Fig. 1: Schematic of the system.

A Schematic of the SINIS junction coupled to the KPO. V_B is the bias voltage applied to the SINIS junction. C_c is the coupling capacitance. The arrows indicate quasiparticle tunnelings. B Energy diagram for the single-quasiparticle tunneling at a bias voltage V_B < 2Δ/e. The black solid curves at the normal metal and the superconductors represent the Fermi-Dirac distribution function and the density of states in the superconductors, respectively. The colored and shaded areas represent the occupied and unoccupied states, respectively. The straight arrows indicate the quasiparticle tunnelings from an initial energy state (beginning of the arrow) to a final energy state (end of the arrow). The wavy arrows indicate energy absorption from the KPO. C Effective circuit of the system composed of a NIS junction, the KPO, and the coupling capacitance C_c. The part in orange is the normal-metal island of the NIS junction with q excess quasiparticles, and the part shaded by light blue is the KPO formed with capacitance C and a SQUID with the Josephson energy E_J and an external magnetic flux Φ_ex(t). The circuit has only one of the NIS junctions with the junction capacitance C_j and the tunneling resistance R_T. C_m is the capacitance of the metallic island to the ground, which includes the capacitance of another junction. V = V_B/2 is the bias voltage applied to a single NIS junction. D Energy diagram of the KPO. The horizontal solid and dashed lines represent even and odd parity energy eigenstates, respectively. The two lowest degenerate levels shaded by light blue are qubit states. The red arrows indicate excitations induced by the pure dephasing. The blue solid (dashed) arrows indicate deexcitations by the QCR to opposite (the same) parity energy levels. E Bloch sphere of the Kerr-cat qubit. Schematic drawings of the Wigner functions corresponding to the eigenstates of X, Y, and Z Pauli operators and excited states outside the qubit subspace are presented.