Fig. 1: Simulating two-dimensional lattice QED with matter fields.

a, The lattice for 2D-QED comprises vertices containing matter particles (blue) connected by links carrying the gauge field (red). b, The gauge field at each link has an infinite discrete spectrum, simulated using a truncated representation using a d-level qudit. c, Qudits (red) are encoded in different Zeeman levels of the S_1/2 ground state and the D_5/2 excited state of trapped ⁴⁰Ca⁺ ions. Matter particles are faithfully represented by qubits (blue). d, By employing Gauss’s law on a plaquette with open boundary conditions, three of the four gauge fields can be eliminated. The remaining field is truncated to at most one energy quantum. Our variational ansatz is a mixed-dimensional circuit in which the quantum register contains one qutrit (red) and four qubits (blue). A fanning out of the qudit circuit line illustrates the action of the qudit entangling gates (‘Variational circuit’ in Methods). A classical optimizer varies the gate angles θ_j to minimize the energy of the quantum state. e, An exemplary optimization run for Ω = 5, m = 0.1 and g⁻² = 10². The red line highlights the current lowest energy found by the algorithm. The initial evaluations explore the variational landscape. Subsequent blocks of evaluations ((1), (2), …) optimize for decreasing values of the coupling (‘A VQE for qudits’ in Methods). Shaded regions correspond to one standard deviation of statistical uncertainty from Monte Carlo resampling around the measured value, averaged over 150 repetitions. f, Expectation value of the plaquette operator \(\langle \hat{\square }\rangle\). The triangular data points were measured for VQE-optimized states, with error bars representing one standard deviation of statistical uncertainty from Monte Carlo resampling around the measured value, averaged over 150 repetitions. The squares are from a simulation of an ideal VQE, with an experimentally motivated noise model applied to the final state (Supplementary Note V). The line shows the ground state from exact diagonalization. The dashed line was obtained from the pure gauge model \({g}^{2}{\hat{H}}_\mathrm{E}+(1/g^{2}){\hat{H}}_\mathrm{B}\): the presence of dynamical matter noticeably affects the slope of \(\langle \hat{\square }\rangle\) when varying g⁻².