Using bi-fluxon tunneling to protect the Fluxonium qubit

Encoding quantum information in quantum states with disjoint wave-function support and noise insensitive energies is the key behind the idea of qubit protection. While fully protected qubits are expected to offer exponential protection against both energy relaxation and pure dephasing, simpler circuits may grant partial protection with currently achievable parameters. Here, we study a fluxonium circuit in which the wave-functions are engineered to minimize their overlap while benefiting from a first-order-insensitive flux sweet spot. Taking advantage of a large superinductance ($L\sim 1~\mu \rm{H}$), our circuit incorporates a resonant tunneling mechanism at zero external flux that couples states with the same fluxon parity, thus enabling bifluxon tunneling. The states $|0\rangle$ and $|1\rangle$ are encoded in wave-functions with parities 0 and 1, respectively, ensuring a minimal form of protection against relaxation. Two-tone spectroscopy reveals the energy level structure of the circuit and the presence of $4 \pi$ quantum-phase slips between different potential wells corresponding to $m=\pm 1$ fluxons, which can be precisely described by a simple fluxonium Hamiltonian or by an effective bifluxon Hamiltonian. Despite suboptimal fabrication, the measured relaxation ($T_1 = 177\pm 3 ~\mu s$) and dephasing ($T_2^E = 75\pm 5~\mu \rm{s}$) times not only demonstrate the relevance of our approach but also opens an alternative direction towards quantum computing using partially-protected fluxonium qubits.