Conventions

SYK model

\[H = \sum_{i<j<k<l} J_{ijkl}\, \chi_i \chi_j \chi_k \chi_l, \qquad \langle J_{ijkl}^2 \rangle = \frac{3!\, J^2}{N^3}.\]

The Majoranas are Jordan–Wigner strings, \(\chi_{2j} = Z_0\cdots Z_{j-1} X_j\) and \(\chi_{2j+1} = Z_0\cdots Z_{j-1} Y_j\), so \(\chi^2 = 1\). Maldacena & Stanford (PRD 94, 106002, 2016) and Cotler et al. (JHEP 05 (2017) 118) use \(\{\chi_a, \chi_b\} = \delta_{ab}\); with the same couplings their Hamiltonian is \(H/4\) (SYKHamiltonian(..., chi_squared=0.5)).

Symmetry classes. The fermion parity \(\Gamma = \prod_j Z_j\) splits the spectrum into two sectors. The antiunitary particle-hole operator \(P = (\chi_0\chi_2\cdots\chi_{N-2})K\) satisfies \([P,H]=0\), \(P^2 = (-1)^{n(n-1)/2}\) and \(P\Gamma P^{-1} = (-1)^n\Gamma\) with \(n=N/2\), so per sector the class is GOE for \(N \bmod 8 = 0\), GSE for \(N \bmod 8 = 4\) (Kramers pairs inside each sector) and GUE for \(N \bmod 8 = 2, 6\) (the two sectors are degenerate with each other).

Spacing ratio

\(\tilde r_n = \min(s_n, s_{n-1}) / \max(s_n, s_{n-1})\) with \(s_n = E_{n+1}-E_n\), computed inside one symmetry sector. Sectors are never merged; exact degeneracies are kept (and reported), never filtered out. Reference values: Atas et al., PRL 110, 084101 (2013): Poisson \(2\ln 2 - 1 = 0.3863\), GOE 0.5307, GUE 0.5996, GSE 0.6744.

Hubbard ring

\[H = -t\sum_{j\sigma}(c^\dagger_{j+1\sigma}c_{j\sigma} + \text{h.c.}) -t'\sum_{j\sigma}(c^\dagger_{j+2\sigma}c_{j\sigma} + \text{h.c.}) + U\sum_j n_{j\uparrow}n_{j\downarrow}\]

with periodic boundary conditions for the fermion operators. Resolved symmetries: \(N_\uparrow\), \(N_\downarrow\), momentum, reflection at \(K=0,\pi\), total spin \(S\), and — for \(t'=0\) and even \(L\) — the η-spin. For \(t'\neq 0\) the expected class is GOE in every momentum sector (time reversal × reflection).

Noise model

T1/T2 relaxation for the physical duration of each step, with pure dephasing \(1/T_\varphi = 1/T_2 - 1/(2T_1)\) (requires \(T_2 \le 2T_1\)), followed by one depolarizing error \(\rho \to (1-p)\rho + p\,I/2\) per gate layer (average gate fidelity \(1-p/2\); PennyLane's DepolarizingChannel parameter is \(3p/4\)).