QMBLthe quantum many-body leaderboardbeta

The best published ground-state energies of open problems

QMBL: Open Problems is a record book of the state of the art in quantum many-body simulation for problems with no exact ground-state energy. Scientists and AI systems should use the shared results to build on each other's work more efficiently and advance frontier science.

The best published energies on the open problems, by system size Results are placed by their relative gap to the instance's energy to beat, QMBL's best variational energy, so that different Hamiltonians share one axis. A better energy is lower. Variational bound Projected (fixed-node) Extrapolated Flagged, see the row energy to beat estimates below it 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 sites relative gap to the energy to beat (lower is better) 35 18 27 14 3 3 10 3 4 8 3 7 14 24 5 19 34 17 9 13 37 10 7 15 6 14 19 2 2 6 2 3 4 3 2 2 2 11 4 6 6 4 2 3 5 7 3 3 5 7 4 3 7 19 4 5 5 5 3 9 4 7 13 6 6 21 9 3 12 6 12 4 2 14 7 25 7 7 3 3 22 27 11 24 10 4 5 18 4 3 37 2 2 25 7 11 18 9 7 18 7 5 31 21 7 3 2 9 11 22 2 17 2 18 3 13 5 15 6 6 7 13 2 2 3 19 4 12 5 6 9 10 9 4 7 3 41 instances have no variational energy yet and are not drawn. Height is (E − E_target) / |E_target| on a log scale, so that a better energy is lower. Hovering over a dot lists the result and clicking leads to the table entry and source.

The published energies, by Hamiltonian

periodic

Heisenberg triangular, periodic Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.58 −0.56 −0.54 −0.52 −0.50 −0.48 −0.46 16 36 48 108 144 324 576 900 1764 sites

open

Heisenberg triangular, open Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.52 −0.50 −0.48 −0.46 100 144 196 256 sites
Heisenberg pyrochlore Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 32 108 128 256 432 1024 sites
Heisenberg kagome Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.46 −0.44 −0.42 −0.40 12 18 24 30 36 42 48 108 192 sites

periodic

Heisenberg chain, periodic Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.450 −0.445 −0.440 −0.435 −0.430 20 200 sites

open

Heisenberg chain, open Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.445 −0.440 −0.435 −0.430 20 200 sites
Heisenberg shuriken Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.450 −0.445 −0.440 −0.435 24 96 216 384 sites
−0.8 −0.7 −0.6 −0.5 −0.4 E/N (S.S) 38312315352550101320138942816319314319 400 324 256 196 144 100 64 40 36 16 record at each size: J2
J2 =
J1-J2 square, J2 = 0.05 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.6810 −0.6805 −0.6800 16 sites
J1-J2 square, J2 = 0.1 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.67 −0.66 −0.65 −0.64 −0.63 −0.62 16 36 40 64 100 sites
J1-J2 square, J2 = 0.15 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.6400 −0.6395 −0.6390 16 sites
J1-J2 square, J2 = 0.2 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.63 −0.62 −0.61 −0.60 −0.59 16 36 40 64 100 sites
J1-J2 square, J2 = 0.25 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.6015 −0.6010 −0.6005 16 sites
J1-J2 square, J2 = 0.3 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.59 −0.58 −0.57 −0.56 −0.55 −0.54 16 36 40 64 100 sites
J1-J2 square, J2 = 0.35 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.5665 −0.5660 16 sites
J1-J2 square, J2 = 0.4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.56 −0.54 −0.52 −0.50 16 36 40 64 100 196 324 sites
J1-J2 square, J2 = 0.44 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.5130 −0.5125 −0.5120 −0.5115 100 144 196 256 324 sites
J1-J2 square, J2 = 0.45 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.55 −0.54 −0.53 −0.52 −0.51 −0.50 16 36 64 100 144 196 256 324 sites
J1-J2 square, J2 = 0.46 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.5080 −0.5075 −0.5070 −0.5065 −0.5060 100 144 196 256 324 sites
J1-J2 square, J2 = 0.48 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.504 −0.503 −0.502 −0.501 −0.500 64 100 144 196 256 324 sites
J1-J2 square, J2 = 0.5 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.54 −0.52 −0.50 −0.48 −0.46 16 36 40 64 100 144 196 256 324 400 sites
J1-J2 square, J2 = 0.52 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.495 −0.494 −0.493 −0.492 −0.491 64 144 256 sites
J1-J2 square, J2 = 0.55 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 36 40 64 100 144 196 256 324 sites
J1-J2 square, J2 = 0.6 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.55 −0.50 −0.45 −0.40 16 36 40 64 100 144 196 256 324 sites
J1-J2 square, J2 = 0.65 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.56 −0.54 −0.52 −0.50 −0.48 16 36 40 64 100 sites
J1-J2 square, J2 = 0.7 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.60 −0.55 −0.50 −0.45 −0.40 16 36 40 64 100 sites
J1-J2 square, J2 = 0.75 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.5945 −0.5940 16 sites
J1-J2 square, J2 = 0.8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.65 −0.60 −0.55 −0.50 16 36 40 64 100 sites
J1-J2 square, J2 = 0.85 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.6620 −0.6615 −0.6610 16 sites
J1-J2 square, J2 = 0.9 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.75 −0.70 −0.65 −0.60 −0.55 16 36 40 64 100 sites
J1-J2 square, J2 = 0.95 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.7330 −0.7325 −0.7320 16 sites
J1-J2 square, J2 = 1 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.80 −0.75 −0.70 −0.65 −0.60 16 36 40 64 100 sites
J1-J2 triangular, J2 = 0.125 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (S.S) −0.520 −0.515 −0.510 −0.505 48 108 144 sites
−5 −4 −3 −2 −1 0 E/site periodic, n = 1 periodic, n = 0.875 periodic, t2 = -0.2, n = 0.875, t12 periodic/antiperiodic, n = 1 1636641001441962563311181035 642561026229 6414425640057637 16366410014419625611121116 576 400 256 196 144 100 64 36 16 record at each size: U
U =U =U =U =
Hubbard square, periodic, n = 1, U = -8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −4.530 −4.525 −4.520 64 256 sites
Hubbard square, periodic, n = 1, U = -4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −2.862 −2.860 −2.858 64 256 sites
Hubbard square, periodic, n = 1, U = 2 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −1.18 −1.16 −1.14 −1.12 16 36 64 100 144 196 256 sites
Hubbard square, periodic, n = 1, U = 4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.865 −0.860 −0.855 −0.850 −0.845 16 36 64 100 144 196 256 sites
Hubbard square, periodic, n = 1, U = 6 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.660 −0.658 −0.656 −0.654 16 36 64 100 144 196 256 sites
Hubbard square, periodic, n = 1, U = 8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.54 −0.52 −0.50 −0.48 −0.46 16 36 64 100 144 196 256 sites
Hubbard square, periodic, n = 0.875, U = -8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −4.02 −4.01 −4.00 −3.99 64 256 sites
Hubbard square, periodic, n = 0.875, U = -4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −2.602 −2.600 64 256 sites
Hubbard square, periodic, n = 0.875, U = 4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −1.04 −1.03 −1.02 −1.01 −1.00 64 256 sites
Hubbard square, periodic, n = 0.875, U = 6 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.870 −0.865 −0.860 −0.855 256 sites
Hubbard square, periodic, n = 0.875, U = 8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.77 −0.76 −0.75 −0.74 −0.73 −0.72 64 256 sites
Hubbard square, periodic, t2 = -0.2, n = 0.875, t12, U = 8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.76 −0.75 −0.74 −0.73 −0.72 64 144 256 400 576 sites
Hubbard square, periodic/antiperiodic, n = 1, U = 2 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −1.28 −1.26 −1.24 −1.22 −1.20 −1.18 −1.16 16 36 64 100 144 196 256 sites
Hubbard square, periodic/antiperiodic, n = 1, U = 4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.92 −0.90 −0.88 −0.86 −0.84 16 36 64 100 144 196 256 sites
Hubbard square, periodic/antiperiodic, n = 1, U = 6 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.69 −0.68 −0.67 −0.66 −0.65 16 36 64 100 144 196 256 sites
Hubbard square, periodic/antiperiodic, n = 1, U = 8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.55 −0.54 −0.53 −0.52 −0.51 16 36 64 100 144 196 256 sites

periodic, 4 × L, U = 8, n = 0.875

Hubbard rectangular, periodic, 4 × L, U = 8, n = 0.875 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.80 −0.75 −0.70 −0.65 −0.60 −0.55 32 48 64 sites

periodic, 6 × L, U = 8, n = 0.875

Hubbard rectangular, periodic, 6 × L, U = 8, n = 0.875 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.755 −0.750 −0.745 −0.740 96 192 288 384 sites

periodic, 8 × L, U = 8, n = 0.875

Hubbard rectangular, periodic, 8 × L, U = 8, n = 0.875 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.758 −0.756 −0.754 −0.752 −0.750 −0.748 128 192 256 sites

periodic/open, 4 × L, U = 8, n = 0.875

Hubbard rectangular, periodic/open, 4 × L, U = 8, n = 0.875 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site −0.76 −0.75 −0.74 −0.73 −0.72 32 64 sites
Transverse-field Ising square, open, h = 3 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Pauli) −3.20 −3.15 −3.10 −3.05 64 100 144 400 sites

TB-DMFT

Impurity TB-DMFT, TB-DMFT Variational bound Projected Extrapolated Exact Flagged, see the row Record E (total) −4000 −3000 −2000 −1000 0 1000 9 19 119 309 sites

TB-DMFT-SOC

Impurity TB-DMFT, TB-DMFT-SOC Variational bound Projected Extrapolated Exact Flagged, see the row Record E (total) −4000 −3000 −2000 −1000 0 1000 9 19 119 309 sites
−50 0 50 100 150 E/N (Ha) ζ = 1 ζ = 0 71927111111119211229 31115811117259817211114774111171111923 246 162 128 114 66 54 38 27 19 14 7 record at each size: rs
rs =rs =
Electron gas, 3D sc, ζ = 1, rs = 1 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 1.04 1.06 1.08 1.10 1.12 1.14 7 19 27 sites
Electron gas, 3D sc, ζ = 1, rs = 2 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 0.095 0.100 0.105 0.110 0.115 7 19 27 sites
Electron gas, 3D sc, ζ = 1, rs = 5 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.0675 −0.0670 −0.0665 −0.0660 7 19 27 sites
Electron gas, 3D sc, ζ = 1, rs = 10 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.0532 −0.0530 −0.0528 −0.0526 −0.0524 7 19 27 sites
Electron gas, 3D sc, ζ = 1, rs = 20 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.0323 −0.0322 −0.0321 −0.0320 −0.0319 7 19 sites
Electron gas, 3D sc, ζ = 1, rs = 30 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.02284 −0.02282 −0.02280 27 sites
Electron gas, 3D sc, ζ = 1, rs = 50 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.01470 −0.01465 −0.01460 −0.01455 7 19 27 sites
Electron gas, 3D sc, ζ = 1, rs = 70 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.01076 −0.01075 −0.01074 27 sites
Electron gas, 3D sc, ζ = 1, rs = 90 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.008535 −0.008530 27 sites
Electron gas, 3D sc, ζ = 1, rs = 100 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.00775 −0.00774 −0.00773 −0.00772 −0.00771 7 19 sites
Electron gas, 3D sc, ζ = 0, rs = 0.1 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 106.8 106.9 107.0 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.2 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 25.40 25.42 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.3 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 10.69 10.70 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 5.675 5.680 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.5 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 3.1 3.2 3.3 3.4 3.5 3.6 14 38 54 114 sites
Electron gas, 3D sc, ζ = 0, rs = 0.6 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 2.214 2.216 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.7 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 1.512 1.513 1.514 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 1.069 1.070 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.9 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 0.7745 0.7750 14 sites
Electron gas, 3D sc, ζ = 0, rs = 1 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 0.50 0.52 0.54 0.56 0.58 0.60 0.62 14 38 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 2 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.02 −0.01 0.00 0.01 0.02 0.03 14 38 54 66 114 sites
Electron gas, 3D sc, ζ = 0, rs = 3 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.075 −0.074 −0.073 −0.072 −0.071 −0.070 14 sites
Electron gas, 3D sc, ζ = 0, rs = 4 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.07930 −0.07925 −0.07920 14 sites
Electron gas, 3D sc, ζ = 0, rs = 5 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.09 −0.08 −0.07 −0.06 −0.05 14 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 6 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.07070 −0.07065 14 sites
Electron gas, 3D sc, ζ = 0, rs = 7 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.06515 −0.06510 −0.06505 14 sites
Electron gas, 3D sc, ζ = 0, rs = 8 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.05995 −0.05990 14 sites
Electron gas, 3D sc, ζ = 0, rs = 9 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.05540 −0.05535 14 sites
Electron gas, 3D sc, ζ = 0, rs = 10 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.060 −0.055 −0.050 −0.045 −0.040 −0.035 14 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 15 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.042 −0.040 −0.038 −0.036 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 20 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.035 −0.030 −0.025 −0.020 14 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 30 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.02034 −0.02032 14 sites
Electron gas, 3D sc, ζ = 0, rs = 40 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.01559 −0.01558 14 sites
Electron gas, 3D sc, ζ = 0, rs = 50 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.015 −0.014 −0.013 −0.012 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 60 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.01063 −0.01062 −0.01061 14 sites
Electron gas, 3D sc, ζ = 0, rs = 70 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.009125 −0.009120 −0.009115 14 sites
Electron gas, 3D sc, ζ = 0, rs = 80 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.008045 −0.008040 −0.008035 14 sites
Electron gas, 3D sc, ζ = 0, rs = 90 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.007150 −0.007145 14 sites
Electron gas, 3D sc, ζ = 0, rs = 100 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.0080 −0.0075 −0.0070 −0.0065 −0.0060 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 110 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.00715 −0.00710 −0.00705 −0.00700 128 sites
Electron gas, 3D sc, ζ = 0, rs = 200 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.00408 −0.00406 −0.00404 −0.00402 −0.00400 −0.00398 128 sites
Electron gas, 3D fcc, rs = 2, ζ = 0 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) 0.000 0.005 0.010 54 102 178 226 sites
Electron gas, 2D triangular, rs = 30, ζ = 1 Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) −0.03205 −0.03200 −0.03195 −0.03190 −0.03185 −0.03180 16 91 169 sites
Hubbard: Hamiltonians published at one size Every Hamiltonian of this model with energies published at only one size and no strip to join, grouped where they share lattice, boundary and size; the coupling or filling that differs between them runs along x. Colour is the kind of number, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact Flagged, see the row Record E/site chain, 14 sites, n ≈ 0.5714 −1.0 −0.9 −0.8 −0.7 −0.6 1 1.67 2.78 4.64 7.74 U rectangular 4x16, periodic/open, U = 2 −1.30 −1.25 −1.20 −1.15 0.813 0.875 0.938 1 filling n rectangular 4x16, periodic/open, U = 8 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 0.813 0.938 1 filling n rectangular 6x48, U = 8 −0.85 −0.80 −0.75 −0.70 −0.65 0.833 0.917 filling n rectangular 6x48, U = 12 −0.635 −0.630 −0.625 −0.620 0.875 filling n rectangular 8x48, U = 8, n ≈ 0.9167 −0.686 −0.685 −0.684 −0.683 −0.682 −0.681 rectangular 14x16, U = 8, n = 0.875 −0.7585 −0.7580 square 4x4, n = 0.5 −1.4 −1.2 −1.0 −0.8 −0.6 0.1 1 2 3.59 4 6 7.74 10 U square 4x4, n = 0.625 −1.4 −1.3 −1.2 −1.1 −1.0 2 2.15 3.6 4 6 7.74 10 U square 6x6, U = 4 −1.5 −1.0 −0.5 0.0 0.5 0.667 1.33 filling n square 8x8, n = 0.4375 −1.072 −1.071 4 U square 8x8, n = 0.6875 −1.186 −1.185 −1.184 −1.183 4 U square 8x8, n ≈ 0.7813 −4 −3 −2 −1 0 -8 -4 4 8 U square 8x8, periodic/open, U = 8, n = 1 −0.501 −0.500 −0.499 −0.498 square 10x10, U = 4, n = 0.8 −1.114 −1.112 −1.110 −1.108 square 12x12, U = 4, n ≈ 0.8056 −1.110 −1.108 −1.106 −1.104 −1.102 square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359 −0.83 −0.82 −0.81 −0.80 −0.79 none variant square 16x16, periodic/antiperiodic, U = 8, t2 = -0.25, n ≈ 0.8359 −1 0 1 2 t12 t12, UV1V2 variant square 16x16, U = 4 −1.010 −1.005 −1.000 −0.995 0.891 filling n square 16x16, U = 6 −0.845 −0.840 −0.835 −0.830 −0.825 0.891 filling n square 16x16, U = 8 −0.90 −0.85 −0.80 −0.75 −0.70 −0.65 −0.60 0.828 0.859 0.891 0.914 filling n square 16x16, U = 8, t2 = -0.1, t12 −0.7375 −0.7370 −0.7365 0.875 filling n square 16x16, U = 8, t2 = -0.3, t12 −0.740 −0.735 −0.730 −0.725 −0.720 0.875 filling n square, 50 sites, U = 8, n = 0.84 −0.8235 −0.8230
Transverse-field Ising: Hamiltonians published at one size Every Hamiltonian of this model with energies published at only one size and no strip to join, grouped where they share lattice, boundary and size; the coupling or filling that differs between them runs along x. Colour is the kind of number, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Pauli) chain, 10 sites, open, h = 1 −1.239 −1.238 −1.237 chain, 32 sites, h = 0.5 −1.064 −1.063 −1.062 square 6x6, h = 3 −3.202 −3.200 −3.198 square 12x12, open −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 2 4 h
Electron gas, 3D: Hamiltonians published at one size Every Hamiltonian of this model with energies published at only one size and no strip to join, grouped where they share lattice, boundary and size; the coupling or filling that differs between them runs along x. Colour is the kind of number, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) bcc cell, N = 27, ζ = 1 −20 0 20 40 60 0.2 1 2 3 5 10 30 50 70 90 rs bcc cell, N = 64, rs = 100, ζ = 1 −0.00770 −0.00769 −0.00768 −0.00767 −0.00766 rect-3x3x2 cell, N = 36, ζ = 0 −0.10 −0.08 −0.06 −0.04 −0.02 0.00 5 10 50 86.6 86.8 87 100 rs
Electron gas, 2D: Hamiltonians published at one size Every Hamiltonian of this model with energies published at only one size and no strip to join, grouped where they share lattice, boundary and size; the coupling or filling that differs between them runs along x. Colour is the kind of number, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact Flagged, see the row Record E/N (Ha) square cell, N = 37, rs = 5, ζ = 1 −0.1438 −0.1437 square cell, N = 90, rs = 30, ζ = 0 −0.034 −0.032 −0.030 −0.028 −0.026 −0.024 triangular cell, N = 91, rs = 40, ζ = 1 −0.02446 −0.02444 −0.02442 −0.02440 −0.02438 −0.02436

The best published energies, by cost

Skim through three kinds of cost: compute hours, estimated FLOPs and number of parameters. Papers state compute cost in hours on GPU/CPU on 14 instances. Many more papers state the number of parameters, samples and optimisation steps, and public DMRG run scripts state their sweeps, which imply a count of floating-point operations that we estimate (how). On top, we plot the parameter count alone.

triangular
Heisenberg triangular 324: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.553 −0.552 −0.551 −0.550 −0.549 −0.548 10² 10³ 10⁴ 10⁵ 2D RNN ViT ViT hours, as reported Every mark is GPU-hours.
Heisenberg triangular 576: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.553 −0.552 −0.551 −0.550 −0.549 −0.548 −0.547 10² 10³ 10⁴ 10⁵ 2D RNN ViT ViT hours, as reported Every mark is GPU-hours.
Heisenberg triangular 900: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.553 −0.552 −0.551 −0.550 −0.549 −0.548 −0.547 10² 10³ 10⁴ 10⁵ 2D RNN ViT ViT hours, as reported Every mark is GPU-hours.
Heisenberg triangular 1764: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.5518 −0.5517 −0.5516 −0.5515 −0.5514 10³ 10⁴ 10⁵ ViT ViT hours, as reported Every mark is GPU-hours.
J1-J2 8×8, J2 = 0.5: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.500 −0.495 −0.490 −0.485 −0.480 −0.475 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ RBM (QMBL) RBM (QMBL) DMRG (QMBL) DMRG (QMBL) DMRG (QMBL) ViT (QMBL) GCNN (QMBL) hours, as reported record: −0.499200 (DMRG), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (3 rows). Labels marked QMBL are QMBL's own reference runs.
J1-J2 10×10, J2 = 0.5: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ RBM (QMBL) RBM (QMBL) RBM (QMBL) DMRG (QMBL) RBM (QMBL) DMRG (QMBL) DMRG (QMBL) ViT (QMBL) ViT ×2 GCNN (QMBL) ViT ViT GCNN (QMBL) ViT (QMBL) ViT hours, as reported record: −0.498800 (DMRG), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (3 rows). For 5 rows the hours are devices × wall-clock, multiplied here. Labels marked QMBL are QMBL's own reference runs.
J1-J2 12×12, J2 = 0.5: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10⁻¹ 10⁰ 10¹ 10² 10³ RBM (QMBL) DMRG (QMBL) DMRG (QMBL) RBM (QMBL) DMRG (QMBL) GCNN (QMBL) ViT (QMBL) hours, as reported record: −0.496989 (CNN), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (3 rows). Labels marked QMBL are QMBL's own reference runs.
J1-J2 14×14, J2 = 0.5: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10⁻¹ 10⁰ 10¹ 10² RBM (QMBL) GCNN (QMBL) RBM (QMBL) ViT (QMBL) hours, as reported record: −0.497220 (VMC), no cost stated, not drawn Every mark is GPU-hours. Labels marked QMBL are QMBL's own reference runs.
J1-J2 16×16, J2 = 0.5: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ RBM (QMBL) RBM (QMBL) ViT (QMBL) mVMC + RBM GCNN hours, as reported record: −0.496914 (CNN-MPS), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (1 row). Labels marked QMBL are QMBL's own reference runs.
J1-J2 16×16, J2 = 0.55: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.4857 −0.4856 −0.4855 −0.4854 −0.4853 −0.4852 −0.4851 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ mVMC + RBM GCNN hours, as reported record: −0.485900 (TPS), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (1 row).
J1-J2 20×20, J2 = 0.5: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.49685 −0.49680 −0.49675 −0.49670 10³ 10⁴ 10⁵ ViT ViT hours, as reported Every mark is GPU-hours.
J1-J2 triangular 108, J2 = 0.125: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.5130 −0.5125 −0.5120 −0.5115 10¹ 10² 10³ GCNN GCNN hours, as reported Every mark is GPU-hours. For 2 rows the hours are devices × wall-clock, multiplied here.
J1-J2 triangular 144, J2 = 0.125: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/N (S.S) −0.5125 −0.5120 −0.5115 −0.5110 −0.5105 10¹ 10² 10³ GCNN GCNN hours, as reported Every mark is GPU-hours. For 2 rows the hours are devices × wall-clock, multiplied here.
Hubbard 4x16, U = 8, n = 0.875: the best energies at each cost Every energy on this instance whose paper, or QMBL's own run, states what it cost in hours. Variational bound Projected Extrapolated Exact Frontier E/site −0.766 −0.765 −0.764 −0.763 −0.762 10¹ 10² 10³ 10⁴ NNBF NNBF NNBF NNBF hours, as reported record: −0.767480 (NNBF), no cost stated, not drawn Every mark is GPU-hours.
Heisenberg chain 200, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.445 −0.440 −0.435 −0.430 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ Jastrow RBM DMRG FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg chain 200: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.445 −0.440 −0.435 −0.430 10¹⁴ 10¹⁵ 10¹⁶ Jastrow RBM DMRG FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg triangular 100, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ Jastrow RBM RNN DMRG FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg triangular 144, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ Jastrow RBM DMRG 2D RNN DMRG FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg triangular 144: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.53 −0.52 −0.51 −0.50 −0.49 −0.48 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ Jastrow RBM DMRG FLOPs, estimated record: −0.549329 (2D RNN), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg triangular 196, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ RBM Jastrow DMRG DMRG 2D RNN FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg triangular 256, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.50 −0.48 −0.46 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM Jastrow DMRG 2D RNN FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg triangular 324: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.5521 −0.5520 −0.5519 −0.5518 −0.5517 −0.5516 10¹⁸ 10¹⁹ 10²⁰ ViT ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Heisenberg triangular 576: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.5520 −0.5519 −0.5518 −0.5517 −0.5516 −0.5515 10¹⁹ 10²⁰ 10²¹ ViT ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Heisenberg triangular 900: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.5519 −0.5518 −0.5517 −0.5516 −0.5515 10¹⁹ 10²⁰ 10²¹ ViT ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Heisenberg triangular 1764: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.5518 −0.5517 −0.5516 −0.5515 −0.5514 10²⁰ 10²¹ 10²² ViT ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
square
J1-J2 8×8, J2 = 0.5: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.51 −0.50 −0.49 −0.48 −0.47 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ Jastrow RBM (QMBL) RBM DMRG (QMBL) RBM (QMBL) DMRG DMRG (QMBL) DMRG (QMBL) GCNN (QMBL) RBM ViT (QMBL) ViT FLOPs, estimated record: −0.499200 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run). Labels marked QMBL are QMBL's own reference runs (7 rows).
J1-J2 10×10, J2 = 0.3: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.560 −0.555 −0.550 −0.545 −0.540 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM Jastrow DMRG ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 10×10, J2 = 0.4: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.530 −0.525 −0.520 −0.515 −0.510 −0.505 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM Jastrow DMRG ViT FLOPs, estimated record: −0.525300 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 10×10, J2 = 0.5: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ RBM (QMBL) ×2 Jastrow RBM DMRG (QMBL) MLP DMRG RBM (QMBL) ×2 DMRG (QMBL) DMRG (QMBL) ViT (QMBL) ×2 RNN ConvNeXt ViT GCNN GCNN (QMBL) ×2 CNN-MPS ResNet ViT ViT ×2 FLOPs, estimated record: −0.498800 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run). Labels marked QMBL are QMBL's own reference runs (11 rows).
J1-J2 10×10, J2 = 0.6: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.49 −0.48 −0.47 −0.46 −0.45 −0.44 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM DMRG ViT FLOPs, estimated record: −0.478390 (CNN), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 10×10, J2 = 0.7: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.55 −0.50 −0.45 −0.40 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM Jastrow DMRG ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 10×10, J2 = 0.8: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.58 −0.57 −0.56 −0.55 −0.54 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM Jastrow DMRG ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 10×10, J2 = 0.9: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.64 −0.62 −0.60 −0.58 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM DMRG ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 10×10, J2 = 1: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.72 −0.70 −0.68 −0.66 −0.64 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow DMRG RBM ViT FLOPs, estimated record: −0.696700 (RBM), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. 1 of these estimate needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 12×12, J2 = 0.5: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM (QMBL) DMRG (QMBL) DMRG (QMBL) RBM (QMBL) DMRG (QMBL) GCNN (QMBL) ViT (QMBL) FLOPs, estimated record: −0.496989 (CNN), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. Labels marked QMBL are QMBL's own reference runs (7 rows).
J1-J2 14×14, J2 = 0.5: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM (QMBL) RBM Jastrow DMRG GCNN (QMBL) RBM (QMBL) ViT (QMBL) FLOPs, estimated record: −0.497220 (VMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this. Labels marked QMBL are QMBL's own reference runs (4 rows).
J1-J2 16×16, J2 = 0.5: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ RBM (QMBL) RBM (QMBL) ViT (QMBL) GCNN CNN-MPS FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. Labels marked QMBL are QMBL's own reference runs (3 rows).
J1-J2 20×20, J2 = 0.5: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.49685 −0.49680 −0.49675 −0.49670 10¹⁸ 10¹⁹ 10²⁰ 10²¹ ViT CNN-MPS ViT FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
U = 8, n = 0.875
Hubbard 4x16, U = 8, n = 0.875: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.770 −0.765 −0.760 −0.755 −0.750 −0.745 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ 10²¹ NNBF HFPS NNBF NNBF NNBF HFPS NNBF ACE FLOPs, estimated record: −0.767480 (NNBF), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Hubbard 4x16, U = 8, n = 0.875, cylinder: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.7512 −0.7511 −0.7510 −0.7509 −0.7508 10¹⁷ 10¹⁸ 10¹⁹ DMRG DMRG FLOPs, estimated Every mark is an estimate, good to an order of magnitude. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Hubbard 4x8, U = 8, n = 0.875, cylinder: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.73644 −0.73642 −0.73640 −0.73638 10¹⁷ 10¹⁸ 10¹⁹ DMRG DMRG FLOPs, estimated Every mark is an estimate, good to an order of magnitude. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Hubbard 8x16, U = 8, n = 0.875: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.758 −0.757 −0.756 −0.755 −0.754 −0.753 −0.752 10¹⁷ 10¹⁸ 10¹⁹ Tensor BF Tensor BF FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Hubbard 8x24, U = 8, n = 0.875: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.756 −0.754 −0.752 −0.750 −0.748 10¹⁸ 10¹⁹ 10²⁰ Tensor BF Tensor BF FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Hubbard 8x32, U = 8, n = 0.875: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.756 −0.754 −0.752 −0.750 10¹⁸ 10¹⁹ 10²⁰ Tensor BF Tensor BF FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
Hubbard 16×16, U = 8, n = 0.875: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/site −0.758 −0.756 −0.754 −0.752 −0.750 10¹⁸ 10¹⁹ 10²⁰ 10²¹ 10²² Tensor BF Tensor BF ACE FLOPs, estimated record: −0.758652 (VAFQMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted.
square, open, h = 3
TFIM 12×12, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (Pauli) −3.17395 −3.17390 −3.17385 −3.17380 −3.17375 −3.17370 −3.17365 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ Jastrow RBM RNN DMRG 2D RNN FLOPs, estimated Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
TFIM 20×20, open: the best energies at each estimated cost in FLOPs Every energy on this instance whose paper or run script states enough to estimate its cost in floating-point operations. Variational bound Projected Extrapolated Frontier E/N (Pauli) −3.185 −3.180 −3.175 −3.170 −3.165 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM DMRG Jastrow FLOPs, estimated record: −3.181970 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates: iterations × samples × (connected configurations + sampler proposals + 3) × forward-pass FLOPs × evaluations per amplitude, stage by stage, from the counts the paper or run script states, with the optimizer's solve where the code is known; pre-training is not counted. DMRG estimates: the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations; the saving from conserved quantum numbers is not counted, so a symmetric code does less arithmetic than this.
Heisenberg chain 200: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.445 −0.440 −0.435 −0.430 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM VMC variational parameters record: −0.443168 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
Heisenberg triangular 100, open: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.51 −0.50 −0.49 −0.48 −0.47 10³ 10⁴ 10⁵ Jastrow RBM RNN variational parameters record: −0.514767 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
Heisenberg triangular 144, open: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 10³ 10⁴ 10⁵ 10⁶ 10⁷ Jastrow RBM 2D RNN variational parameters record: −0.515065 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
Heisenberg triangular 144: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.56 −0.54 −0.52 −0.50 −0.48 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ Jastrow RBM VMC variational parameters record: −0.549329 (2D RNN), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
Heisenberg triangular 196, open: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 10³ 10⁴ 10⁵ 10⁶ 10⁷ Jastrow RBM 2D RNN variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
Heisenberg triangular 256, open: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.52 −0.50 −0.48 −0.46 10⁴ 10⁵ 10⁶ 10⁷ Jastrow RBM 2D RNN variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
square
J1-J2 8×8, J2 = 0.5: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.51 −0.50 −0.49 −0.48 −0.47 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM (QMBL) RBM RBM (QMBL) GCNN (QMBL) VMC ViT (QMBL) RBM + PP ViT RBM variational parameters record: −0.499200 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.3: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.560 −0.555 −0.550 −0.545 −0.540 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM VMC ViT variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.4: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.5088 −0.5086 −0.5084 −0.5245 −0.5240 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM CNN aCNN aCNN ViT variational parameters record: −0.525300 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.5: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.49 −0.48 −0.47 −0.4980 −0.4975 −0.4970 −0.4965 −0.4960 −0.4955 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ RBM (QMBL) ×2 Jastrow RBM CNN MLP RBM (QMBL) ×2 PEPS + CNN VMC CNN RNN GCNN (QMBL) PEPS + CNN aCNN GCNN (QMBL) ResNet CNN ResNet ×2 ViT (QMBL) ×2 ViT ViT ViT CNN ViT GCNN CNN ConvNeXt ResNet ViT ResNet CNN-MPS CTWF 1 variational parameters record: −0.498800 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.55: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.487 −0.486 −0.485 −0.484 −0.483 −0.482 10³ 10⁴ 10⁵ 10⁶ aCNN aCNN CNN variational parameters record: −0.488000 (DMRG), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.6: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.49 −0.48 −0.47 −0.46 −0.45 −0.44 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM CNN CNN ViT CNN variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.7: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.55 −0.50 −0.45 −0.40 10³ 10⁴ 10⁵ 10⁶ RBM Jastrow ViT variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.8: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.58 −0.57 −0.56 −0.55 −0.54 10³ 10⁴ 10⁵ 10⁶ RBM Jastrow CNN ViT variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 0.9: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.64 −0.62 −0.60 −0.58 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM ViT variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 10×10, J2 = 1: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.72 −0.70 −0.68 −0.66 −0.64 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM CNN ViT variational parameters record: −0.696700 (RBM), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 12×12, J2 = 0.5: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10² 10³ 10⁴ 10⁵ 10⁶ RBM (QMBL) RBM (QMBL) GCNN (QMBL) ViT (QMBL) CNN variational parameters record: −0.496989 (CNN), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 14×14, J2 = 0.5: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.50 −0.49 −0.48 −0.47 −0.46 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ RBM (QMBL) RBM Jastrow GCNN (QMBL) RBM (QMBL) VMC ViT (QMBL) CNN variational parameters record: −0.497220 (VMC), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 16×16, J2 = 0.5: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.48 −0.46 −0.498 −0.497 −0.496 −0.495 10² 10³ 10⁴ 10⁵ 10⁶ RBM (QMBL) RBM (QMBL) ViT (QMBL) CNN GCNN ResNet CNN-MPS variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
J1-J2 20×20, J2 = 0.5: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (S.S) −0.49685 −0.49680 −0.49675 −0.49670 10⁴ 10⁵ 10⁶ 10⁷ ViT CNN-MPS ViT variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
U = 8, n = 0.875
Hubbard 4x16, U = 8, n = 0.875: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/site −0.751 −0.750 −0.766 −0.765 −0.764 −0.763 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ NNBF HFPS NNBF NNBF HFPS ACE NNBF NNBF variational parameters record: −0.767480 (NNBF), no cost stated, not drawn Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
Hubbard 16×16, U = 8, n = 0.875: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/site −0.760 −0.755 −0.750 −0.745 −0.740 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ VMC VMC Tensor BF SCALE Tensor BF Transformer BF ACE VAFQMC variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.
TFIM 12×12, open: the best energies at each parameter count Every energy on this instance whose paper states the ansatz's parameter count. Variational bound Projected Extrapolated Frontier E/N (Pauli) −3.17395 −3.17390 −3.17385 −3.17380 −3.17375 −3.17370 −3.17365 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM RNN 2D RNN variational parameters Parameter counts as the papers print them; a count evaluated from a printed formula is marked medium confidence in the row. Tensor-network bond dimensions and Monte Carlo sample counts are other costs and are not on this axis.

Citing QMBL: Open Problems

Cite QMBL: Open Problems by its concept DOI, which always resolves to the latest release:

T. Zerweck and J. Carrasquilla, QMBL: Open Problems, Zenodo (2026). 10.5281/zenodo.23439787

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