  In superconducting circuits interrupted by Josephson junctions, the
dependence of the energy spectrum on offset charges on different islands is
$2e$ periodic through the Aharonov-Casher effect and resembles a crystal band
structure that reflects the symmetries of the Josephson potential. We show that
higher-harmonic Josephson elements described by a $\cos(2\varphi)$ energy-phase
relation provide an increased freedom to tailor the shape of the Josephson
potential and design spectra featuring multiplets of flat bands and Dirac
points in the charge Brillouin zone. Flat bands provide noise-insensitive
quantum states, and band engineering can help improve the coherence of the
system. We discuss a modified version of a flux qubit that achieves in
principle no decoherence from charge noise and introduce a flux qutrit that
shows a spin-one Dirac spectrum and is simultaneously quote robust to both
charge and flux noise.
