Flat bands in superconducting circuits with <i>e</i> and <i>4e </i>tunnel junctions

In superconducting circuits interrupted by Josephson junctions, the energy spectrum is determined by the symmetries of the Josephson potential and depends on offset charges on different islands modulo 2e via the Aharonov-Casher effect, resembling a crystal band structure. We show that by exploiting 4e tunnel junctions effectively described by a cos(2φ) energy-phase relation, where tunneling of pairs of Cooper pairs dominates, or by introducing semiconducting wires hosting Majorana end fermions realizing single e tunnel junction via the 4π Josephson effect, we can engineer the Josephson potential and design spectra featuring flat bands and multiple pairs of Dirac points, that can be used to park the qubits in charge noise free states or manipulate Majorana qubits.