QMBLthe quantum many-body leaderboardbeta

All instances / Hubbard

Hubbard square 8x8, U = 8, n = 0.875, t12 (t2 = -0.2)

Record

-0.7540(4) E/site

Transformer backflow (Pfaffian; extrapolated: variance → 0 (linear))

Viteritti et al. (2026) · lowest unflagged energy published, an extrapolation, not a 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↓ = 28 on 64 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
64
boundary
periodic
Nf
28
U
8
t1
1
t2
-0.2
energies
19

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 (17)

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
-0.75058 3.4e-3 1.0e-5 n/a n/a Transformer backflow (Pfaffian; projected: translations, C4)
preprint
 Viteritti et al. (2026) 2026
-0.75026 3.7e-3 1.0e-5 n/a n/a Transformer backflow (particle-hole transformed determinant; projected: translations, C4)
preprint
 Viteritti et al. (2026) 2026
-0.74993 4.1e-3 1.0e-5 n/a n/a Transformer backflow (Slater determinant; projected: translations, C4)
preprint
 Viteritti et al. (2026) 2026
-0.74952 4.5e-3 1.0e-5 n/a n/a Transformer backflow (Pfaffian; projected: translations)
preprint
 Viteritti et al. (2026) 2026
-0.74934 4.7e-3 1.0e-5 n/a n/a Transformer backflow (particle-hole transformed determinant; projected: translations)
preprint
 Viteritti et al. (2026) 2026
-0.74834 5.7e-3 1.0e-5 n/a n/a Transformer backflow (Slater determinant; projected: translations)
preprint
 Viteritti et al. (2026) 2026
-0.74705 7.0e-3 1.0e-5 n/a n/a Transformer backflow (Pfaffian)
preprint
 Viteritti et al. (2026) 2026
-0.74649 7.5e-3 1.0e-5 n/a n/a Transformer backflow (particle-hole transformed determinant)
preprint
 Viteritti et al. (2026) 2026
-0.74512 8.9e-3 1.0e-5 n/a n/a Transformer backflow (Slater determinant)
preprint
 Viteritti et al. (2026) 2026
-0.7441 9.9e-3 8.0e-4 n/a n/a Tensor backflow (1 Lanczos step; projected: translations, C4)
preprint
 Liang (2026) 2026
-0.7432 1.1e-2 8.0e-4 n/a n/a Tensor backflow (1 Lanczos step; projected: translations)
preprint
 Liang (2026) 2026
-0.7431 1.1e-2 8.0e-4 n/a n/a Tensor backflow (1 Lanczos step; projected: C4)
preprint
 Liang (2026) 2026
-0.7424 1.2e-2 3.0e-4 n/a n/a Tensor backflow (1 Lanczos step)
preprint
 Liang (2026) 2026
-0.7395 1.4e-2 4.0e-4 n/a n/a Tensor backflow (projected: translations, C4)
preprint
 Liang (2026) 2026
-0.7384 1.6e-2 4.0e-4 n/a n/a Tensor backflow (projected: translations)
preprint
 Liang (2026) 2026
-0.7383 1.6e-2 4.0e-4 n/a n/a Tensor backflow (projected: C4)
preprint
 Liang (2026) 2026
-0.7374 1.7e-2 4.0e-4 n/a n/a Tensor backflow
preprint
 Liang (2026) 2026

Extrapolations (2)

Zero-variance, bond-dimension or Trotter-error extrapolations. Not a bound: no ansatz reached the number itself. 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
-0.7540record 0 4.0e-4 n/a n/a Transformer backflow (Pfaffian; extrapolated: variance → 0 (linear))
preprint
 Viteritti et al. (2026) 2026
-0.7455 8.5e-3 n/a n/a n/a Tensor backflow (extrapolated: variance → 0 (linear))
preprint
 Liang (2026) 2026

How these numbers were read

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

-0.7540(4) — Transformer backflow (Pfaffian; extrapolated: variance → 0...

Checked 2026-10-16 · 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. -0.7540(4)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row 'Zero-Var.Extr.'; sources/2604.21978.txt line 199; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page); the table prints -0.7540(4), the text -0.7540(1) three times; the table's error bar is kept | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.75058(1) — Transformer backflow (Pfaffian; projected: translations, C4)

Checked 2026-10-16 · 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. -0.75058(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Pfaffian, Symm T+R; sources/2604.21978.txt line 198; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.75026(1) — Transformer backflow (particle-hole transformed...

Checked 2026-10-16 · 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. -0.75026(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det-PH, Symm T+R; sources/2604.21978.txt line 195; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.74993(1) — Transformer backflow (Slater determinant; projected:...

Checked 2026-10-16 · 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. -0.74993(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det, Symm T+R; sources/2604.21978.txt line 192; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.74952(1) — Transformer backflow (Pfaffian; projected: translations)

Checked 2026-10-16 · 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. -0.74952(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Pfaffian, Symm T; sources/2604.21978.txt line 197; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.74934(1) — Transformer backflow (particle-hole transformed...

Checked 2026-10-16 · 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. -0.74934(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det-PH, Symm T; sources/2604.21978.txt line 194; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.74834(1) — Transformer backflow (Slater determinant; projected:...

Checked 2026-10-16 · 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. -0.74834(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det, Symm T; sources/2604.21978.txt line 191; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.74705(1) — Transformer backflow (Pfaffian)

Checked 2026-10-16 · 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. -0.74705(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Pfaffian, Symm -; sources/2604.21978.txt line 196; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.74649(1) — Transformer backflow (particle-hole transformed determinant)

Checked 2026-10-16 · 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. -0.74649(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det-PH, Symm -; sources/2604.21978.txt line 193; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.7455 — Tensor backflow (extrapolated: variance → 0 (linear))

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. -0.7455

Section III.A text after Fig. 2 discussion (also Fig. 2 caption); sources/2609.22618.txt line 293; 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.

-0.74512(1) — Transformer backflow (Slater determinant)

Checked 2026-10-16 · 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. -0.74512(1)

Table I (8x8, PBC, U=8, t'=-0.2, delta=1/8); columns Ansatz|Symm|Energy|Variance|Rel.Error; row Det, Symm -; sources/2604.21978.txt line 190; version read: arXiv v1 (sources/2604.21978.txt); no journal version (Crossref, arXiv abs page) | Eq. (1): H = -sum t_rr' (c+c + h.c.) + U sum n_up n_dn, t_rr' = t' on next-nearest bonds, t' = -0.2, U = 8, doping 1/8, 8x8 periodic; energies per site.

-0.7441(8) — Tensor backflow (1 Lanczos step; projected: translations...

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. -0.7441(8)

TABLE II, row E_p=1, column 'T+R'; sources/2609.22618.txt line 201; 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.

-0.7432(8) — Tensor backflow (1 Lanczos step; projected: translations)

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. -0.7432(8)

TABLE II, row E_p=1, column 'T'; sources/2609.22618.txt line 201; 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.

-0.7431(8) — Tensor backflow (1 Lanczos step; projected: C4)

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. -0.7431(8)

TABLE II, row E_p=1, column 'R'; sources/2609.22618.txt line 201; 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.

-0.7424(3) — Tensor backflow (1 Lanczos step)

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. -0.7424(3)

TABLE II, row E_p=1, column 'No Sym'; sources/2609.22618.txt line 201; 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.

-0.7395(4) — Tensor backflow (projected: translations, C4)

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. -0.7395(4)

TABLE II, row E_p=0, column 'T+R'; sources/2609.22618.txt line 200; 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.

-0.7384(4) — Tensor backflow (projected: translations)

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. -0.7384(4)

TABLE II, row E_p=0, column 'T'; sources/2609.22618.txt line 200; 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.

-0.7383(4) — Tensor backflow (projected: C4)

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. -0.7383(4)

TABLE II, row E_p=0, column 'R'; sources/2609.22618.txt line 200; 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.

-0.7374(4) — Tensor backflow

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. -0.7374(4)

TABLE II, row E_p=0, column 'No Sym'; sources/2609.22618.txt line 200; 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.

Coverage

Newest published number here: 2026.

This instance as data

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