QMBLthe quantum many-body leaderboard

The best published ground-state energies

QMBL is a record book of the state of the art in quantum many-body simulation: one row per published energy, ranked within each Hamiltonian instance, every row citing the paper that produced the number and declaring what kind of quantity it is. Where an instance is solved, the exact energy is the record; everywhere else the best variational bound is. All 341 instances and their 1215 energies →

How far published energies sit from the exact answer, by system size 417 energies on the 229 instances that have an exact ground-state energy, each as its relative distance from that energy; the closer to exact, the higher. Colour is the kind of number; filled marks can hold a record, hollow ones cannot. Variational bound Projected (fixed-node) Extrapolated Listed, cannot hold a record 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 sites relative distance from the exact energy (closer is higher) Distance is |E − E_exact| / |E_exact|, drawn upside down so that a better energy is higher; 34 rows closer than 10⁻⁸ sit on the top line. No line joins the sizes: the instances at one size are different Hamiltonians, and a frontier across them would compare a Hubbard energy with a Heisenberg one. Exact references are exact diagonalization or sign-problem-free QMC; flagged rows are not shown.
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The same distances, one panel per method family Each panel colours one family's energies over all the others in grey and joins the family's best distance at each size. A method that only ever ran at one size is a lone mark. tensor network (82) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 transformer / ViT (20) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 CNN / ResNet (31) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 RNN (54) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 RBM (60) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 mean field (12) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 backflow / Pfaffian (46) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 other NQS (11) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 classic VMC (48) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 AFQMC / GFMC (3) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 VQE / circuit (45) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 sites other (5) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 sites Families are assigned from the method string (views.mjs), first match wins; 'other' is what none of the patterns name. Filled marks can hold a record, hollow ones cannot. Axes as in the figure above: closer to exact is higher.
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Accuracy as the same Hamiltonian grows Six families of instances that differ only in size. Distance is from the exact energy where one exists (sizes in grey), otherwise from the standing record (sizes in orange), whose holder then sits in the band above the plot. One line per method family, through its best energy at each size; a family present at one size only is a lone mark. Closer to the reference is higher. Best at each size One method family Can hold a record Cannot J1-J2 square, J2 = 0.5, periodic (119 energies, 10 sizes) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 16 36 64 100 144 196 324 holds the record transformer / ViT tensor network CNN / ResNet other RBM classic VMC other NQS Heisenberg square, periodic (52 energies, 29 sizes) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 16 36 64 144 256 484 900 1936 4096 9216 RNN transformer / ViT CNN / ResNet classic VMC RBM tensor network Heisenberg triangular, periodic (22 energies, 5 sizes) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 16 36 48 108 144 holds the record tensor network RBM classic VMC Heisenberg kagome, periodic (11 energies, 8 sizes) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 12 18 24 30 42 192 holds the record tensor network classic VMC Heisenberg pyrochlore, periodic (26 energies, 6 sizes) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 32 108 256 432 1024 holds the record RBM tensor network classic VMC Hubbard U = 8, n = 0.875, periodic (64 energies, 5 sizes) 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 32 64 224 288 holds the record CNN / ResNet AFQMC / GFMC transformer / ViT backflow / Pfaffian mean field Sizes are numbers of sites. A family's line joins its lowest distance at each size, whichever paper set it, so a line is a family's reach rather than one calculation. Where the reference is the record, the distances say how far behind the others are and nothing about the record itself.
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The best energies at each cost, instance by instance Every published energy whose paper states what it cost in hours, on the 9 instances with at least two such rows. The line is the frontier: the results nothing beats for less. Colour is the kind of number; filled marks can hold a record, hollow ones cannot. Variational bound Projected Extrapolated Exact Cannot hold a record Frontier Hubbard 4x16, U = 8, n = 0.875 E/site −0.767 −0.766 −0.765 −0.764 −0.763 −0.762 10¹ 10² 10³ 10⁴ record: ACE MLP-NNBF NNBF hours, as reported J1-J2 10×10, J2 = 0.5 E/N (S.S) −0.4978 −0.4976 −0.4974 −0.4972 −0.4970 −0.4968 10⁰ 10¹ 10² 10³ 10⁴ record: CNN-MPS Factored attention fViT hours, as reported J1-J2 6×6, J2 = 0.5 E/N (S.S) −0.5040 −0.5038 −0.5036 −0.5034 −0.5032 −0.5030 10⁰ 10¹ 10² exact: Exact diagonalization Factored attention Decoupled attention hours, as reported Heisenberg triangular 36 E/N (S.S) −0.562 −0.560 −0.558 −0.556 −0.554 10⁰ 10¹ 10² 10³ 10⁴ exact: Exact diagonalization Group CNN hours, as reported J1-J2 16×16, J2 = 0.5 E/N (S.S) −0.4970 −0.4968 −0.4966 −0.4964 −0.4962 −0.4960 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ record: CNN-MPS GCNN hours, as reported J1-J2 16×16, J2 = 0.55 E/N (S.S) −0.4857 −0.4856 −0.4855 −0.4854 −0.4853 −0.4852 −0.4851 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ record: GCNN hours, as reported J1-J2 20×20, J2 = 0.5 E/N (S.S) −0.49685 −0.49680 −0.49675 −0.49670 10³ 10⁴ 10⁵ record: CNN-MPS ViT with symmetry restoration hours, as reported J1-J2 triangular 108, J2 = 0.125 E/N (S.S) −0.5130 −0.5125 −0.5120 −0.5115 10¹ 10² 10³ record: GCNN + Lanczos step hours, as reported J1-J2 triangular 144, J2 = 0.125 E/N (S.S) −0.5125 −0.5120 −0.5115 −0.5110 −0.5105 10¹ 10² 10³ record: GCNN + Lanczos step hours, as reported Circles are GPU-hours, squares CPU core-hours (2 rows); a whisker under a mark means the hours are devices × wall-clock, multiplied here (12 rows) - the stored fields are never derived, GPU generations are not normalised, and a CPU core-hour is not converted into a GPU-hour. Projections and extrapolations are not bounds and never on the frontier. The horizontal line is the instance's record, costed or not.
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How current this is

114 of 341 instances carry a result published in 2025 or 2026; the rest stand where the VarBench compilation left them on 2024-10-22. Every instance page says which of the two it is, because a website that says nothing about its own currency reads as more authoritative than it is.

Separately, 141 sampled variational energies across 84 instances carry no error bar, so they are listed and rank for nothing — and 17 of them sit below their instance's current record. If one of those is your paper, the error bar is the only thing missing.

What a row claims

Every row is one published claim about one Hamiltonian instance: the energy, its error bar, the method, the primary reference, and a declared bound_type saying what the number actually is — a strict variational bound, a projected or fixed-node estimate, a zero-variance extrapolation, or a numerically exact result. An exact result is the record wherever one exists; on every other instance only strict variational bounds compete for it, which is what keeps an extrapolated number from beating a measured one. The rules are in RULES.md, the row format in DATA.md.

Citing QMBL

Cite the dataset by its concept DOI, which always resolves to the latest release:

Tristan Zerweck, QMBL - the Quantum Many-Body Leaderboard, v0.2.0, Zenodo (2026). 10.5281/zenodo.22753734

Individual energies should cite the primary paper named on the row, not this site.