$4\leq p \leq 10$ and $m=10$ for $p=11$. The asymptotic estimate of $E_0^{(\infty)}$ corresponds to the model on the Bethe lattice.

\[Tab2\]

The ground-state energies $E_0^{(p)}$ obtained by the TPVF algorithm for both the XY and the Heisenberg models are summarized in table \[Tab2\]. The energies $E_0^{(p)}$ remained identical even if the larger set of $N_{\rm Ising}^{(p)}$ free variational parameters $W_p(\theta_j)$ in TPVF was used, whereby the optimal values of the parameters $W^{*}_p(\theta_j)$ coupled by spin-inversion symmetry were equal. These results witness the spin-inversion symmetry of the models on hyperbolic lattices. Recall that $E_0^{(p)}$ represents only an upper estimate of the true ground-state energy ${\cal E}_0^{(p)}$. We have shown that the energies $E_0^{(4)}$ of the referencing Euclidean square lattice calculated by TPVF were higher if compared to the Monte Carlo simulations (the relative errors for the XY and the Heisenberg models, respectively, are $1.2\%$ and $2.2\%$) [@DaniskaGendiar]. This observation can be explained by suppression of the quantum long-range correlations induced by the TPS approximation of the low-dimensional uniform tensors $W_4$, which cannot correctly reproduce the divergence of the correlation length in the models on the square lattice. On the other hand, any quantum spin model on hyperbolic lattice belongs to 