  Superconductivity and ferroelectricity,representing two distinct forms of
ordered states, are typically not found together in the same system, making it
even more difficult to create a connection between them. Here, supported by
first-principles calculations, we propose that Anderson-Blount's
ferroelectric-like metal can be manifested in electron-doped ferroelectric
Pca2$_1$ HfO$_2$. In this system, polar phonons and consequently
ferroelectricity are not affected by the presence of itinerant electrons. We
find that a nonpolar optical phonon, being strongly coupled to doped electrons,
can acquire a pronounced electron-phonon coupling strength to activate
conventional Bardeen-Cooper-Schrieffer superconductivity. The displacements of
polar oxygen atoms in Pca2$_1$ HfO$_2$ create a link between ferroelectricity
and superconductivity, enabling a tunable superconducting temperature ranging
approximately from 10 to 30 Kelvin. Owing to hafnia's compatibility with
silicon, we suggest HfO$_2$-based ferroelectric superconductors present an
opportunity to construct high-performing hybrid integrated systems utilizing
switchable quantum states.
