QMBLthe quantum many-body leaderboard

All instances / Hubbard

Hubbard square 16x16, U = 8, n = 0.875

Hubbard/square_256_P_112_8

Record

-0.75865(3) E/site

VAFQMC stripe length 8

Sorella (2023) · lowest eligible strict variational bound (rules §6)

The Hamiltonian

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

t = 1, U = 8, N↑ = N↓ = 112 on 256 sites.

lattice
square
sites
256
boundary
periodic
Nf
112
U
8
energies
13

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

Every published energy

Grouped by what the number is. Ranking happens only inside the first group.

Strict variational upper bounds (9)

The record is the lowest eligible energy in this group, and only this group.

E/siteσVar(E)V-score methodsourceyear
-0.75865(3)record 2.7e-5 1.50e+0 9.8e-4 VAFQMC stripe length 8 Sorella (2023) 2023
-0.7573000 ○ n/a n/a n/a ACE (16 conv layers), no symmetry projection
preprint ineligible: sampled, no error bar
 Gu et al. (2026) 2026
-0.7563000 ○ n/a n/a n/a Transformer backflow
quoted from another paper preprint ineligible: sampled, no error bar
 Gu et al. (2026) 2026
-0.7552000 ○ n/a n/a n/a Tensor-Backflow + Lanczos
quoted from another paper preprint ineligible: sampled, no error bar
 Gu et al. (2026) 2026
-0.7529000 ○ n/a n/a n/a SCALE (1 conv layer), no symmetry projection
preprint ineligible: sampled, no error bar
 Gu et al. (2026) 2026
-0.7515(1) 1.0e-4 n/a n/a HFPS + symmetry projection
quoted from another paper preprint
 Gu et al. (2026) 2026
-0.7509000 ○ n/a n/a n/a Tensor-Backflow
quoted from another paper preprint ineligible: sampled, no error bar
 Gu et al. (2026) 2026
-0.74885(2) 1.6e-5 1.52e+1 1.0e-2 VMC with stripe of wavelength 8 (+Jastrow and backflow)
VarBench reference run
 run script, no paper cited n/a
-0.74395(3) 3.1e-5 2.66e+1 1.8e-2 VMC with uniform BCS pairing (+Jastrow and backflow)
VarBench reference run
 run script, no paper cited n/a

Projected or fixed-node estimates (4)

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 the group above, and never the record.

E/siteσVar(E)V-score methodsourceyear
-0.7583000 ○ n/a n/a n/a ACE (16 conv layers) trial state + fixed-node GFMC
preprint
 Gu et al. (2026) 2026
-0.7560000 ○ n/a n/a n/a SCALE (1 conv layer) trial state + fixed-node GFMC
preprint
 Gu et al. (2026) 2026
-0.75438(4) 3.9e-5 n/a n/a FN on the state above
VarBench reference run
 run script, no paper cited n/a
-0.74991(3) 2.7e-5 n/a n/a FN on the state above
VarBench reference run
 run script, no paper cited n/a

How these numbers were read

Rows added after the VarBench import record where the number came from.

-0.7583000 ○ — ACE (16 conv layers) trial state + fixed-node GFMC

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7583 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7573000 ○ — ACE (16 conv layers), no symmetry projection

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7573 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7563000 ○ — Transformer backflow

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7563 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7560000 ○ — SCALE (1 conv layer) trial state + fixed-node GFMC

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.756 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7552000 ○ — Tensor-Backflow + Lanczos

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7552 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7529000 ○ — SCALE (1 conv layer), no symmetry projection

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7529 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7515(1) — HFPS + symmetry projection

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7515 (+/- 0.0001) per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

-0.7509000 ○ — Tensor-Backflow

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7509 per site

Read directly from the Table 1 HTML of arXiv:2604.25775, column t'=0 PBC (16x16, U=8, delta=1/8). ACE Table 1 energies are WITHOUT symmetry projection (Sec. III.C.2), so they are not the ansatz floor.

Coverage

Newest published number here: 2026. 8 rows added from the 2026-09 literature sweep, on top of the VarBench snapshot of 2024-10-22.

This instance as data

JSON · source file on GitHub · open an issue about this instance