Viewing protected superconducting qubits through the lens of the cat qubit

We draw analogies between protected superconducting qubits and bosonic qubits by studying the fluxonium Hamiltonian in its Fock basis. The mean-field phase diagram of fluxonium (at the sweet spot) is identified, with a region in parameter space that is characterized by $\mathbb{Z}_2$-symmetry-broken ground states. In the heavy fluxonium limit, these ground states are well approximated by squeezed coherent states in a Fock basis (corresponding to persistent current states with definite flux but indefinite charge), and simple expressions are provided for them in terms of the circuit parameters. We study the noise bias in fluxonium via a universal Lindblad master equation and find that the bit-flip rate is exponentially small in $E_j/(k_B T)$, while the phase-flip rate does not get worse with this ratio. Analogous behavior is found in $\cos(2 \theta)$ qubits. We describe cat-qubit-inspired bias-preserving $X$ and $CX$ gates for fluxonium. We discuss first steps towards generating an Ising interaction between protected superconducting qubits on a two-dimensional lattice, with the aim of achieving a passive quantum memory by coupling a static Hamiltonian to a generic thermal bath.