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.
-0.6399410(2), SSE QMC
Checked 2026-09-15 · arXiv HTML parsed locally, one value per table cell; no LLM transcription
Reported as. E0/N_b = 0.3445836(1) (magnitude, per bond, N_b = 364), L = 14, open
Table 2, "SSE data for L x L systems with open boundary conditions. The ground state energy is normalized by the number of interaction bonds N_b = 2(L^2 - L)", row L = 14. Printed as a magnitude; E/N = -(E0/N_b) N_b / N = -0.63994097 in S.S units, Pauli total = 4 N E/N. The L = 6 value agrees with the exact-diagonalization row on square_36_O to 3.3e-7 (2 sigma), and L = 10 and 16 reproduce the rows QMBL already quotes from this paper via arXiv:2605.13807.
-0.639929(6), VMC (2D RNN)
Checked 2026-09-16 · authors' data release read locally (pickle via numpy), values copied by key, no transcription; the file is committed as sources/2502.17144-repo-final_energy_data.json and reproduces the four values Table V prints
Reported as. -0.6399285841961295 +/- 0.0000056564089371291915 per site (S.S), Var(E/N) = 3.1994962064034987e-7
github.com/mschuylermoss/HeisenbergRNN, commit 29bf62ac28 (2025-02-24, 'reproduce figures'), Square/final_plotting/final_data/final_energy_data_plotting.pkl: ['open']['rate=0.25']['scale=4'], L = 14; the authors' best run, the lowest of the s = 2 and 4 training schedules (s = 4, r = 0.25). Plotted as a point in Fig. 12(a), not printed in the paper; Table III prints the zero-variance extrapolation over the s >= 1 runs at this L, carried on this instance as an `extrapolated` row. Variance is Var(H)/N^2 in S.S units (see script header); total = var x N^2 x 16.
-0.639842(6), VMC (minGRU)
Checked 2026-10-27 · arXiv:2605.13807: arXiv:2605.13807v1 (only version; no journal reference on 2026-10-27): pypdf layout text sources/2605.13807.txt read locally, checked against Table 6 of the arXiv HTML sources/2605.13807.html; no LLM transcription; single reading
Reported as. -0.639842(6)
TABLE VI, row N = 14^2 (line 686 of sources/2605.13807.txt) | energy per site in S.S units, J = 1, open boundaries; the same table row prints the QMC value equal to the instance's stored SSE row