Fig. 1: Numerical results beyond our analytical guarantees.

Plotting the gap Δ of the full Lindbladian \({{{\mathcal{L}}}}^{{\dagger} }\) associated with the spinless D = 2 Fermi-Hubbard thermal state with design choices beyond our analytical results—when using the Metropolis filter function and single site Pauli jump operators instead—as a function of the coupling strength U. Here we plot different system sizes separately, for the case β = 1 and t = 1, demonstrating a large spectral gap in the regime of intermediate coupling 2 ≲ U/t ≲ 6, which also does not seem to close with growing inverse temperature β (see Supplementary Figs. 7 and 9). At what coupling strength the gap closes is controlled by the support of the filter function, incurring only polylogarithmic additional algorithmic cost in U/t.