Every method family measured on every exactly solvable model: 70 combinations — 51 completed, 17 unsuitable, 2 failed. Each completed entry is the largest size that method reached within the compute budget, with its relative error against the exact solution.
Benchmark — largest size each method reached within 100 CPU core-hours per run; relative error vs the exact solution at that size. Accuracy bar: 10-8 for deterministic routes, 10-3 for variational, 2σ for stochastic; entries above their bar report the method's intrinsic limit at this size (2026-07).
✓ completed — largest size · relative error (hover for
the measured observable) ∅ unsuitable — the method cannot express
this model (hover for why) ✗ failed — the software route broke
(largest size attempted).
model \ method
ed
mps
peps
ltrg
mcrg
qmc
vmc
mf
polyopt
qcs
tfim-chain (L sites)
✓ 34 · 1.4×10-12
✓ 256 · 3.7×10-12
∅
✓ 1.0×106 · 3.4×10-5
✓ 9 · 0.22
✓ 128 · 6.0×10-5
✓ 128 · 0.0034
✓ 1.1×109 · 0.018
✓ 1.1×105 · 0.57
✓ 1024 · 0.0085
ising-2d-onsager (L (L×L))
✓ 5 · 0.0024
∅
✓ 160 · 2.1×10-4
✓ 1024 · 0.002
✓ 6561 · 9.8×10-4
✓ 256 · 0.0016
∅
✓ 32 768 · 0.76
∅
✗ 4
toric-code (L (L×L, 2L² qubits))
✓ 5 · 6.9×10-12
✓ 9 · 5.5×10-12
✓ ∞ · 2.2×10-16
✓ 28 · 3.2×10-14
∅
∅
✓ 5 · 0.035
✓ 88 000 · 0.5
✓ 112 · 1.1×10-10
✓ 576 · 0
aklt-chain (L sites)
✓ 20 · 5.0×10-16
✓ ∞ · 1.7×10-16
✓ ∞ · 1.6×10-15
✓ 1.1×109 · 1.8×10-8
∅
✓ 16 · 0.046
✓ 16 · 0.078
✓ 3.2×109 · 0.5
✓ 32 768 · 0.12
✓ 16 384 · 1.2×10-14
hubbard-1d-lieb-wu (L sites)
✓ 20 · 0.0027
✓ 256 · 2.7×10-6
∅
✓ 8192 · 0.0084
∅
✓ 254 · 0.0021
✗ 24
✓ 2.2×1012 · 0.18
✓ 16 384 · 2.5
✓ 14 · 0.25
lmg (N spins)
✓ 109 · 0
✓ 1.3×105 · 1.9×10-12
∅
∅
∅
✓ 3.0×105 · 2.4×10-5
✓ 16 384 · 1.7×10-8
✓ 2.5×1010 · 8.6×10-12
✓ 1011 · 1.4×10-11
✓ 4.3×108 · 3.3×10-11
kicked-ising-floquet (L qubits)
✓ 18 · 0.013
✓ ∞ · 1.4×10-15
✓ ∞ (t=11) · 3.2×10-11
∅
∅
∅
∅
✓ 760 · 1.1×10-13
∅
✓ 109 · 3.1×10-6
Frontier figure — each point is one completed model × method entry: the largest size reached within the compute budget (x) against the relative error at that size (y); color = method, shape = model. The x axis converts each model's size to the log₂ of its Hilbert-space dimension: L (TFIM chain, kicked Ising), 1.58 L (AKLT, 3 states per site), 2 L (Hubbard), L² (2D Ising, classical configurations), 2 L² (toric code), N (LMG). Entries that reached the infinite-size limit sit in the ∞ band past the axis break; points on the bottom edge are machine-exact (error 0). The dashed lines mark the deterministic (10-8) and variational (10-3) accuracy bars; the stochastic bar — agreement with the exact value within 2σ of the run's own statistical error bar — is a per-point criterion and has no fixed level to draw.