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VALIDATION

What has been tested, with which tolerance, what came out, and what is not yet verified. Status at version 0.4.0 (2026-10-16).

Environment of the recorded runs: macOS (Apple M1, 16 GB), Python 3.13.5 and 3.14.6, NumPy 2.5.4, SciPy 1.18.1, PennyLane 0.45.1. GitHub Actions (first run on 2026-10-15, commit fc38cc8): lint, type check, docs build and the fast test suite pass on Ubuntu and macOS for Python 3.11, 3.12 and 3.13, and on Python 3.14 (Ubuntu).

Code quality at 0.2.0: 170 fast tests + 4 slow tests pass, line coverage 95 % (gate 85 %), ruff and mypy clean, mkdocs build --strict succeeds with the tutorials executed.

Reproduce:

pytest                                        # fast suite
pytest -m slow                                # statistical tests (~1-2 min)
python scripts/validate_level_statistics.py   # section 2 tables (~40 min)
make figures                                  # figures/*.png (+ JSON sidecars)

1. Exact and deterministic checks (unit tests)

Each bug fix of phase 0 was preceded by a test that failed on the original code (git log shows the failing-first commits with the original output).

Area Check Tolerance Test file
Partial trace Tr_B(ρ_A⊗ρ_B)=ρ_A for 2×2, 2×3, 3×2 and three subsystems; Tr=1; product state → projector (old code: zero matrix) 1e-12 test_partial_trace.py
Entropy product 0, Bell 1 bit, GHZ 1 bit, S_A=S_B for pure states 1e-12 / 1e-10 test_partial_trace.py
Fidelity F(ρ,ρ)=1 (pure, rank 1, 2, full; old NISQ code gave 4), pure overlap, symmetry, bounds, commuting states 1e-10 test_fidelity.py
Floquet static H: quasi-energies = folded eigenvalues (old code: sign inverted); F = exp(−iHT); unitarity; midpoint propagator converges as n⁻² 1e-10 / 1e-12 / slope ±0.15 test_floquet.py
Noise Kraus completeness; |ρ01| = ½e^{−t/T2} and P1 = e^{−t/T1} (old: rate 1/(2T1)+1/(2T2)); T2>2T1 rejected; H=0 result independent of step count; NumPy ≡ PennyLane default.mixed 1e-10 – 1e-14 test_noise_channels.py, test_no_fake_features.py
PennyLane 0.45 all subpackages import (old: 3 of 6 failed on Python ≤3.13); Pauli Hamiltonian matrix; StatePrep evolution; set_shots; Trotter error slopes 1 and 2 1e-12 / 1e-8 / ±0.15 test_pennylane_compat.py
OTOC circuit Hadamard-test circuit = exact OTOC, T=∞ (Bell-pair purification) and pure state (old circuit: 1 for all t) 1e-8 test_no_fake_features.py
Kernels K(x,x)=1, symmetric PSD; every feature changes the state (old SYK kernel: mean(x) only; old Hubbard kernel: K≡1); circuit = state-vector kernel 1e-12 / 1e-8 test_quantum_kernel.py
Effective Hamiltonian order 0 = period average; higher orders raise (old: silently ignored) 1e-12 test_no_fake_features.py
SYK construction Pauli-string build = Jordan–Wigner product build (N=4…12); χ²=½ gives H/4; Tr H²/D = Σ J² 1e-12 test_syk_symmetry.py
SYK symmetry P = (∏χ_{2j})K: [P,H]=0, P²=(−1)^{n(n−1)/2}, PΓP⁻¹=(−1)^nΓ (N=8…16); GUE sectors degenerate; GSE Kramers pairs; GOE no degeneracy 1e-12 / 1e-10 test_syk_symmetry.py
Hubbard ED [H,T]=[H,R]=[H,S²]=0; [H,η²]=0 only for t'=0; union of blocks = full sector spectrum; U=0 free fermions with t'; t=0 doublon counting; full Fock spectrum = Jordan–Wigner model (L=3,4); k ≡ −k 1e-10 – 1e-12 test_hubbard_ed.py
RMT references sampled GOE/GUE/GSE/Poisson reproduce Atas et al. (dim 64); surmise densities integrate to 1 and reproduce the tabulated surmise means 3σ+1e-3 / 1e-10 test_spectral.py
README the quick-start block executes — test_readme.py

2. Level statistics

Produced by scripts/validate_level_statistics.py (raw numbers and provenance in results/level_statistics.json; package source tree 03c1a9a). Errors are one standard error. Same-size RMT: random GOE/GUE/GSE matrices with the same number of distinct levels as one SYK sector, analysed identically (syk_rmt_reference); z is the difference in units of the combined standard error.

2.1 SYK (sector resolved; class from N mod 8)

N class levels/sector realisations ⟨r⟩ SYK ⟨r⟩ RMT same size z Atas (N→∞)
10 GUE 16 400 0.6024 ± 0.0037 0.5972 ± 0.0030 +1.10 0.5996
12 GSE 16 400 0.6731 ± 0.0023 0.6720 ± 0.0026 +0.31 0.6744
14 GUE 64 300 0.5997 ± 0.0019 0.5990 ± 0.0015 +0.31 0.5996
16 GOE 128 200 0.5305 ± 0.0015 0.5293 ± 0.0012 +0.65 0.5307
18 GUE 256 100 0.5983 ± 0.0019 0.5996 ± 0.0012 −0.58 0.5996
20 GSE 256 40 0.6743 ± 0.0016 0.6758 ± 0.0011 −0.74 0.6744
22 GUE 1024 20 0.5997 ± 0.0023 0.6000 ± 0.0006 −0.13 0.5996

All seven sizes, covering the three classes, agree with the same-size RMT reference within |z| ≤ 1.1 and with the large-N values of Atas et al. within the statistical error plus the finite-size offset (≤ 0.005 for 16 levels). For comparison, the original code compared N = 6, 8, 10 with GOE; N = 6 and 10 are GUE.

Tolerance used in the tests: |z| < 3 against the same-size reference and |⟨r⟩ − Atas| < 3σ + 0.005 (N = 12, 14, 16 in the fast suite; 18, 20 in -m slow).

2.2 Hubbard ring (t = 1, U = 4; K, reflection, S and — for t' = 0 — η resolved)

L N↑+N↓ t' sector dim blocks exact degeneracies left ⟨r⟩ all blocks (≥ 10 levels) ⟨r⟩ blocks ≥ 400 levels, 20 % edge trim
8 4+4 0.0 4900 89 1.7 % 0.3863 ± 0.0053 (n=2840) —
8 4+4 0.5 4900 29 0.1 % 0.5173 ± 0.0048 (n=2986) —
10 3+3 0.0 14400 71 1.1 % 0.3794 ± 0.0030 (n=8443) 0.3762 ± 0.0057 (n=2402)
10 3+3 0.5 14400 32 0.2 % 0.5148 ± 0.0028 (n=8564) 0.5290 ± 0.0048 (n=2840)
10 4+4 0.0 44100 110 0.4 % 0.3840 ± 0.0017 (n=26172) 0.3866 ± 0.0025 (n=12000)
10 4+4 0.5 44100 39 0.0 % 0.5291 ± 0.0016 (n=26369) 0.5313 ± 0.0020 (n=15412)
10 5+5 0.0 63504 150 0.2 % 0.3857 ± 0.0014 (n=37720) 0.3860 ± 0.0021 (n=17508)
10 5+5 0.5 63504 41 0.0 % 0.5287 ± 0.0013 (n=38012) 0.5296 ± 0.0017 (n=22087)

References: Poisson 0.3863, GOE 0.5307.

Interpretation:

  • Integrable chain (t' = 0): consistent with Poisson at L = 10, N = 4+4 and 5+5 (0.3840(17), 0.3857(14)); the low-density sector 3+3 lies 2.3σ below Poisson, partly because exactly degenerate spacings remain (1.1 %), which contribute r = 0.
  • t' = 0.5 (non-integrable): at L = 10 with N = 4+4 and 5+5 the result is consistent with GOE (0.5291(16), 0.5287(13); −1.0σ and −1.5σ), and closer still for large blocks with edge trimming. At L = 8 and at the low density L = 10, N = 3+3, ⟨r⟩ is clearly below GOE (0.515–0.517), approaching it when only large blocks are used — a finite-size effect at these sizes.
  • The standard errors treat the ratios as independent; neighbouring ratios are weakly correlated, so the errors are slight underestimates.
  • The original code reported ⟨r⟩ = 0.7808 for L = 2 with all sectors pooled; that number has no meaning and the method was removed.

3. Chaos diagnostics (phase 3B)

Figures figures/sff.png and figures/chaos_dynamics.png (numbers in the JSON sidecars). Majorana operators with χ² = 1 unless stated; OTOCs use χ² = ½.

3.1 Spectral moments and ground-state energy

Check Result Status
Disorder-averaged kurtosis M4/M2² − 3 from Wick contractions vs García-García & Verbaarschot, PRD 94, 126010 (2016), Eq. (8) identical formula (tested N = 6…34); disorder averages agree (N = 8: −0.922 vs −0.914; N = 12: −1.029 vs −1.034; N = 16: −1.019 vs −1.020) verified
Second moment M2 = C(N,4)·6J²/N³·(χ²)⁴ per-realisation identity Tr H²/D = Σ J² and disorder average verified
Ground-state energy vs the same paper's fit E0 = −0.055 − 0.029 N (Eq. 14, quoted verbatim) our ensemble ⟨E_min⟩ (χ² = ½): −0.568(3), −0.783(3), −0.963(3), −1.149(3) at N = 12, 16, 20, 24, i.e. −2.14 σ at N = 24 with σ from the exact M2; the fit corresponds to −1.4 σ not reconciled — PERLU VERIFIKASI (convention difference not identifiable from the text)

3.2 Spectral form factor (β = 0)

N (class) realisations plateau × L (exact degeneracies) late-time g × L dip time Thouless τ (10 % or 2σ of RMT)
16 (GOE) 60 1.000 0.94 4.1 0.75
18 (GUE) 40 2.000 1.89 3.8 0.51
20 (GSE) 30 2.000 2.06 4.9 not resolved

The plateau heights 1/L (N mod 8 = 0) and 2/L (otherwise) are those stated by Cotler et al. (JHEP 05 (2017) 118, sec. 3.1). The analytic GOE/GUE/GSE form factors used for the ramp were checked against sampled random matrices. With ≤ 60 realisations the unfolded ramp agrees with RMT within errors, but the Thouless time is not determined reliably at these sizes.

3.3 OTOCs and Lyapunov exponent (N = 14, 6 realisations, χ² = ½)

model β late-time F/F(0) fitted early-time rate 2πT
SYK4 0 0.009 1.96 ± 0.21 (stat) ± 0.75 (window) ∞
SYK4 10 0.076 1.52 ± 0.14 ± 0.56 0.63
SYK2 (free) 0 0.90 −0.01 ± 0.09 ± 0.41 ∞
SYK2 (free) 10 1.29 1.73 ± 0.27 ± 1.41 0.63
  • The late-time OTOC cleanly separates interacting SYK (decays to ≈ 0) from the free SYK2 control.
  • Negative result: no Lyapunov exponent can be extracted at these sizes. The early-time rate exceeds 2πT at low temperature, depends strongly on the fit window, and the free model gives rates of the same size at β = 10. The chaos bound λ ≤ 2πT is therefore not tested here.
  • Larger N with dynamical typicality (T = ∞, results/large_n_syk.json; N = 20, 24, 28 with 6, 3, 2 realisations): the local slope d ln(1 − F/F(0))/dt falls monotonically like ≈ 2/t (e.g. N = 28: 4.37, 2.00, 1.30, 0.95, 0.74 at t = 0.5, 1, 1.5, 2, 2.5), i.e. 1 − F/F(0) ∝ t² over the whole early-time window, nearly independent of N (1 − F/F(0) at t = 1: 0.051, 0.045, 0.041). There is no regime of constant slope: the window fits (1.74, 1.74, 1.60 ± ~0.5) measure the curvature of a t² law, not a Lyapunov exponent. The finite-size rescaling analysis up to N = 34 is in section 3.6.

3.4 Krylov complexity and operator size

  • b₁² = 4 Σ_{α∋0} J_α² holds exactly per realisation; the Krylov amplitude φ₀(t) equals the exact autocorrelation function.
  • SYK4 (N = 10–14): b_n rises over the first ~3 steps and then saturates (≈ 2.7 at N = 14); the linear-growth regime of Parker et al. is too short to be resolved at these sizes.
  • Free SYK2: the Krylov space of χ₀ ends after N − 1 steps (N = 10: b₁₀ ≈ 2·10⁻⁸). Numerical limitation: the residual b_N grows with N (1·10⁻⁵ at N = 12, 4·10⁻³ at N = 14) and termination is not detected at N = 16 in double precision (Lanczos error amplification); the figure therefore uses N = 10.
  • Mean Majorana size of χ₀(t): stays exactly 1 for SYK2 and saturates at N/2 for SYK4.

3.5 ETH and eigenstate entanglement (even parity sector)

N class sector dim diagonal fluctuation of χ₀χ₁χ₂χ₃ ⟨|O|⁴⟩/⟨|O|²⟩² off-diagonal
12 GSE 32 0.164 2.22
14 GUE 64 0.114 1.96 (Gaussian complex: 2)
16 GOE 128 0.103 2.90 (Gaussian real: 3)
18 GUE 256 0.061 1.98
  • Diagonal fluctuations scale as D^−0.485 (N = 10–20; ETH: D^−1/2).
  • The particle-hole-odd operator −iχ₀χ₁ has an exactly vanishing diagonal for N mod 8 = 0 — a symmetry effect, not ETH.
  • N = 20 mid-spectrum eigenstates follow the Page curve within 0.002 for l = 1…5 qubits (Page formula checked against Haar-random states); at N = 16 the half-system value is 2.22 vs Page 2.27.

3.6 Lyapunov exponent from finite-size rescaling (N = 20–34)

Data: results/syk_lyapunov.json (scripts/syk_lyapunov.py), figure figures/syk_lyapunov.png (configs/figures/syk_lyapunov.yaml). SYK4 with χ² = ½ and ⟨J²⟩ = 3!J²/N³ (J = 1); fully regularised OTOC Tr(y χ₀(t) y χ₁ y χ₀(t) y χ₁)/Tr(y⁴) with y = ρ^{1/4}, one Haar-random vector per disorder realisation; t = 0…4 in steps of 0.2; βJ = 0 and 10. Realisations: 64, 48, 32, 24, 16, 10, 6, 4 for N = 20, 22, …, 34.

Method (Kobrin et al., PRL 126, 030602 (2021)): the crossing time t* where 1 − F(t)/F(0) = F0 (F0 = 0.25, check 0.16) should shift by (1/λ) ln r under N → rN. Three estimators, errors from 400 bootstrap resamplings of the realisations:

  • pairs: λ(N) = Δ ln N/Δt* for neighbouring sizes, extrapolated linearly in 1/N (the paper's estimator);
  • ln N: one fit t* = A + (ln N)/λ;
  • ln N + 1/N: t* = A + (ln N)/λ + B/N. For F = C0 + C1 e^{λt}/N normalised by F(0) the plain ln N fit is biased upwards by 1 + c(1 − F0)/(F0 N), c = C1/C0 (derived and confirmed on synthetic data in the tests); the extra 1/N term removes this bias (synthetic data: within 5 %).
βJ sizes F0 pairs → ∞ ln N ln N + 1/N power law t* ∝ N^a (χ²/dof vs ln N + 1/N) 2π/β
0 20–34 0.25 0.2 ± 19.7 1.09 ± 0.04 0.62 ± 0.26 a = 0.37 (0.38 vs 0.25) ∞
0 20–34 0.16 −0.1 ± 20.9 1.30 ± 0.06 0.66 ± 0.29 a = 0.39 (0.41 vs 0.18) ∞
10 24–34 0.25 0.6 ± 119 0.67 ± 0.05 0.7 ± 16.8 a = 0.43 (0.30 vs 0.33) 0.628
10 24–34 0.16 −0.2 ± 63 0.84 ± 0.09 0.5 ± 5.3 a = 0.45 (0.29 vs 0.37) 0.628
  • Validity cut (Kobrin et al.: the lowest 20 states within 1/β of the ground state, even sector, 3 realisations): at βJ = 10, E₂₀ − E₀ = 0.138–0.147 (N = 20), 0.106–0.114 (N = 22), 0.079–0.095 (N = 24), 0.044–0.046 (N = 28). N = 20 and 22 are therefore excluded at βJ = 10.
  • The neighbouring-pair estimator is uninformative at these sizes: Δ ln N ≈ 0.08 between neighbours while t* is known to ±0.013–0.04, so single λ(N) carry errors of 30 % to > 100 %.
  • βJ = 0: the bias-corrected fit gives λ ≈ 0.6 ± 0.3, the same for both thresholds, whereas the plain ln N fit changes with F0 (1.09 → 1.30) as the bias formula predicts. However, a power law t ∝ N^0.37 describes the crossing times equally well* (χ²/dof < 1 for both models), as does the t² early-time growth of section 3.3. Exponential growth is therefore not established, and 0.6 ± 0.3 is at most an estimate conditional on the large-N ansatz.
  • βJ = 10: no bias-free estimate is possible (ln N and 1/N are almost collinear over N = 24–34). The plain ln N fits, 0.67 ± 0.05 and 0.84 ± 0.09, are biased upwards and are not a measurement of λ. The chaos bound λ ≤ 2π/β is not tested.
  • No large-N (Schwinger–Dyson) value of λ is computed or quoted here. Kobrin et al. needed up to N = 46 (single curves at 60) and hundreds of realisations; the sizes reachable on a 16 GB laptop are not sufficient.
  • Numerics: Chebyshev propagation (spectral bounds from Lanczos plus a 2 % margin; the propagator checks norm conservation) agrees with dense exponentials to 1e-11 (tests). One N = 28 OTOC curve (33 times, β = 0) takes 35 s instead of ≈ 1000 s with expm_multiply; one N = 34 realisation (21 times, both temperatures) takes 1480–1950 s on an Apple M1.

4. Large-N numerics (phase 3A)

Benchmarks on the recording machine (benchmarks/results/*.json; one parity sector, complex128, χ² = ½):

N sector dim flip masks matrix memory build mat-vec Lanczos E0 peak RSS
24 2 048 562 0.023 GB 0.13 s 1 ms 0.16 s 0.10 GB
28 8 192 1 093 0.18 GB 0.84 s 8 ms 1.1 s 0.28 GB
30 16 384 1 471 0.48 GB 2.8 s 24 ms 4.9 s 0.61 GB
32 32 768 1 941 1.27 GB 8.2 s 66 ms 15 s 1.51 GB
34 65 536 2 517 3.30 GB 24 s 0.38 s 66 s 3.49 GB

The memory prediction D/2·(C(n,4)+C(n,2)+1)·(itemsize+4) bytes agrees with the measured matrix size to < 0.1 %. Dense diagonalisation of a sector is limited to N ≈ 28 on 16 GB; shift-invert/POLFED for mid-spectrum statistics at N ≥ 30 is not implemented.

Checks: sector matrices equal the dense construction exactly; Lanczos E0 equals dense ED (N = 16, 1e-10); the KPM second moment reproduces Σ J²(χ²)⁴ (3 %, stochastic trace); typicality autocorrelations and (regularised, β = 2) OTOCs agree with exact results within 5σ of the random-vector error (N = 12).

Ensemble-averaged ground-state energy (χ² = ½, results/large_n_syk.json):

N 12 16 20 24 28 30 32 34
realisations 20 20 10 8 6 4 3 3
⟨E0⟩ −0.566(6) −0.783(4) −0.961(4) −1.149(3) −1.308(3) −1.391(5) −1.478(5) −1.563(3)
⟨E0⟩/N −0.0471 −0.0490 −0.0480 −0.0479 −0.0467 −0.0464 −0.0462 −0.0460

A 1/N fit through N = 20 and 34 gives E0/N → ≈ −0.043; no verified literature value is available here for comparison (the García-García & Verbaarschot fit remains unreconciled, section 3.1).

5. Floquet dynamics (phase 3C)

  • High-frequency expansions validated against exact Floquet operators (L = 5 Hubbard ring, J0 = J1 = 1, U = 4): the van Vleck spectral error scales as ω^−2.07 at order 0 and ω^−4.09 at order 2 (odd orders vanish for a cosine drive); the Floquet–Magnus operator error scales as ω^−1 / ω^−2 at orders 0 / 1 for a sine drive. Propagators: midpoint (2nd order) and 4th-order Magnus with tested convergence orders.
  • Heating (figures/floquet_heating.png; J(t) = 1 + cos ωt, t' = 0.5, U = 4, half filling, K = 0, 2·10⁴ periods from the ground state of H0):
ω 4 8 10 12 14 16 20 24
L = 6 (dim 68, W = 19.5) 0.986 0.894 0.284 0.581 0.026 0.008 0.0025 0.0010
L = 8 (dim 618, W = 25.3) 0.980 0.873 0.752 0.703 0.020 0.007 0.0022 0.0009

(time-averaged absorbed energy fraction; W = many-body bandwidth.) Heating to infinite temperature within 2–50 periods for ω ≤ 12 and less than 2 % absorption for ω ≥ 14, i.e. already below W. Not established: whether the suppression follows the exponential law of Abanin et al. / Mori et al.; at L ≤ 8 the residual absorption is not separable from perturbative dressing and finite-size resonances (L = 6 is non-monotonic at ω = 10–12). - The choice of drive protocol is still open; the code supports J(t) = J0 + J1 cos ωt (J0 = 0 is the original model).

6. Circuits and NISQ (phase 3D)

  • First-order commutator bound of Childs et al. (PRX 11, 011020, Prop. 15, verified) holds for SYK N = 8 at all r and is ~8.5× above the actual error; both scale as 1/r. The second-order bound was not implemented.
  • CNOTs per first-order Trotter step (CNOT-ladder decomposition): dense SYK 300 (N = 8) → 1.2·10⁵ (N = 24); sparse SYK with k = 4: 138 → 1.0·10³.
  • Sparse SYK (Xu, Susskind, Su, Swingle; definition verified): ⟨r⟩ recovers the RMT value for k ≥ 4 (N = 16 GOE: 0.5306(33); N = 20 GSE: 0.6719(30)), deviates at k = 2 and fails for k ≤ 1.
  • Interferometric OTOC with Trotter evolution reproduces the exact OTOC (N = 6, 1e-3 with 12 second-order steps). Under depolarizing noise (p = 0.005 per qubit and layer, figures/nisq.png), normalising by the echo with W = identity and zero-noise extrapolation (folds 1, 3, 5, Richardson) both move the signal towards the noiseless value for SYK4 and for the SYK2 control; in the unit test ZNE reduces the error ≥ 3×. These are small-N, single-noise-model statements, not hardware results.

7. Quantum kernels (phase 3E)

  • Haar fidelity moments (mean 1/D, variance (D−1)/(D²(D+1))) checked by sampling.
  • SYK Hamiltonian-encoding kernels concentrate exponentially with the number of qubits at long evolution times (Var[K] ∝ 2^(−2.06 n) at t = 10, n = 3–7; ~4.5× above Haar at n = 7), not at short times (t = 0.1: K ≈ 1).
  • Hubbard kernels concentrate much less (L = 4: 1.2·10⁻² vs 7·10⁻⁴ for Haar in the (N, S_z) sector), because conserved quantities restrict the explored space.
  • Classification (20 seeds, cross-validated hyperparameters for every model, paired Wilcoxon tests): both quantum kernels are significantly worse than an RBF-SVM — moons: RBF 0.921, SYK 0.775 (p = 1e-4), Hubbard 0.835 (p = 2e-4); circles: RBF 0.988, SYK 0.929 (p = 0.013), Hubbard 0.7875 (p = 1e-4). No quantum or "chaos" advantage is observed.

8. Literature values used

Quantity Value Source Status
⟨r̃⟩ Poisson 2 ln 2 − 1 = 0.38629 Atas et al., PRL 110, 084101 (2013), Table I verified (arXiv:1212.5611)
⟨r̃⟩ GOE / GUE / GSE (numerics, 1000×1000) 0.5307(1) / 0.5996(1) / 0.6744(1) same verified
⟨r̃⟩ surmise 0.53590 / 0.60266 / 0.67617 same, Eq. (7) verified; reproduced by numerical integration
SYK class per parity sector N mod 8 = 0: GOE, 2/6: GUE, 4: GSE Cotler et al., JHEP 05 (2017) 118, §3.1 verified
SYK conventions {ψ_a,ψ_b}=δ_ab, ⟨J²⟩=3!J²/N³ Cotler et al. §2; Maldacena & Stanford, PRD 94, 106002 (2016) verified (code: χ²=1 → H_code = 4 H_MS)
SYK kurtosis M4/M2² − 3 = −32(N−4)(N²−11N+36)/[N(N−1)(N−2)(N−3)] exact García-García & Verbaarschot, PRD 94, 126010 (2016), Eq. (8) verified; re-derived from Wick contractions
SYK E0 fit −0.055 − 0.029 N fit same, Eq. (14) quoted verbatim; not reproduced (section 3.1)
First-order Trotter commutator bound (t²/2) Σ ‖[Σ_{γ2>γ1} H_γ2, H_γ1]‖ Childs, Su, Tran, Wiebe, Zhu, PRX 11, 011020 (2021), Prop. 15, Eq. (145) verified
Sparse SYK pruning p = kN/C(N,4), ⟨J²⟩ = 3!J²/(N³p) Xu, Susskind, Su, Swingle, arXiv:2008.02303 verified
Crossing-time rescaling N → rN, t → t + (1/λ) ln r; F0 = 0.25; validity: 20 states within 1/β of E0 method Kobrin, Yang, Kahanamoku-Meyer, Olund, Moore, Stanford, Yao, PRL 126, 030602 (2021), Supp. Sec. II.3 verified (arXiv:2002.05725v3)

9. Not yet verified / open

  • The weekly CI jobs (slow statistical tests, newest PennyLane) have not run yet.
  • SYK ground-state energy vs García-García & Verbaarschot's fit: ~1.5× discrepancy not reconciled (section 3.1).
  • Lyapunov exponent and the chaos bound: with finite-size rescaling up to N = 34 (section 3.6) exponential growth cannot be distinguished from a power law at βJ = 0 (bias-corrected estimate 0.6 ± 0.3 if the large-N ansatz is assumed), and no bias-free estimate exists at βJ = 10; the chaos bound is not tested. Larger N (≳ 46, Kobrin et al.) and a Schwinger–Dyson reference value would be needed.
  • Thouless time and the linear growth of Lanczos coefficients: not resolved.
  • Hubbard ring: GOE only reached at L = 10 for N = 4+4, 5+5; residual degeneracies of the integrable chain unexplained.
  • Floquet: exponential suppression of heating not established at L ≤ 8; drive protocol not decided.
  • Mid-spectrum level statistics beyond N ≈ 28 (shift-invert / POLFED) not implemented; second-order Trotter commutator bound not implemented.
  • NISQ results use a single depolarizing model on simulators; no hardware, no transpilation to a device topology.
  • References from memory not yet checked: Essler et al. (2005), Yang PRL 63, 2144 (1989), Kobrin et al. (2021), Roberts–Stanford–Streicher (2018), Parker et al. (2019), Abanin et al. (2015), Mori et al. (2016), Eckardt & Anisimovas (2015), Bukov et al. (2015), Bartsch & Gemmer (2009), Thanasilp et al. (2022) — cited for context only, no numbers taken from them.
  • Platform note: Apple Accelerate's zheevd fails for complex matrices with n ≳ 3000; dense Hermitian problems use *heevr (utils.helpers.hermitian_eigh).