  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.
