Valley-polarized Josephson Junctions as gate-tunable $0$-$\pi$ qubit platforms

Recently, gate-defined Josephson junctions based on magic-angle twisted bilayer graphene (MATBG) have been fabricated. In such a junction, local electrostatic gating can create two superconducting regions connected by an interaction-driven valley-polarized state as the weak link. Due to the spontaneous time-reversal and inversion symmetry breaking of the valley-polarized state, novel phenomena such as the Josephson diode effect have been observed without applying external fields. Importantly, when the so-called nonreciprocity efficiency (which measures the sign and strength of the Josephson effect) changes sign, the energy-phase relation of the junction is approximate $F(\phi) \approx \cos(2\phi)$ where $F$ is the free energy and $\phi$ is the phase difference of the two superconductors. In this work, we show that such a MATBG-based Josephson junction, when shunted by a capacitor, can be used to realize the long-sought-after $0$-$\pi$ qubits which are protected from local perturbation-induced decoherence. Interestingly, by changing the junction parameters, transmon-like qubits with large anharmonicity can also be realized. In short, by utilizing the novel interaction-driven valley-polarized state in MATBG, a single gate-defined Josephson junction can be used to replace complicated superconducting circuits for realizing qubits that are protected from local perturbations.