Zero-Point Quantum Diffusion of Proton in Hydrogen-rich Superconductor $LaH_{10}$

$LaH_{10}$, as a member of hydrogen-rich superconductors, has a superconducting critical temperature of 250 K at high pressures, which exhibits the possibility of solving the long-term goal of room temperature superconductivity. Considering the extreme pressure and low mass of hydrogen, the nuclear quantum effects in $LaH_{10}$ should be significant and have an impact on its various physical properties. Here, we adopt the method combines deep-potential (DP) and quantum thermal bath (QTB), which was verified to be able to account for quantum effects in high-accuracy large-scale molecular dynamics simulations. Our method can actually reproduce pressure-temperature phase diagrams of $LaH_{10}$ consistent with experimental and theoretical results. After incorporating quantum effects, the quantum fluctuation driven diffusion of proton is found even in the absence of thermal fluctuation near 0 K. The high mobility of proton is found to be compared to liquid, yet the structure of $LaH_{10}$ is still rigid. These results would greatly enrich our vision to study quantum behavior of hydrogen-rich superconductors.