  Increasing numerical studies showed that the simplest Hubbard model on the
square lattice with strong repulsion may not exhibit high-temperature
superconductivity (SC). It is desired to look for other possible microscopic
mechanism of realizing high-temperature SC. Here, we explore the interplay
between the Su-Schrieffer-Heeger (SSH) electron-phonon coupling (EPC) and the
Hubbard repulsion by density-matrix-renormalization-group (DMRG) simulations.
Our state-of-the-art DMRG study showed convincingly that the interplay between
strong Hubbard $U$ and moderate Su-Schrieffer-Heeger EPC $\lambda$ can induce
robust $d$-wave SC. The SSH-type EPC can generates effective antiferromagnetic
spin-exchange interactions between neighboring sites, which plays a crucial
role in the interplay of inducing robust $d$-wave SC. Specifically, for $U=8t$,
we find that $d$-wave SC emerges when $\lambda>\lambda_c$ with a moderate
critical value $\lambda_c=0.1\sim 0.2$. Our results might shed new light to
understanding high-temperature SC in cuprates as well as pave a possible new
route in looking for high-temperature SC in other quantum materials with both
strong $U$ and moderate $\lambda$.
