Fig. 2: Refining the gauge-field discretization.

a, We consider pure gauge QED in two spatial dimensions with periodic boundary conditions, that is on a lattice on the surface of a torus. As before, the gauge field resides on the links of the lattice, although the vertices remain empty. b, We consider the smallest instance of such a torus. It has four empty sites and eight gauge-field links. The ground state of this particular system can be described with three separate circulation paths of the gauge field, which are called rotators, as discussed in ‘Pure gauge 2D-QED’ in Methods. Each rotator satisfies an eigenvalue equation equivalent to a single-link gauge field and can, thus, be subject to the same truncation rules as discussed in the main text by employing a d-level qudit. Here, we demonstrate the difference between a realization employing qutrits and ququints. c, The variational circuit in the electric representation (see main text) for the qutrit truncation (solid lines) and the ququint truncation (all, except shaded box marked with qutrit symbol). The explicit form of the gates employed is given in ‘Variational circuit’ in Methods. d, Experimentally measured expectation values of the plaquette operator \(\hat{\square }\) in the VQE-optimized ground states using qutrits (light blue and orange triangles) compared to ququints (dark blue and red pentagons). The error bars indicate one standard deviation of statistical uncertainty from Monte Carlo resampling around the measured value, averaged over 150 (300) repetitions for qutrits (ququints). The black line represents numerical results obtained for d = 21 using the electric (magnetic) representation for small (large) values of g⁻². The dashed lines are exact numerical results for qutrits and ququints. e, The duality between the electric representations (orange bars) and the magnetic representations (blue bars) is clearly seen in the experimentally measured populations of the eigenvectors of the yellow rotator from b for the qutrit VQE experiment and ququint experiment. The grey bars were obtained with exact diagonalization. In the regime dominated by the electric Hamiltonian (small g⁻²), a qutrit representation (light orange) is enough to approximate the correct ground state, whereas for larger g⁻², truncation errors become more relevant and a ququint representation (dark orange) becomes advantageous. A complementary argument applies to the magnetic qutrit (light blue) and ququint (dark blue) representations.