Every published energy
Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (4)
Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.
How these numbers were read
Where each number below was read, and how it was checked.
-0.4471262, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Heisenberg--kagome-2x3_18_P.json (checks/cost/ed/run_ed.py: NetKet 3.22.4 operator, sparse CSR built in blocks, SciPy eigsh/ARPACK k=1 SA; commit 716e043); results file read locally, no rerun
Reported as. energy -32.193083094164926 (stored_exact -32.19308309416487, relative_difference 1.77e-15, reproduces_stored true)
|reproduced - stored| = 5.68e-14 (relative 1.77e-15); within 1e-9 relative. Sector total Sz = 0, dimension 48620, 221 Lanczos steps (the file states no residual). The stored value is the source's own ED (varbench@2024-10-22), so two codes agree; a third code, E1's node Lanczos (qmbl-runs/qmbl-verify/out/E1-exact-recompute/results/Heisenberg__kagome-2x3_18_P.json), gives -32.19308309416476 (1.63e-13 from the cost run). Also read by VB3-varbench-code-heisenberg#19 (scope config: vendor/varbench/Heisenberg/kagome-2x3_18_P.md (snapshot 390a21e), row 'Exact Diagonalization'; linked scripts/Heisenberg/kagome-2x3_18_P/ed_netket.sh).
-0.4471262, DMRG
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -32.19308309416483 | | 1e-14 | 18 | 0 | DMRG (bond dimension = 368) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/kagome-2x3_18_P/dmrg.sh) |
dmrg.sh: heisenberg_kagome.jl --L 0 --zero_mag --max_B 512: explicit 36-edge list (lines 17-55, isomorphic to the ED cluster, work/kagome18.mjs), Pauli (J *= 4, line 63), Sz = 0, 50 sweeps, cutoff 1e-12: 368 is the reached bond dimension under the cutoff (cap 512). Best variational bound on the solved instance: 4e-14 above the ED row and 1e-13 above QMBL's ED, i.e. exact to machine precision. M6 (V-score 1.7e-16 outlier): genuine. Entered bbcfdc1 (2023-01-15, Dian Wu), never changed; linked f88f9ba/cd362c4 (2024-07-19).
-0.4467304, VQE
Checked 2026-09-29 · VarBench snapshot 390a21e (vendor/varbench) and full upstream history (varbench-history, git log --follow on the file, every version); varbench/methods ed31bb0 run script and program read as text; no output log or result file exists in the methods repo for any Heisenberg row
Reported as. | -32.1645862 | | 1.20 | 18 | 0 | VQE (SR + symm. + 108 variational pars) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/kagome-2x3_18_P/vqe.sh) |
Program VQE: H = sum 2 P_ij - N_bonds = sigma.sigma over the 36 Kagome18 bonds (hamiltonians.py lines 26-27, 548-568), isomorphic to the instance's cluster; Sz = 0; symmetry projection; the logged energy is the exact statevector expectation of the projected state (observables.py lines 196-199, 237), deterministic. Script defect, value unaffected: vqe.sh passes --lattice=kagome3x2, which main.py's argparse rejects (choices square4x4, triangle4x4, kagome2x3, square6x4; line 12), so the script as committed exits before building anything; with the parser's own spelling the only kagome branch (config.py line 108) builds this Hamiltonian. 0.0285 above the exact energy. Entered 2015a4d (2022-10-18, N. Astrakhantsev), never changed; linked by him 5285adc (2024-08-12).