Fig. 4. Experimental energies, chemical potentials and charge correlations.

“VQE”: experimental data. “Simulated”: the lowest energy achievable in the VQE ansatz. “Ground state”: energy in the true ground state within each occupation number subspace. “Slater determinant”: the energy achieved by an optimised Slater determinant state as detailed in Supplementary Note 5. Dashed lines are exact numerical calculations, solid line is experimental data. Plots show a piecewise linear interpolation between integer occupations. a–c Energies E(N_occ) produced by VQE experiments compared with exact results (U = 4). VQE results for 1 × 8 and 2 × 4 use one variational layer; 1 × 4 has two variational layers. Inset shows zoomed-in region around half-filling. d, e Chemical potentials μ for a 1 × 8 system, where μ(N_occ) = E(N_occ) − E(N_occ − 1). Inset shows the derivative ${\mu }^{\prime }\left(N_{\mathrm{occ}}\right)=E\left(N_{\mathrm{occ}}+1\right)-2E\left(N_{\mathrm{occ}}\right)+E\left(N_{\mathrm{occ}}-1\right)$ of the chemical potential at even occupations. f Decay of normalised charge correlations $C^c\left(1,i\right)=\left(⟨n_1{n}_i⟩-⟨n_1⟩⟨n_i⟩\right)/\left(⟨n_1^2⟩-{⟨n_1⟩}^2\right)$ for even occupation numbers. Solid lines: experimental results. Dashed lines: correlations in ground state. Error bars were calculated according to the procedure described in Methods section and are often too small to be visible.