Validation¶
python benchmarks/validation_suite.py runs every check below against an exact, analytic or independent numerical reference and writes the report reproduced on this page. The suite runs in continuous integration on every push and pull request. A tolerance tied to floating-point roundoff (for example norm conservation) is derived from the number of operations rather than fixed, because rounding differs between CPU architectures.
In addition, the unit-test suite contains more than 100 physics and API tests, and the GPU path has been verified on an NVIDIA Tesla T4 (see GPU acceleration).
Generated by python benchmarks/validation_suite.py. Every row compares a
simulation with an exact, analytic or independent numerical reference.
Platform: Python 3.14.6, NumPy 2.4.6, arm64.
Total runtime: 73 s.
| Category | Check | Reference | Metric | Error | Tolerance | Result |
|---|---|---|---|---|---|---|
| Dynamics | Free Gaussian spreading (1D) | σ(t) = σ₀√(1+(t/2σ₀²)²), exact density | max |ρ − ρ_exact| | 1.37e-15 | 1e-09 | pass |
| Dynamics | Temporal convergence, Strang splitting | global error ∝ dt^2 | observed order 2.001 | 9.10e-04 | 5e-02 | pass |
| Dynamics | Temporal convergence, Yoshida splitting | global error ∝ dt^4 | observed order 3.997 | 3.47e-03 | 5e-02 | pass |
| Dynamics | Driven harmonic oscillator, t ≤ 20 | Ehrenfest: ⟨x⟩ = classical forced trajectory | max |⟨x⟩ − x_cl| | 5.60e-09 | 1e-07 | pass |
| Dynamics | Norm conservation, 3D, 500 steps | unitary evolution: ‖ψ‖ = 1 up to roundoff | |‖ψ‖ − 1| | 2.55e-15 | 6e-12 | pass |
| Dynamics | Energy conservation, 3D, 500 steps (order 4) | E(t) = E(0) | |ΔE/E| | 2.18e-10 | 1e-08 | pass |
| Observables | Minimum-uncertainty Gaussian (3D) | Δr_j Δp_j = ℏ/2 for each axis | max |Δr Δp − ½| | 4.60e-12 | 1e-08 | pass |
| Observables | Angular momentum of a vortex state | ⟨L_z⟩ = ℏ for ψ ∝ (x+iy)e^{−r²/2} | |⟨L_z⟩ − 1| | 2.22e-16 | 1e-10 | pass |
| Stationary states | Harmonic spectrum 1D, 10 levels (Lanczos) | E_n = n + ½ | max |E_n − exact| | 7.28e-14 | 1e-10 | pass |
| Stationary states | Anisotropic 3D harmonic spectrum, 6 levels (LOBPCG) | E = Σ ω_j (n_j + ½) | max |E − exact| | 1.42e-13 | 1e-08 | pass |
| Stationary states | 3D ground state by imaginary time | E₀ = ½(ω_x+ω_y+ω_z) | |E₀ − exact| | 6.95e-10 | 1e-07 | pass |
| Stationary states | Spectrum from autocorrelation (Feit–Fleck) | peaks of FFT[C(t)] at n + ½ | max peak offset (resolution 2π/T = 0.031) | 1.62e-03 | 3e-03 | pass |
| Gross–Pitaevskii | Moving bright soliton, t = 20 | exact sech solution of the 1D NLS | max |ρ − ρ_exact| | 1.46e-09 | 1e-08 | pass |
| Gross–Pitaevskii | Virial theorem, 3D GP ground state | 2⟨T⟩ − 2⟨V⟩ + 3E_int = 0 | |2T − 2V + 3E_int| / V | 2.32e-06 | 1e-04 | pass |
| Gross–Pitaevskii | Imaginary-time GP ground state accuracy | residual ∝ dτ² (ratio 4 when halving dτ) | observed ratio 3.99 | 2.23e-03 | 1e-01 | pass |
| Rotating frame | Coherent state in a frame rotating at Ω = 0.6 | ⟨r'⟩ = R(−Ωt)·(x₀ cos ωt, 0) | max |⟨r'⟩ − exact| | 1.89e-08 | 1e-07 | pass |
| Rotating frame | Vortex eigenstate in the rotating frame | C(t) = e^{−i(2ℏω − Ωℏ)t/ℏ} | |C(t) − exact| | 1.15e-08 | 1e-07 | pass |
| Strong-field physics | 1D soft-core hydrogen ground state | E₀ = −½ a.u. (I_p = 13.6 eV) | |E₀ + ½| | 1.94e-14 | 1e-09 | pass |
| Strong-field physics | Three-step model maximum return energy | max E_return = 3.17 U_p | |max/U_p − 3.1731| | 3.63e-05 | 2e-04 | pass |
| Strong-field physics | HHG: only odd harmonics (800 nm, 10¹⁴ W/cm²) | inversion symmetry forbids even orders | even/odd intensity, H5–H20 (odd/even = 2.3e+03) | 4.30e-04 | 1e-02 | pass |
| Strong-field physics | Attosecond pulse train (H11–H21) | one burst per half laser cycle | max |spacing − ½| (cycles) | 3.08e-03 | 1e-02 | pass |
| Two-component BEC | Bogoliubov spin mode, miscible mixture (t ≤ 60) | δ(n₁−n₂) ∝ cos ω_s t, ω_s² = ε_k(ε_k + 2n(g − g₁₂)) | max relative deviation | 8.53e-10 | 1e-06 | pass |
| Two-component BEC | Phase-separation instability, g₁₂ = 1.3 g | growth rate γ = √(ε_k(2n(g₁₂ − g) − ε_k)) | |γ_measured/γ − 1| | 9.39e-05 | 1e-03 | pass |
| Two-component BEC | Manakov point g₁₁ = g₂₂ = g₁₂ | n₁ + n₂ equals the one-component GP ground state | max |n₁ + n₂ − n| | 1.76e-07 | 1e-06 | pass |
| Two-component BEC | Virial theorem, coupled GP ground state | 2⟨T⟩ − 2⟨V⟩ + 2E_int = 0 (2D) | |2T − 2V + 2E_int| / V | 2.27e-06 | 1e-04 | pass |
| Stationary states | Conjugate-gradient minimiser, hydrogen (V < 0) | E₀ = −½ a.u. | |E₀ + ½| | 7.77e-16 | 1e-12 | pass |
| Gross–Pitaevskii | Virial theorem, 3D GP ground state (conjugate gradient) | 2⟨T⟩ − 2⟨V⟩ + 3E_int = 0, no dτ error | |2T − 2V + 3E_int| / V | 5.79e-10 | 1e-08 | pass |
27/27 checks passed.