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The recently discovered nickelate superconductor La$_3$Ni$_2$O$_7$ has a hightransition temperature near 80 K under pressure, which offers additionalavenues of unconventional superconductivity. Here with state-of-the-arttensor-network methods, we study a bilayer $t$-$J$-$J_\perp$ model forLa$_3$Ni$_2$O$_7$ and find a robust $s$-wave superconductive (SC) ordermediated by interlayer magnetic couplings. Large-scale density matrixrenormalization group calculations find algebraic pairing correlations withLuttinger parameter of $K_{\rm SC} \simeq 1$. Infinite projected entangled-pairstate method obtains a nonzero SC order directly in the thermodynamic limit,and estimates a strong pairing strength $\bar{\Delta}_z \sim \mathcal{O}(0.1)$.Tangent-space tensor renormalization group simulations further determine a highSC temperature $T_c^*/J \sim \mathcal{O}(0.1)$ and clarify the temperatureevolution of SC order. Because of the intriguing orbital selective behaviorsand strong Hund's rule coupling in the compound, $t$-$J$-$J_\perp$ model hasstrong interlayer spin exchange (while negligible interlayer hopping), whichgreatly enhances the SC pairing in the bilayer system. Such a magneticallymediated strong pairing has also been observed recently in the optical latticeof ultracold atoms. Our accurate and comprehensive tensor-network calculationsreveal robust SC order in the bilayer $t$-$J$-$J_\perp$ model and shed light onthe high-$T_c$ superconductivity in the pressurized nickelateLa$_3$Ni$_2$O$_7$.

Bilayer $t$-$J$-$J_\perp$ Model and Magnetically Mediated Pairing in the Pressurized Nickelate La$_3$Ni$_2$O$_7$