QMBLthe quantum many-body leaderboard

Best published ground-state energies

QMBL is a live record book of the state of the art in quantum many-body simulation. Browse through the leaderboard figures below interactively or look at the pure data in the table. 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, by system size Results are placed by their relative gap to the exact ground-state energy so that different Hamiltonians share one axis. A better energy is lower. Above size 242 there are only Heisenberg and Hubbard model results, because they have an exact ground-state energy as reference. For other Hamiltonians at larger system sizes, see below. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row exact 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 instance's exact energy (lower is better) 10 4 9 5 38 2 3 4 2 2 14 14 15 23 25 17 10 2 5 10 3 2 2 2 5 4 2 2 2 4 3 2 2 4 2 2 3 2 3 3 3 14 2 2 2 2 3 2 7 4 6 5 6 2 3 8 6 2 2 4 2 4 76 2 3 4 5 11 21 33 20 15 13 15 12 13 7 4 4 2 2 3 3 2 5 2 2 34 2 3 7 9 10 7 16 19 16 24 27 32 26 18 10 7 2 2 2 2 18 4 4 4 2 2 3 12 5 3 2 2 2 15 2 6 6 2 3 2 3 3 2 2 2 3 12 8 19 33 18 2 Height is |E − E_exact| / |E_exact| on a log scale, so that a better energy is lower. On the site, hovering over a dot lists the result, and clicking it opens the table entry and source.
Best published energies by method family
The best energies by system size, one panel per method family Each panel colours one family's energies over all the others in grey and joins the family's best energy at each size. A method that only ever ran at one size is a lone mark. tensor network (141) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 3 3 5 2 2 4 2 2 2 2 3 7 7 5 16 9 3 2 4 16 2 2 2 7 2 2 3 4 4 transformer / ViT (37) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 4 3 8 7 3 4 CNN / ResNet (50) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 2 6 20 12 2 RNN (62) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 3 6 6 2 2 2 2 3 2 2 2 5 4 2 3 3 2 RBM (107) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 2 2 3 2 2 2 2 2 2 4 13 8 9 7 2 2 3 3 4 2 8 2 2 mean field (45) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 17 3 2 2 2 2 4 3 3 2 backflow / Pfaffian (124) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 6 4 8 19 22 20 6 2 5 4 2 10 12 other NQS (16) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 2 3 2 2 classic VMC (125) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 2 2 2 2 3 4 2 5 2 3 2 3 8 7 16 2 5 3 8 2 4 3 5 2 8 16 FCIQMC / CC (77) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 12 20 3 15 12 3 2 2 2 2 AFQMC / GFMC (74) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 5 11 2 2 7 4 39 3 VQE / circuit (51) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 2 2 7 30 3 2 other (67) exact 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10 30 100 300 1000 sites 13 25 29 Each ansatz belongs to one family, and a row without one goes by its method (scripts/method_names.mjs); 'other' holds exact methods and names no family covers. A slashed mark is a flagged row, and a number counts the rows a mark holds. Axes and the exact line as in the figure above: a better energy is lower. No dot sits closer to exact than its own error bar. 69 rows lie below the exact energy by more than twice the error bar and are not drawn.

The published energies, by Hamiltonian

periodic

Heisenberg square, periodic Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.72 −0.70 −0.68 −0.66 −0.64 16 36 50 64 100 144 196 256 324 484 676 900 1296 1936 2704 3600 5184 7744 sites

open

Heisenberg square, open Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.68 −0.66 −0.64 −0.62 −0.60 −0.58 36 64 100 144 196 256 324 484 676 900 1296 1936 2704 4096 9216 16384 sites

periodic

Heisenberg triangular, periodic Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.52 −0.50 −0.48 −0.46 100 144 196 256 sites
Heisenberg pyrochlore Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.52 −0.51 −0.50 −0.49 −0.48 −0.47 32 48 64 96 108 128 192 256 432 1024 sites
Heisenberg kagome Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.445 −0.440 −0.435 −0.430 20 200 sites
Heisenberg rectangular, periodic/open, L × 2L Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.670 −0.665 −0.660 −0.655 32 72 128 200 288 392 512 648 800 1152 2048 4608 8192 sites
Heisenberg shuriken Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 lowest energy at each size: J2
J2 =
J1-J2 square, J2 = 0.05 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.6810 −0.6805 −0.6800 16 sites
J1-J2 square, J2 = 0.1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.6400 −0.6395 −0.6390 16 sites
J1-J2 square, J2 = 0.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.6015 −0.6010 −0.6005 16 sites
J1-J2 square, J2 = 0.3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.5665 −0.5660 16 sites
J1-J2 square, J2 = 0.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.5945 −0.5940 16 sites
J1-J2 square, J2 = 0.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.6620 −0.6615 −0.6610 16 sites
J1-J2 square, J2 = 0.9 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.7330 −0.7325 −0.7320 16 sites
J1-J2 square, J2 = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (S.S) −0.520 −0.515 −0.510 −0.505 48 108 144 sites

periodic, K = -1, J = 0, Gamma = 0

Kitaev (K-J-Γ) honeycomb, periodic, K = -1, J = 0, Gamma = 0 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.200 −0.195 −0.190 −0.185 24 32 sites

open, K = -1, J = 0, Gamma = 0

Kitaev (K-J-Γ) honeycomb, open, K = -1, J = 0, Gamma = 0 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.190 −0.185 −0.180 −0.175 50 98 242 sites
−0.50 −0.45 −0.40 −0.35 E/N (S.S) periodic, Jd = 1 1111111111111216271 324 256 196 144 100 64 36 32 16 lowest energy at each size: J
periodic, Jd = 1, J
Shastry-Sutherland square, periodic, Jd = 1, J = 0.5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 36 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.667 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 36 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.674 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 36 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.675 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3754 −0.3752 36 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.677 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.6772 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.67724 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.677244 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.677246 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.677248 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.6773 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 −0.3748 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.6775 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3752 −0.3750 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.678 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3754 −0.3752 −0.3750 32 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.68 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.3780 −0.3775 −0.3770 −0.3765 −0.3760 32 36 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.7 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.40 −0.39 −0.38 −0.37 −0.36 −0.35 16 32 36 144 256 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.453 −0.452 −0.451 −0.450 −0.449 −0.448 36 64 100 144 196 256 324 sites
Shastry-Sutherland square, periodic, Jd = 1, J = 0.85 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) −0.4845 −0.4840 36 sites
−5 −4 −3 −2 −1 0 E/site periodic, n = 1 periodic, n = 0.875 periodic, n ≈ 0.8333 periodic/antiperiodic, n = 1 1636641001441962563311181035 1664144256102414286234 14428827211 16366410014419625610101014 1024 288 256 196 144 100 64 36 16 lowest energy at each size: U
U =U =U =U =
Hubbard square, periodic, n = 1, U = -8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −4.530 −4.525 −4.520 64 256 sites
Hubbard square, periodic, n = 1, U = -4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −2.862 −2.860 −2.858 64 256 sites
Hubbard square, periodic, n = 1, U = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/site −4.02 −4.01 −4.00 −3.99 64 144 256 sites
Hubbard square, periodic, n = 0.875, U = -4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −2.602 −2.600 64 256 sites
Hubbard square, periodic, n = 0.875, U = -2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −2.042 −2.040 −2.038 −2.036 −2.034 −2.032 64 144 sites
Hubbard square, periodic, n = 0.875, U = 4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/site −0.77 −0.76 −0.75 −0.74 −0.73 −0.72 16 64 144 256 1024 sites
Hubbard square, periodic, n ≈ 0.8333, U = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −1.296 −1.295 −1.294 144 288 sites
Hubbard square, periodic, n ≈ 0.8333, U = 4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −1.076 −1.074 −1.072 −1.070 −1.068 144 288 sites
Hubbard square, periodic, n ≈ 0.8333, U = 6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.926 −0.924 −0.922 −0.920 −0.918 144 288 sites
Hubbard square, periodic, n ≈ 0.8333, U = 10 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.7620 −0.7615 144 sites
Hubbard square, periodic, n ≈ 0.8333, U = 15 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.6675 −0.6670 −0.6665 144 sites
Hubbard square, periodic/antiperiodic, n = 1, U = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/site −0.55 −0.54 −0.53 −0.52 −0.51 16 36 64 100 144 196 256 sites
Hubbard square, t12, periodic, U = 8, t2 = -0.2, n = 0.875 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.76 −0.75 −0.74 −0.73 −0.72 64 144 256 400 576 sites

periodic, 4 × L, U = 8, n = 0.875

Hubbard rectangular, periodic, 4 × L, U = 8, n = 0.875 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/site −0.755 −0.750 −0.745 −0.740 −0.735 −0.730 32 64 sites
−0.80 −0.75 −0.70 −0.65 −0.60 E/site 6 × L, U = 12, n = 0.875 8 × L, U = 8, n = 0.875 2402882222222 809612814416019225612 288 256 240 192 160 144 128 96 80 lowest energy at each size: t2
t2 =t2 =
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.6265 −0.6260 −0.6255 240 288 sites
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.45 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.625 −0.624 −0.623 −0.622 −0.621 −0.620 −0.619 240 288 sites
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.624 −0.622 −0.620 −0.618 −0.616 −0.614 240 288 sites
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.35 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.622 −0.620 −0.618 −0.616 −0.614 −0.612 240 288 sites
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.620 −0.618 −0.616 −0.614 −0.612 −0.610 240 288 sites
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.620 −0.618 −0.616 −0.614 −0.612 −0.610 240 288 sites
Hubbard rectangular, t12, 6 × L, U = 12, n = 0.875, t2 = -0.1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.622 −0.620 −0.618 −0.616 −0.614 240 288 sites
Hubbard rectangular, t12, 8 × L, U = 8, n = 0.875, t2 = -0.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site −0.755 −0.750 −0.745 −0.740 −0.735 −0.730 80 96 128 144 160 192 256 sites
Transverse-field Ising square, open, h = 3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Pauli) −3.20 −3.15 −3.10 −3.05 64 100 144 400 sites
Transverse-field Ising chain, periodic, h = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Pauli) −1.280 −1.278 −1.276 −1.274 −1.272 −1.270 10 32 sites

TB-DMFT

Impurity TB-DMFT, TB-DMFT Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 1299958999977631497488188601352772377771523 246 162 128 114 66 54 38 27 19 14 7 lowest energy at each size: rs
ζ = 1, rs
ζ = 0, rs
Electron gas, 3D sc, ζ = 1, rs = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (Ha) −0.02284 −0.02282 −0.02280 27 sites
Electron gas, 3D sc, ζ = 1, rs = 50 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (Ha) −0.01076 −0.01075 −0.01074 27 sites
Electron gas, 3D sc, ζ = 1, rs = 90 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.008535 −0.008530 27 sites
Electron gas, 3D sc, ζ = 1, rs = 100 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (Ha) 25.36 25.38 25.40 25.42 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 10.67 10.68 10.69 10.70 10.71 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 5.65 5.66 5.67 5.68 5.69 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (Ha) 2.19 2.20 2.21 2.22 2.23 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.7 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 1.48 1.49 1.50 1.51 1.52 1.53 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 1.04 1.05 1.06 1.07 1.08 1.09 14 sites
Electron gas, 3D sc, ζ = 0, rs = 0.9 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 0.75 0.76 0.77 0.78 0.79 14 sites
Electron gas, 3D sc, ζ = 0, rs = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 0.50 0.55 0.60 0.65 14 38 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.06 −0.04 −0.02 0.00 0.02 0.04 14 38 54 66 114 sites
Electron gas, 3D sc, ζ = 0, rs = 3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.14 −0.12 −0.10 −0.08 −0.06 −0.04 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.14 −0.12 −0.10 −0.08 −0.06 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.14 −0.12 −0.10 −0.08 −0.06 −0.04 14 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.12 −0.10 −0.08 −0.06 −0.04 14 sites
Electron gas, 3D sc, ζ = 0, rs = 7 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.12 −0.10 −0.08 −0.06 −0.04 14 sites
Electron gas, 3D sc, ζ = 0, rs = 7.5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.06030 −0.06025 54 sites
Electron gas, 3D sc, ζ = 0, rs = 8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.10 −0.09 −0.08 −0.07 −0.06 −0.05 14 sites
Electron gas, 3D sc, ζ = 0, rs = 9 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.09 −0.08 −0.07 −0.06 −0.05 −0.04 14 sites
Electron gas, 3D sc, ζ = 0, rs = 10 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.10 −0.08 −0.06 −0.04 −0.02 14 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 15 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.07 −0.06 −0.05 −0.04 −0.03 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 20 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.05 −0.04 −0.03 −0.02 14 54 66 114 162 246 sites
Electron gas, 3D sc, ζ = 0, rs = 30 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.035 −0.030 −0.025 −0.020 −0.015 14 sites
Electron gas, 3D sc, ζ = 0, rs = 40 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.030 −0.025 −0.020 −0.015 −0.010 14 sites
Electron gas, 3D sc, ζ = 0, rs = 50 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.025 −0.020 −0.015 −0.010 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 60 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.018 −0.016 −0.014 −0.012 −0.010 −0.008 14 sites
Electron gas, 3D sc, ζ = 0, rs = 70 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.016 −0.014 −0.012 −0.010 −0.008 −0.006 14 sites
Electron gas, 3D sc, ζ = 0, rs = 80 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.014 −0.012 −0.010 −0.008 −0.006 14 sites
Electron gas, 3D sc, ζ = 0, rs = 90 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.014 −0.012 −0.010 −0.008 −0.006 −0.004 14 sites
Electron gas, 3D sc, ζ = 0, rs = 100 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.012 −0.010 −0.008 −0.006 −0.004 14 54 sites
Electron gas, 3D sc, ζ = 0, rs = 110 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 and exact (stochastic) Flagged, see the row E/N (Ha) −0.00408 −0.00406 −0.00404 −0.00402 −0.00400 −0.00398 128 sites
−20 0 20 40 60 E/N (Ha) ζ = 1 ζ = 0 5555555555617136 642165126666 512 216 64 27 lowest energy at each size: rs
rs =rs =
Electron gas, 3D bcc, ζ = 1, rs = 0.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 44.15 44.20 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 0.5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 6.315 6.320 6.325 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 1.261 1.262 1.263 1.264 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 0.1520 0.1521 0.1522 0.1523 0.1524 0.1525 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.0070 −0.0069 −0.0068 −0.0067 −0.0066 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.0578 −0.0577 −0.0576 −0.0575 −0.0574 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 10 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.05035 −0.05030 −0.05025 −0.05020 −0.05015 −0.05010 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 30 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.02260 −0.02255 −0.02250 −0.02245 −0.02240 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 50 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.01450 −0.01445 −0.01440 −0.01435 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 70 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.01072 −0.01070 −0.01068 −0.01066 −0.01064 −0.01062 27 sites
Electron gas, 3D bcc, ζ = 1, rs = 80 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00946 −0.00944 −0.00942 −0.00940 −0.00938 64 216 512 sites
Electron gas, 3D bcc, ζ = 1, rs = 90 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00855 −0.00850 −0.00845 −0.00840 27 64 216 512 sites
Electron gas, 3D bcc, ζ = 1, rs = 100 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00770 −0.00769 −0.00768 −0.00767 −0.00766 −0.00765 64 216 512 sites
Electron gas, 3D bcc, ζ = 1, rs = 125 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00627 −0.00626 −0.00625 −0.00624 −0.00623 64 216 512 sites
Electron gas, 3D bcc, ζ = 0, rs = 80 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00946 −0.00944 −0.00942 −0.00940 −0.00938 −0.00936 64 216 512 sites
Electron gas, 3D bcc, ζ = 0, rs = 90 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00850 −0.00848 −0.00846 −0.00844 −0.00842 64 216 512 sites
Electron gas, 3D bcc, ζ = 0, rs = 100 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00772 −0.00770 −0.00768 −0.00766 −0.00764 64 216 512 sites
Electron gas, 3D bcc, ζ = 0, rs = 125 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.00627 −0.00626 −0.00625 −0.00624 −0.00623 −0.00622 64 216 512 sites
Electron gas, 3D fcc, rs = 2, ζ = 0 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 0.000 0.005 0.010 54 102 178 226 sites
−0.3 −0.2 −0.1 0.0 0.1 E/N (Ha) ζ = 1 ζ = 0 21375769113766666 26425874901146666136 114 113 90 74 69 58 57 42 37 26 21 lowest energy at each size: rs
rs =rs =
Electron gas, 2D square, ζ = 1, rs = 5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.146 −0.145 −0.144 −0.143 −0.142 −0.141 21 37 57 69 113 sites
Electron gas, 2D square, ζ = 1, rs = 10 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.0855 −0.0850 −0.0845 −0.0840 −0.0835 21 37 57 69 113 sites
Electron gas, 2D square, ζ = 1, rs = 20 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.0464 −0.0463 −0.0462 −0.0461 −0.0460 21 37 57 69 113 sites
Electron gas, 2D square, ζ = 1, rs = 30 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.03200 −0.03195 −0.03190 −0.03185 −0.03180 21 37 57 69 113 sites
Electron gas, 2D square, ζ = 1, rs = 40 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.02444 −0.02442 −0.02440 −0.02438 −0.02436 −0.02434 21 37 57 69 113 sites
Electron gas, 2D square, ζ = 1, rs = 75 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.01348 −0.01346 −0.01344 −0.01342 21 37 57 69 113 sites
Electron gas, 2D square, ζ = 0, rs = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.23 −0.22 −0.21 −0.20 −0.19 −0.18 26 42 58 74 114 sites
Electron gas, 2D square, ζ = 0, rs = 5 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.151 −0.150 −0.149 −0.148 −0.147 −0.146 26 42 58 74 114 sites
Electron gas, 2D square, ζ = 0, rs = 10 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.0860 −0.0855 −0.0850 −0.0845 −0.0840 26 42 58 74 114 sites
Electron gas, 2D square, ζ = 0, rs = 20 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.0466 −0.0464 −0.0462 −0.0460 −0.0458 −0.0456 26 42 58 74 114 sites
Electron gas, 2D square, ζ = 0, rs = 30 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.034 −0.032 −0.030 −0.028 −0.026 −0.024 26 42 58 74 90 114 sites
Electron gas, 2D square, ζ = 0, rs = 50 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.01980 −0.01975 −0.01970 −0.01965 −0.01960 26 42 58 74 114 sites
Electron gas, 2D triangular, rs = 30, ζ = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.03205 −0.03200 −0.03195 −0.03190 −0.03185 −0.03180 16 91 169 sites
−0.6 −0.5 −0.4 −0.3 E/N (Ha) basis = cc-pV5Z basis = cc-pVDZ basis = cc-pVQZ basis = cc-pVTZ basis-free basis = STO-6G basis = STO-3G 1030509778888899 323337354337393363630 10305018172222232322191917 10305021242626282828242522 1030508210222222119242121212119 5451541557515149503456713 46810121416711414 102 82 70 50 30 22 20 16 14 12 10 8 6 4 lowest energy at each size: R (bohr)
basis = cc-pV5Z, R (bohr)
basis = cc-pVDZ, R (bohr)
basis = cc-pVQZ, R (bohr)
basis = cc-pVTZ, R (bohr)
basis-free, R (bohr)
basis = STO-6G, R (bohr)
basis = STO-3G, R (bohr)
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.46 −0.44 −0.42 −0.40 −0.38 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 1.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 1.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 1.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 −0.52 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 1.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 2.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 2.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.55 −0.54 −0.53 −0.52 −0.51 −0.50 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 3.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.53 −0.52 −0.51 −0.50 −0.49 −0.48 10 30 50 atoms
Hydrogen chain, open, basis = cc-pV5Z, R (bohr) = 3.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.44 −0.42 −0.40 −0.38 −0.36 −0.34 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 1.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 −0.44 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 1.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.55 −0.54 −0.53 −0.52 −0.51 −0.50 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 1.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 −0.52 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 1.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 2.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 2.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.54 −0.53 −0.52 −0.51 −0.50 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.5275 −0.5270 −0.5265 −0.5260 6 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 3.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.53 −0.52 −0.51 −0.50 −0.49 −0.48 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVDZ, R (bohr) = 3.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.46 −0.44 −0.42 −0.40 −0.38 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 1.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 1.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 1.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 −0.52 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 1.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 2.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 2.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.55 −0.54 −0.53 −0.52 −0.51 −0.50 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 3.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.53 −0.52 −0.51 −0.50 −0.49 −0.48 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVQZ, R (bohr) = 3.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.46 −0.44 −0.42 −0.40 −0.38 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 1.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 1.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 1.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 −0.52 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 1.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 2.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 2.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.55 −0.54 −0.53 −0.52 −0.51 −0.50 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 3.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.53 −0.52 −0.51 −0.50 −0.49 −0.48 10 30 50 atoms
Hydrogen chain, open, basis = cc-pVTZ, R (bohr) = 3.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.45 −0.44 −0.43 −0.42 −0.41 10 30 50 82 102 atoms
Hydrogen chain, open, basis-free, R (bohr) = 1.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.51 −0.50 −0.49 −0.48 10 30 50 82 102 atoms
Hydrogen chain, open, basis-free, R (bohr) = 1.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 1.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 1.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 102 atoms
Hydrogen chain, open, basis-free, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.57 −0.56 −0.55 −0.54 −0.53 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 2.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.56 −0.55 −0.54 −0.53 −0.52 −0.51 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 2.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.55 −0.54 −0.53 −0.52 −0.51 −0.50 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 3.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.53 −0.52 −0.51 −0.50 −0.49 −0.48 10 30 50 atoms
Hydrogen chain, open, basis-free, R (bohr) = 3.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.40 −0.38 −0.36 −0.34 −0.32 10 22 30 50 70 102 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.48 −0.47 −0.46 −0.45 −0.44 10 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.525 −0.520 −0.515 −0.510 −0.505 −0.500 −0.495 10 22 30 50 70 102 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1.42 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.5230 −0.5225 10 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.540 −0.535 −0.530 −0.525 −0.520 −0.515 10 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.55 −0.54 −0.53 −0.52 4 6 8 10 12 22 30 50 70 102 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 1.89 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.5420 −0.5415 −0.5410 10 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.545 −0.540 −0.535 −0.530 −0.525 −0.520 −0.515 10 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 2.4 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.53 −0.52 −0.51 −0.50 −0.49 10 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 2.8 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 10 22 30 50 70 102 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 2.83 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.5040 −0.5035 −0.5030 6 8 10 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 3.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.65 −0.60 −0.55 −0.50 −0.45 −0.40 10 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 3.6 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.7 −0.6 −0.5 −0.4 −0.3 4 6 8 10 12 20 30 50 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 3.78 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.4795 −0.4790 10 atoms
Hydrogen chain, open, basis = STO-6G, R (bohr) = 4.2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.50 −0.45 −0.40 −0.35 50 atoms
Hydrogen chain, open, basis = STO-3G, R (bohr) = 1.51 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.545 −0.540 −0.535 −0.530 −0.525 −0.520 4 6 8 10 12 14 16 atoms
Hydrogen chain, open, basis = STO-3G, R (bohr) = 1.89 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.5380 −0.5375 12 atoms
Hydrogen chain, open, basis = STO-3G, R (bohr) = 2 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.540 −0.535 −0.530 −0.525 −0.520 −0.515 6 8 atoms
Hydrogen chain, open, basis = STO-3G, R (bohr) = 3 Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) −0.52 −0.50 −0.48 −0.46 −0.44 6 8 atoms
Heisenberg: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) honeycomb 7x7 (98 sites) −0.5425 −0.5420 rectangular 6x8 −0.6765 −0.6760 −0.6755
J1-J2: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) honeycomb 7x7 (98 sites), J2 = 0.2 −0.456 −0.454 −0.452 −0.450 −0.448 rectangular 4x6, J2 = 0.5 −0.525 −0.520 −0.515 −0.510 −0.505
Kitaev (K-J-Γ): Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) honeycomb 6x6 (72 sites), periodic/open, K = 1, J = 0, Gamma = 0 −0.1892 −0.1891
Shastry-Sutherland: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (S.S) square 20x20, open, J = 0.8, Jd = 1 −0.4430 −0.4425 −0.4420
Hubbard: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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 4x24, U = 4, n ≈ 0.8333 −1.079 −1.078 −1.077 rectangular 6x48, U = 8, n ≈ 0.8333 −0.828 −0.826 −0.824 −0.822 −0.820 −0.818 0 t2 rectangular 6x48, U = 8, n ≈ 0.9167 −0.69 −0.68 −0.67 −0.66 0 t2 rectangular 6x48, U = 12, n = 0.875 −0.635 −0.630 −0.625 −0.620 0 t2 rectangular 6x48, U = 12, n = 0.875, t12 −0.67 −0.66 −0.65 −0.64 −0.63 0.1 0.15 0.2 0.25 t2 rectangular 8x48, U = 8, n ≈ 0.9167 −0.686 −0.685 −0.684 −0.683 −0.682 −0.681 rectangular 10x16, U = 8, t2 = -0.2, n = 0.875, t12 −0.7440 −0.7435 −0.7430 rectangular 12x16, U = 8, n = 0.875 −0.7565 −0.7560 0 t2 rectangular 12x16, U = 8, n = 0.875, t12 −0.7445 −0.7440 −0.7435 -0.2 t2 rectangular 14x16, U = 8, n = 0.875 −0.7585 −0.7580 rectangular 16x32, U = 8, n = 0.875 −0.756 −0.755 −0.754 −0.753 −0.752 −0.751 −0.750 0 t2 rectangular 16x32, U = 8, n = 0.875, t12 −0.742 −0.740 −0.738 −0.736 −0.734 -0.2 t2 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, open, U = 8, n = 0.875 −0.730 −0.728 −0.726 −0.724 −0.722 0 t2 square 16x16, open, U = 8, n = 0.875, t12 −0.712 −0.710 −0.708 −0.706 −0.704 -0.2 t2 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
Spinless t-V: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site chain, 32 sites, n = 0.5 −0.6 −0.5 −0.4 −0.3 −0.2 1 2 4 V square 4x4, n = 0.3125 −0.8 −0.6 −0.4 −0.2 0.0 0.01 0.1 1 10 V square 6x6, n ≈ 0.3611 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.01 0.1 1 10 V square 8x8, n = 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 1 2 4 V
t-J: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/site honeycomb-flake24, 24 sites, open, J = 0.3333333333333333 −0.45 −0.40 −0.35 −0.30 −0.25 0.917 0.958 filling n ladder 2x11, J = 0.5, n ≈ 0.9091 −0.5910 −0.5905 ladder 2x13, J = 0.5, n ≈ 0.9231 −0.5735 −0.5730 ladder 3x6, antiperiodic legs, J = 0.5 −0.68 −0.66 −0.64 −0.62 −0.60 −0.58 −0.56 0.889 0.944 filling n rectangular 6x16, periodic/open, J = 0.35, n ≈ 0.9167 −0.549 −0.548 −0.547 −0.546 −0.545 −0.544 square 4x4, open, J = 0.4, n = 0.875 −0.5645 −0.5640 square 8x8, n = 0.875 −0.68 −0.67 −0.66 −0.65 −0.64 0.4 J square 8x8, n ≈ 0.7813 −0.70 −0.68 −0.66 −0.64 0.2 J square 8x8, open, J = 0.3333333333333333, n ≈ 0.9844 −0.39 −0.38 −0.37 −0.36 −0.35 square, 18 sites, J = 0.4, n ≈ 0.8889 −0.67 −0.66 −0.65 −0.64 −0.63 square, 26 sites, n ≈ 0.8462 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 0.3 0.4 0.5 0.6 0.8 1 J square, 26 sites, n ≈ 0.9231 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 0.3 0.4 0.5 0.6 0.8 1 J square, 32 sites, J = 0.3, t2 = 0.3, t3 = -0.2 −0.7 −0.6 −0.5 −0.4 0.875 0.938 0.969 filling n
Transverse-field Ising: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row 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
Impurity: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E (total) SB-DMFT-MI-HF, 10 sites −34.60 −34.58 −34.56 SB-DMFT-MT-AHF, 9 sites −13.90 −13.89 −13.88 SB-DMFT-MT-HF, 9 sites −23.32 −23.30 −23.28 SB-IMP, 9 sites −10.49 −10.48
Electron gas, 3D: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) 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 Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) triangular cell, N = 91, rs = 40, ζ = 1 −0.02446 −0.02444 −0.02442 −0.02440 −0.02438 −0.02436
Hydrogen chain: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) H10 chain, open, R = 1, basis = aug-cc-pVQZ −0.45 −0.44 −0.43 −0.42 −0.41
Hydrogen sheet: Hamiltonians published at one size Hamiltonians of this model published at one size only. Each panel shares lattice, boundary and size; the coupling or filling that differs runs along x. Colour shows the bound type, and a slashed mark is a flagged row. Variational bound Projected Extrapolated Exact and exact (stochastic) Flagged, see the row E/N (Ha) H4 sheet (2x2), open, R = 2.4, basis = none −0.530 −0.525 −0.520 −0.515

The best published energies, by cost

Cost on three axes: stated GPU or CPU hours (17 instances), estimated FLOPs, and parameter count. FLOPs are estimated from the stated parameter, sample and step counts, or from the sweep schedules of public DMRG run scripts (how).

triangular
Heisenberg triangular 36: 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) −0.52 −0.50 −0.565 −0.560 −0.555 −0.550 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ RBM (QMBL) RBM (QMBL) DMRG (QMBL) DMRG (QMBL) ViT (QMBL) 2D RNN DMRG (QMBL) GCNN (QMBL) GCNN hours, as reported exact: −0.560373 (ED), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (3 rows). For 1 row the hours are devices × wall-clock, multiplied here. Labels marked QMBL are QMBL's own reference runs.
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 4×4, 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) −0.530 −0.525 −0.520 −0.515 −0.510 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ RBM (QMBL) RBM (QMBL) ViT (QMBL) GCNN (QMBL) ED (cost measured by QMBL) hours, as reported Circles are GPU-hours, squares CPU core-hours (1 row). Labels marked QMBL are QMBL's own reference runs. An exact energy marked "cost measured by QMBL" is placed at what diagonalizing the instance cost QMBL. QMBL ran it on 1 core of an AMD EPYC 7H12 64-Core Processor, conserving only particle number or Sz.
J1-J2 6×6, 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) −0.49 −0.48 −0.504 −0.503 −0.502 −0.501 −0.500 10⁻² 10⁻¹ 10⁰ 10¹ 10² RBM (QMBL) RBM (QMBL) DMRG (QMBL) DMRG (QMBL) ViT (QMBL) DMRG (QMBL) GCNN (QMBL) ViT ×2 ViT hours, as reported exact: −0.503810 (ED), no cost stated, not drawn Circles are GPU-hours, squares CPU core-hours (3 rows). For 3 rows the hours are devices × wall-clock, multiplied here. Labels marked QMBL are QMBL's own reference runs.
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 best variational: −0.498963 (RBM + PP); best extrapolated: −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 best variational: −0.497694 (CNN-MPS); best projected: −0.495700 (PEPS); best extrapolated: −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 best variational: −0.496989 (CNN); best extrapolated: −0.496300 (TPS), 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 best variational: −0.496740 (CNN); best extrapolated: −0.497220 (Gutzwiller), 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) PP + RBM GCNN hours, as reported best variational: −0.496914 (CNN-MPS); best extrapolated: −0.496600 (TPS), 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 PP + RBM GCNN hours, as reported best variational: −0.485650 (CNN); best extrapolated: −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 best variational: −0.496799 (CNN-MPS), no cost stated, not drawn 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 4×16, 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 best variational: −0.766230 (ACE); best projected: −0.766470 (ACE); best extrapolated: −0.767480 (NNBF), no cost stated, not drawn Every mark is GPU-hours.
Heisenberg chain 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 (S.S) −0.4342 −0.4340 −0.4338 −0.4336 −0.4334 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ Jastrow RBM RNN DMRG FLOPs, estimated exact: −0.434124 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg chain 20: 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) −0.4455 −0.4450 −0.4445 −0.4440 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ Jastrow RBM RNN DMRG FLOPs, estimated exact: −0.445219 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
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 exact: −0.442208 (Exact solution), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 exact: −0.443168 (Exact solution), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Heisenberg kagome-2×3: 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) −0.4472 −0.4471 −0.4470 −0.4469 −0.4468 −0.4467 −0.4466 10¹² 10¹³ 10¹⁴ 10¹⁵ VQE DMRG FLOPs, estimated exact: −0.447126 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg 4×4: 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) −0.701750 −0.701745 −0.701740 −0.701735 −0.701730 −0.701725 −0.701720 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.701780 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg 6×6, 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) −0.605 −0.600 −0.595 −0.590 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ RBM Jastrow Adaptive RNN minGRU 2D RNN RNN Adaptive RNN DMRG RNN FLOPs, estimated exact: −0.603522 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Heisenberg 6×6: 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) −0.680 −0.675 −0.670 −0.665 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ RBM Jastrow CNN BF + RBM DMRG ViT 2D RNN FLOPs, estimated exact: −0.678872 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg 8×8, 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) −0.619040 −0.619035 −0.619030 −0.619025 −0.619020 10¹⁷ 10¹⁸ 10¹⁹ minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.619037 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg 8×8: 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) −0.67347 −0.67346 −0.67345 −0.67344 −0.67343 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ ViT 2D RNN FLOPs, estimated exact (stochastic): −0.673490 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg 10×10, 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) −0.630 −0.625 −0.620 −0.615 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ Jastrow RBM pRNN pRNN minGRU 2D RNN RNN minGRU DMRG ×2 2D RNN FLOPs, estimated exact (stochastic): −0.628656 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg 10×10: 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) −0.674 −0.672 −0.670 −0.668 −0.666 −0.664 −0.662 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ RBM Jastrow DMRG 2D RNN CNN BF + RBM RNN ViT ResNet FLOPs, estimated exact (stochastic): −0.671553 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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).
Heisenberg 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 (S.S) −0.63522 −0.63520 −0.63518 −0.63516 −0.63514 −0.63512 10¹⁷ 10¹⁸ 10¹⁹ minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.635201 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg 14×14, 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) −0.63995 −0.63990 −0.63985 −0.63980 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.639941 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg 14×14: 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) −0.675 −0.670 −0.665 −0.660 −0.655 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ DMRG Jastrow RBM 2D RNN FLOPs, estimated exact (stochastic): −0.670232 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Heisenberg 16×16, 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) −0.64355 −0.64350 −0.64345 −0.64340 −0.64335 10¹⁸ 10¹⁹ 10²⁰ 10²¹ minGRU minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.643532 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg 18×18, 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) −0.64635 −0.64630 −0.64625 −0.64620 −0.64615 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.646345 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg 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 (S.S) −0.64865 −0.64860 −0.64855 −0.64850 −0.64845 −0.64840 −0.64835 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.648609 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg 24×24, 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) −0.65200 −0.65195 −0.65190 −0.65185 −0.65180 −0.65175 10¹⁷ 10¹⁸ 10¹⁹ 10²⁰ minGRU 2D RNN FLOPs, estimated exact (stochastic): −0.652025 (SSE QMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Heisenberg triangular 16: 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) −0.540 −0.535 −0.530 −0.525 −0.520 −0.515 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ Jastrow RBM VQE DMRG FLOPs, estimated exact: −0.534720 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
Heisenberg triangular 36: 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 −0.56 −0.54 −0.52 −0.50 −0.48 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ Jastrow RBM RBM (QMBL) RBM (QMBL) DMRG (QMBL) DMRG DMRG (QMBL) ViT (QMBL) LCN GCNN DMRG (QMBL) GCNN (QMBL) FLOPs, estimated exact: −0.560373 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. Labels marked QMBL are QMBL's own reference runs (7 rows).
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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 best variational: −0.549329 (2D RNN), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
J1-J2 4×6, 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) −0.5226 −0.5224 −0.5222 −0.5220 −0.5218 −0.5216 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ VQE VQE DMRG FLOPs, estimated exact: −0.522525 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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).
J1-J2 4×4, J2 = 0.05: 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) −0.68060 −0.68055 −0.68050 −0.68045 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.680590 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.1: 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) −0.659785 −0.659780 −0.659775 −0.659770 −0.659765 −0.659760 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.659817 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.15: 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) −0.639512 −0.639510 −0.639508 −0.639506 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.639543 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.2: 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) −0.61988 −0.61986 −0.61984 −0.61982 −0.61980 −0.61978 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.619874 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.25: 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) −0.600935 −0.600930 −0.600925 −0.600920 −0.600915 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.600955 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.582955 −0.582950 −0.582945 −0.582940 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.582984 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.35: 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) −0.566235 −0.566230 −0.566225 −0.566220 −0.566215 −0.566210 −0.566205 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.566245 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.551115 −0.551110 −0.551105 −0.551100 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.551148 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.45: 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) −0.538277 −0.538276 −0.538275 −0.538274 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.538300 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.54 −0.53 −0.52 −0.51 −0.50 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ Jastrow RBM (QMBL) RBM RBM (QMBL) ViT (QMBL) GCNN (QMBL) VQE DMRG FLOPs, estimated exact: −0.528620 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 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 (4 rows).
J1-J2 4×4, 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) −0.52582 −0.52581 −0.52580 −0.52579 −0.52578 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.525896 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.65: 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) −0.5394 −0.5393 −0.5392 −0.5391 −0.5390 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.539382 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.563774 −0.563772 −0.563770 −0.563768 −0.563766 −0.563764 −0.563762 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.563858 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.75: 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) −0.59422 −0.59421 −0.59420 −0.59419 −0.59418 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.594273 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.62735 −0.62730 −0.62725 −0.62720 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.627335 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.85: 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) −0.66175 −0.66170 −0.66165 −0.66160 −0.66155 −0.66150 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.661720 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.69685 −0.69680 −0.69675 −0.69670 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.696866 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, J2 = 0.95: 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) −0.73250 −0.73248 −0.73246 −0.73244 −0.73242 −0.73240 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.732499 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 4×4, 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) −0.76850 −0.76845 −0.76840 −0.76835 10¹³ 10¹⁴ 10¹⁵ VQE VQE FLOPs, estimated exact: −0.768468 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. VQE estimates count the state-vector gate, metric-tensor and energy-derivative work of the run script's natural-gradient schedule. 2 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.565 −0.560 −0.555 −0.550 −0.545 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM DMRG ViT FLOPs, estimated exact: −0.562459 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.535 −0.530 −0.525 −0.520 −0.515 −0.510 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM DMRG ViT FLOPs, estimated exact: −0.529745 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.48 −0.46 −0.506 −0.504 −0.502 −0.500 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM (QMBL) RBM RBM (QMBL) DMRG (QMBL) DMRG (QMBL) ViT (QMBL) DMRG (QMBL) DMRG GCNN (QMBL) ViT ×2 ViT ViT RBM FLOPs, estimated exact: −0.503810 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 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 (7 rows).
J1-J2 6×6, 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) −0.495 −0.490 −0.485 −0.480 −0.475 −0.470 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM ViT DMRG FLOPs, estimated exact: −0.493239 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.54 −0.52 −0.50 −0.48 −0.46 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM ViT DMRG FLOPs, estimated exact: −0.530001 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.590 −0.585 −0.580 −0.575 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM ViT DMRG FLOPs, estimated exact: −0.586487 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.650 −0.645 −0.640 −0.635 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM ViT DMRG FLOPs, estimated exact: −0.649052 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
J1-J2 6×6, 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) −0.715 −0.710 −0.705 −0.700 −0.695 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ 10¹⁷ 10¹⁸ Jastrow RBM ViT DMRG FLOPs, estimated exact: −0.714360 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates needed an assumption (an architecture detail, or a Lanczos count taken from the calibration run).
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 best variational: −0.498963 (RBM + PP); best extrapolated: −0.499200 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 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 (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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 best extrapolated: −0.525300 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 best projected: −0.495700 (PEPS); best extrapolated: −0.498800 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 best variational: −0.478390 (CNN), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 best variational: −0.696700 (RBM), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this. 1 of these estimates 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 best variational: −0.496989 (CNN); best extrapolated: −0.496300 (TPS), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 best variational: −0.496740 (CNN); best extrapolated: −0.497220 (Gutzwiller), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, 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 best extrapolated: −0.496600 (TPS), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard chain 14, U = 1, n = 0.571: 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 −0.9116305 −0.9116300 −0.9116295 −0.9116290 10¹³ 10¹⁴ 10¹⁵ DMRG DMRG FLOPs, estimated exact: −0.911630 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard chain 14, U = 1.66810054, n = 0.571: 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 −0.8761540 −0.8761535 −0.8761530 10¹³ 10¹⁴ 10¹⁵ DMRG ×2 FLOPs, estimated exact: −0.876154 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard chain 14, U = 10, n = 0.571: 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 −0.7103640 −0.7103635 10¹³ 10¹⁴ 10¹⁵ DMRG DMRG FLOPs, estimated exact: −0.710364 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 4×16, 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 −0.770 −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 best variational: −0.766230 (ACE); best projected: −0.766470 (ACE); best extrapolated: −0.767480 (NNBF), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard 4×16, U = 8, n = 0.875, periodic/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/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 count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 4×8, U = 8, n = 0.875, periodic/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/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 count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 8×16, 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 −0.757 −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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard 8×24, 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard 8×32, 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard 4×4, U = 4, n = 0.625: 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 −1.2238090 −1.2238085 −1.2238080 −1.2238075 −1.2238070 −1.2238065 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ HFPS DMRG FLOPs, estimated exact: −1.223809 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 4×4, U = 6, n = 0.625: 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 −1.1473980 −1.1473975 −1.1473970 −1.1473965 −1.1473960 −1.1473955 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ HFPS DMRG FLOPs, estimated exact: −1.147398 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 4×4, U = 8, n = 0.625: 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 −1.0943985 −1.0943980 −1.0943975 −1.0943970 −1.0943965 −1.0943960 −1.0943955 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ HFPS DMRG FLOPs, estimated exact: −1.094398 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 4×4, U = 10, n = 0.625: 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 −1.056474 −1.056472 −1.056470 −1.056468 −1.056466 10¹⁶ 10¹⁷ 10¹⁸ 10¹⁹ HFPS DMRG FLOPs, estimated exact: −1.056472 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Hubbard 4×4, U = 4, n = 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/site −0.851145 −0.851140 −0.851135 −0.851130 −0.851125 10¹⁵ 10¹⁶ 10¹⁷ PITQS TQS FLOPs, estimated exact: −0.851366 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard 4×4, U = 8, n = 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/site −0.52920 −0.52915 −0.52910 −0.52905 10¹⁵ 10¹⁶ 10¹⁷ TQS PITQS FLOPs, estimated exact: −0.529305 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Hubbard 8×8, U = 8, n = 1, periodic/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/site −0.5005 −0.5000 −0.4995 −0.4990 −0.4985 −0.4980 10¹⁸ 10¹⁹ 10²⁰ DMRG DMRG FLOPs, estimated exact (stochastic): −0.500453 (AFQMC), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
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 best variational: −0.758652 (VAFQMC); best projected: −0.758300 (ACE), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training.
Spinless t-V chain 32: 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 −0.4983395 −0.4983390 −0.4983385 10¹² 10¹³ 10¹⁴ 10¹⁵ DMRG DMRG FLOPs, estimated exact: −0.498339 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising chain 10, 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) −1.2382 −1.2381 −1.2380 −1.2379 −1.2378 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² Jastrow DMRG FLOPs, estimated exact: −1.238149 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising chain 10: 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) −1.2790 −1.2785 −1.2780 −1.2775 −1.2770 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² Jastrow DMRG FLOPs, estimated exact: −1.278491 (Exact solution), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising chain 32: 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) −1.0636 −1.0634 −1.0632 −1.0630 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ Jastrow DMRG FLOPs, estimated exact: −1.063544 (Exact solution), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising chain 32: 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) −1.2740 −1.2735 −1.2730 −1.2725 −1.2720 −1.2715 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ Jastrow RBM DMRG FLOPs, estimated exact: −1.273751 (Exact solution), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising 6×6: 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 (Pauli) −3.2010 −3.2005 −3.2000 −3.1995 10¹³ 10¹⁴ 10¹⁵ 10¹⁶ Jastrow RBM DMRG FLOPs, estimated exact: −3.200909 (ED), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising 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 count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Transverse-field Ising 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 best variational: −3.181970 (DMRG), no cost stated, not drawn Every mark is an estimate, good to an order of magnitude. Network estimates count 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. They include the optimizer's solve where the code is known, and leave out pre-training. DMRG estimates count the dense tensor contractions of every sweep in the run script's schedule, plus its variance evaluations. They leave out the saving from conserved quantum numbers, so a symmetric code does less arithmetic than this.
Heisenberg chain 20, 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) −0.4342 −0.4340 −0.4338 −0.4336 −0.4334 10¹ 10² 10³ 10⁴ Jastrow RBM RNN variational parameters exact: −0.434124 (ED), 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 chain 20: 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) −0.4455 −0.4450 −0.4445 −0.4440 10⁰ 10¹ 10² 10³ 10⁴ Jastrow RBM Gutzwiller RNN variational parameters exact: −0.445219 (ED), 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 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 Gutzwiller variational parameters exact: −0.443168 (Exact solution), 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 6×6, 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) −0.605 −0.600 −0.595 −0.590 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ RBM Jastrow minGRU RNN RNN Adaptive RNN ×2 2D RNN ×2 variational parameters exact: −0.603522 (ED), 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 6×6: 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) −0.670 −0.668 −0.6790 −0.6789 −0.6788 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ RBM Jastrow CNN CNN ViT 2D RNN ×2 CNN BF + RBM variational parameters exact: −0.678872 (ED), 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 8×8: 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) −0.67352 −0.67350 −0.67348 −0.67346 −0.67344 −0.67342 10⁴ 10⁵ 10⁶ 10⁷ ViT 2D RNN 2D RNN variational parameters exact (stochastic): −0.673490 (SSE QMC), 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 10×10, 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) −0.630 −0.625 −0.620 −0.615 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ Jastrow RBM minGRU pRNN ×2 2D RNN PEPS RNN minGRU PEPS 2D RNN ×2 variational parameters exact (stochastic): −0.628656 (SSE QMC), 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 10×10: 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) −0.674 −0.672 −0.670 −0.668 −0.666 −0.664 −0.662 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ Jastrow RBM Gutzwiller CNN aCNN CNN BF + RBM RNN ViT 2D RNN ×2 ResNet variational parameters exact (stochastic): −0.671553 (SSE QMC), 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 14×14: 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) −0.672 −0.670 −0.668 −0.666 −0.664 −0.662 −0.660 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ Jastrow RBM Gutzwiller 2D RNN ×2 variational parameters exact (stochastic): −0.670232 (SSE QMC), 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 16×16, 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) −0.64355 −0.64350 −0.64345 −0.64340 −0.64335 10⁵ 10⁶ 10⁷ 10⁸ PEPS minGRU minGRU 2D RNN variational parameters exact (stochastic): −0.643532 (SSE QMC), 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 16×16: 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) −0.6700 −0.6699 −0.6698 −0.6697 −0.6696 10³ 10⁴ 10⁵ 10⁶ 10⁷ aCNN 2D RNN 2D RNN variational parameters exact (stochastic): −0.669977 (SSE QMC), 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 24×24: 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) −0.6697 −0.6696 −0.6695 −0.6694 −0.6693 −0.6692 10³ 10⁴ 10⁵ 10⁶ 10⁷ aCNN 2D RNN 2D RNN variational parameters exact (stochastic): −0.669604 (SSE QMC), 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 16: 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) −0.540 −0.535 −0.530 −0.525 −0.520 −0.515 10¹ 10² 10³ Jastrow RBM VQE variational parameters exact: −0.534720 (ED), 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 36: 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 −0.56 −0.54 −0.52 −0.50 −0.48 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM RBM (QMBL) RBM (QMBL) GCNN (QMBL) ViT (QMBL) aCNN LCN GCNN variational parameters exact: −0.560373 (ED), 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 best variational: −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 best variational: −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 Gutzwiller variational parameters best variational: −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 4×4, 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) −0.54 −0.53 −0.52 −0.51 −0.50 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM (QMBL) RBM RBM (QMBL) ViT (QMBL) GCNN (QMBL) VQE variational parameters exact: −0.528620 (ED), 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 6×6, 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) −0.565 −0.560 −0.555 −0.550 −0.545 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM ViT variational parameters exact: −0.562459 (ED), 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 6×6, 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) −0.535 −0.530 −0.525 −0.520 −0.515 −0.510 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM CNN ViT variational parameters exact: −0.529745 (ED), 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 6×6, 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 −0.504 −0.502 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM (QMBL) RBM RBM (QMBL) CNN Gutzwiller CNN ViT (QMBL) GCNN (QMBL) ViT ViT ×2 ViT aCNN RBM variational parameters exact: −0.503810 (ED), 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 6×6, 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) −0.495 −0.490 −0.485 −0.480 −0.475 −0.470 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM CNN ViT variational parameters exact: −0.493239 (ED), 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 6×6, 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) −0.54 −0.52 −0.50 −0.48 −0.46 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM ViT variational parameters exact: −0.530001 (ED), 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 6×6, 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) −0.590 −0.585 −0.580 −0.575 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM ViT CNN variational parameters exact: −0.586487 (ED), 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 6×6, 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) −0.650 −0.645 −0.640 −0.635 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM ViT variational parameters exact: −0.649052 (ED), 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 6×6, 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) −0.715 −0.710 −0.705 −0.700 −0.695 10² 10³ 10⁴ 10⁵ 10⁶ Jastrow RBM ViT CNN variational parameters exact: −0.714360 (ED), 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 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) Gutzwiller ViT (QMBL) RBM + PP ViT RBM variational parameters best variational: −0.498963 (RBM + PP); best extrapolated: −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 Gutzwiller 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 best extrapolated: −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 Gutzwiller 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 best projected: −0.495700 (PEPS); best extrapolated: −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 best variational: −0.486930 (Gutzwiller); best projected: −0.485300 (PEPS); best extrapolated: −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 best variational: −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 best variational: −0.496989 (CNN); best extrapolated: −0.496300 (TPS), 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) Gutzwiller ViT (QMBL) CNN variational parameters best variational: −0.496740 (CNN); best extrapolated: −0.497220 (Gutzwiller), 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 best extrapolated: −0.496600 (TPS), 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 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 4×16, 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 −0.751 −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 best variational: −0.766230 (ACE); best projected: −0.766470 (ACE); best extrapolated: −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 Jastrow + BF Jastrow + BF Tensor BF SCALE Tensor BF Transformer BF ACE VAFQMC variational parameters best projected: −0.758300 (ACE), 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.
open
Transverse-field Ising chain 10, 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) −1.2382 −1.2381 −1.2380 −1.2379 −1.2378 10¹ 10² 10³ Jastrow VQE VQE variational parameters exact: −1.238149 (ED), 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.
Transverse-field Ising 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

T. Zerweck and J. Carrasquilla, QMBL: the Quantum Many-Body Leaderboard, arXiv:2610.22387 (2026).

To cite the data itself, use the concept DOI, which always resolves to the latest release: 10.5281/zenodo.22753734.

Individual energies should be cited to the primary paper named on the row, not to this site.