  The electronic and magnetic states of $\beta$-MnO$_{2}$ in terms of hydrogen
impurities have been investigated by muon spin rotation ($\mu$SR) technique
combined with density-functional theory (DFT) calculations for muon as
pseudo-hydrogen. We found that 85% of implanted muons are localized in the
oxygen channels of the rutile structure and behave as interstitial protons
(Mu$^{+})$ except those (7.6%) forming a charge-neutral state (Mu$^{0})$ at 2.3
K, which indicates that interstitial hydrogen acts as a shallow donor within
less than 0.1 meV of ionization energy. The residual 15% of muons are
attributed to those related to lattice imperfection as Mn vacancies. Detailed
analyses combined with DFT approach suggested that the muon is localized at the
center of the oxygen channel due to its large zero-point vibration energy.
