Quasiparticle density of states (DoS) schematic (light shaded regions) for the bottom and top superconducting films in a junction, characterized by the energy gaps Δ_B>Δ_T. The states occupied by quasiparticles (QP) are identified by darker color. The arrows denote the quasiparticle tunneling events for two choices of gap asymmetry ω_BT exemplifying (a) non-protected design with qubit frequency ω₁₀ larger than asymmetry, and (b) gap-engineered design. The dashed arrows represent the suppressed transition rates due to the gap asymmetry. Adapted from [8].