Set of Holonomic and Protected Gates on Topological Qubits for Realistic Quantum Computer

In recent years qubit designs such as transmons approached the fidelities of up to 0.999. However, even these devices are still insufficient for realizing quantum error correction requiring better than 0.9999 fidelity. Topologically protected superconducting qubits are arguably most prospective for building a realistic quantum computer as they are intrinsically protected from noise and leakage errors that occur in transmons. We propose a topologically protected qubit design based on a $\pi$-periodic Josephson element and a universal set of gates: protected Clifford group and highly robust (with infidelity $\sim 10^{-4}$) non-discrete holonomic phase gate. The qubit is controlled via charge($Q$) and flux($\Phi$)-biases. The holonomic gate is realized by quickly, but adiabatically, going along a particular closed path in the two-dimensional $\{\Phi,Q\}$-space -- a path where computational states are always degenerate, but Berry curvature is localized inside the path. This gate is robust against currently achievable noise levels. This qubit architecture allows building a realistic scalable superconducting quantum computer with leakage and noise-induced errors as low as $10^{-4}$, which allows performing realistic error correction codes with currently available fabrication techniques.