  Theoretical studies suggest that Su-Schrieffer-Heeger-like electron-phonon
($e$-ph) interactions can mediate high-temperature bipolaronic
superconductivity that is robust against repulsive electron-electron
interactions. Here we present a comparative analysis of the pairing and
competing charge/bond correlations in the two-dimensional Holstein and optical
Su-Schrieffer-Heeger (SSH) models using numerically exact determinant quantum
Monte Carlo. We find that the SSH interactions support light bipolarons and
strong superconducting correlations out to relatively large values of the
$e$-ph coupling $\lambda$ and densities near half-filling, while the Holstein
interaction does not due to the formation of heavy bipolarons and competing
charge-density-wave order. We further find that the Holstein and SSH models
have comparable pairing correlations in the weak coupling limit for carrier
concentrations $\langle n \rangle \ll 1$, where competing orders and polaronic
effects are absent. These results support the proposal that SSH (bi)polarons
can support superconductivity to larger values of $\lambda$ in comparison to
the Holstein polaron, but that the resulting $T_\mathrm{c}$ gains are small in
the weak coupling limit. We also find that the SSH model's pairing correlations
are suppressed after including a weak on-site Hubbard repulsion. These results
have important implications for identifying and engineering bipolaronic
superconductivity.
