PF2, U_UHF=0, steps=2020); sources/2609.22618.txt line 406; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7342(2) — Tensor backflow (pinning field PF1 in the optimisation only...

Checked 2026-10-16 · source text of arXiv:2609.22618 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. -0.7342(2)

TABLE III, row E_p=0, column 1 (PF1, U_UHF=4, steps=2484); sources/2609.22618.txt line 406; version read: arXiv v1 (sources/2609.22618.txt); no journal version (Crossref, arXiv abs page) | Eq. (8): H = -t sum_<ij> - t' sum_<<ij>> + U sum n_up n_dn, t = 1, t' = -0.2 (Eq. (S6) form of QMBL's t12 instances, t2 = t'), U = 8, n = 0.875, periodic; energies per site. The 16x16 pinning fields (Eq. 15) act during optimisation only and are removed before the Table III energies are measured.

-0.7261(1) — HFPS (filled stripe)

Checked 2026-10-16 · source text of arXiv:2511.07566 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. -0.7261(1)

Table V (App. D 'Supplementary data': 256-site 16x16 cluster, U=8, delta=1/8; columns Filled Stripe | Half-filled stripe + No SC | Half-filled stripe + SC); row t'=-0.2, column Filled Stripe; sources/2511.07566.txt line 1196; version read: arXiv v1 (sources/2511.07566.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = U sum n_up n_dn - sum t_xx' c+c, t_xx' = t' on next-nearest bonds, periodic; Table V: 16x16, U = 8, doping 1/8, energy per site. The d-wave field Delta_0 enters only the mean-field Hamiltonian of the initial orbitals (Eq. 5, App. D).

Coverage

Newest published number here: 2026.

This instance as data

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