We report first-principles and strongly-correlated calculations of the newly-discovered heavy fermion superconductor UTe$_2$. Our analyses reveal three key aspects of its magnetic, electronic, and superconducting properties, that include: (1) a two-leg ladder-type structure with strong magnetic frustrations, which might explain the absence of long-range orders and the observed magnetic and transport anisotropy; (2) quasi-two-dimensional Fermi surfaces composed of two separate electron and hole cylinders with similar nesting properties as in UGe$_2$, which may potentially promote magnetic fluctuations and help to enhance the spin-triplet pairing; (3) a unitary spin-triplet pairing state of strong spin-orbit coupling at zero field, with point nodes presumably on the heavier hole Fermi surface along the $k_x$-direction, in contrast to the previous belief of non-unitary pairing. Our proposed scenario is in excellent agreement with latest thermal conductivity measurement and provides a basis for understanding the peculiar magnetic and superconducting properties of UTe$_2$.
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