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.
-0.5347197, 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. | -34.222059670065036 | | 1e-15 | 16 | 0 | DMRG (bond dimension = 256) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_16_P/dmrg.sh) |
dmrg.sh: heisenberg_2d.jl --peri --L 4 --J2 1 --zero_mag --max_B 256: 16 sites, D = 256 = 2^8 is an exact MPS; Pauli, J = 1, Sz = 0. Best variational bound on the solved instance: equal to the ED to float precision (1.6e-14 below VarBench's ED row, 5.4e-14 above QMBL's ED; relative 5e-16, far inside the validator's 1e-8). M6 (V-score 1.4e-17 outlier): genuine. Entered ad306c2 (2022-10-19, Dian Wu) with variance 0, set to 1e-15 in 638bcda ('Add note for machine precision'); linked 2024-07-19.
-0.5347197, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Heisenberg--triangular_16_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 -34.22205967006509 (stored_exact -34.22205967006502, relative_difference 2.08e-15, reproduces_stored true)
|reproduced - stored| = 7.11e-14 (relative 2.08e-15); within 1e-9 relative. Sector total Sz = 0, dimension 12870, 121 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__triangular_16_P.json), gives -34.222059670065185 (9.24e-14 from the cost run). Also read by VB3-varbench-code-heisenberg#66 (scope config: vendor/varbench/Heisenberg/triangular_16_P.md (snapshot 390a21e), row 'Exact diagonalization'; linked scripts/Heisenberg/triangular_16_P/ed_netket.sh).
-0.52417(5), VMC (RBM)
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. | -33.5469 | 0.0034 | 12.0932 | 16 | 0 | RBM (alpha = 1) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_16_P/vmc_rbm.sh) |
scripts/Heisenberg/triangular_16_P/vmc_rbm.sh: 'python3 vmc.py --ham heis_tri --boundary peri --sign mars --ham_dim 2 --L 4 --zero_mag --net rbm --seed 123 --show_progress' -> 4x4 triangular lattice (ColoredJ1J2 back_diag=False: square grid + (i,j)-(i+1,j+1) diagonals), periodic; NetKet Heisenberg J = 1 in Pauli form with Marshall rotation on the square bonds only (sign_rule=[True, False], a unitary: the diagonals join same-sublattice sites); Sz = 0 (total_sz=0, MetropolisExchange); real float32 RBM, alpha = 1 (nk.models.RBM, log_cosh); 10^4 SR steps; final estimate over 2^20 samples (vmc.py lines 268-269). sigma/sqrt(Var/2^20) = 1.001: the stored sigma is the error of that final estimate. 0.6752 above the exact -34.22205967006502. M2/M6 flags (if any): real float32 parameters make the amplitudes positive, so on this frustrated lattice the ansatz cannot carry the ground-state signs; no constant shift is involved and the energy is a valid bound. History: entered ad306c2 (2022-10-19, Dian Wu) as -33.5469 | 0.0034 | 12.0932 | RBM (alpha = 1); linked 5047bfd (2024-07-30). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.
-0.51710(4), VMC (Jastrow)
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. | -33.0945 | 0.0027 | 7.37471 | 16 | 0 | Jastrow baseline | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/triangular_16_P/vmc_jastrow.sh) |
scripts/Heisenberg/triangular_16_P/vmc_jastrow.sh: 'python3 vmc.py --ham heis_tri --boundary peri --sign mars --ham_dim 2 --L 4 --zero_mag --net jas --seed 123 --show_progress' -> 4x4 triangular lattice (ColoredJ1J2 back_diag=False: square grid + (i,j)-(i+1,j+1) diagonals), periodic; NetKet Heisenberg J = 1 in Pauli form with Marshall rotation on the square bonds only (sign_rule=[True, False], a unitary: the diagonals join same-sublattice sites); Sz = 0 (total_sz=0, MetropolisExchange); real float32 Jastrow (nk.models.Jastrow); 10^4 SR steps; final estimate over 2^20 samples (vmc.py lines 268-269). sigma/sqrt(Var/2^20) = 1.018: the stored sigma is the error of that final estimate. 1.1276 above the exact -34.22205967006502. M2/M6 flags (if any): real float32 parameters make the amplitudes positive, so on this frustrated lattice the ansatz cannot carry the ground-state signs; no constant shift is involved and the energy is a valid bound. History: entered ad306c2 (2022-10-19, Dian Wu) as -33.0945 | 0.0027 | 7.37471 | Jastrow baseline; linked fdb4bfb (2024-07-30). Program files postdate the value (scripts added to varbench/methods 2024-07-30); they document the settings after the run.