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 (2)
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.
-0.7017802, ED
Checked 2026-09-29 · QMBL ED cost run checks/cost/ed/results/Heisenberg--square_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 -44.91393283371538 (stored_exact -44.9139328, relative_difference 7.51e-10, reproduces_stored true)
|reproduced - stored| = 3.37e-8 (relative 7.51e-10); within 1e-9 relative. Sector total Sz = 0, dimension 12870, 61 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__square_16_P.json), gives -44.91393283371616 (7.82e-13 from the cost run). Also read by VB3-varbench-code-heisenberg#38 (scope config: vendor/varbench/Heisenberg/square_16_P.md (snapshot 390a21e), row 'Exact diagonalization'; linked scripts/Heisenberg/square_16_P/vqe.sh).
-0.7017449, 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. | -44.9116731 | | 0.1064 | 16 | 0 | VQE + symm. circuit (64 pars., exact grad, statevector) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_16_P/vqe.sh) |
Best variational bound on the solved instance. vqe.sh: --j2=0.0 --lattice=square4x4 -> HeisenbergSquare, NN bonds of the periodic 4x4 lattice, H = sum 2 P_ij - N_bonds = sigma.sigma (hamiltonians.py lines 26-27, 170-229), Sz = 0, then --symmetry=1 (projection onto the eigenvalue-1 sector of the listed generators). The logged energy is the exact statevector expectation of the normalised projected state (observables.py lines 196-199, '{:.7f}' line 237): deterministic, so sigma null is right. 'exact grad' is the default 2^36 shots per gradient term (main.py line 13), effectively exact. 0.00226 above the exact energy. Which iteration VarBench took is not stated; every logged line is a strict bound. Entered 2022-11-11 (N. Astrakhantsev), never changed.
-0.7017252, 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. | -44.9104126 | | 0.1599 | 16 | 0 | VQE + symm. circuit (64 pars., 2^14 samples/grad) | [code](https://github.com/varbench/methods/blob/main/scripts/Heisenberg/square_16_P/vqe_noisy.sh) |
vqe_noisy.sh: same Hamiltonian and projection with --log2samples=14. The logged energy is still the exact statevector expectation of the projected state (observables.py lines 196-199: state = self.circuit(noisy=False); circuits.py __call__ returns the noiseless state), so the 2^14 shots enter only the gradient and metric estimates; the energy carries no statistical error (see the ambiguous entry on eligibility). 0.00352 above the exact energy and 0.00126 above the exact-grad variant, so nothing changes on this instance. Entered 2022-11-11 (N. Astrakhantsev), never changed; linked 5285adc (2024-08-12).