magnetic susceptibility $\chi_p$. We have calculated a sequence of the phase transition magnetic fields $h_t^{(p)}$, which is a strictly monotonous and increasing function, which converges exponentially to the asymptotic value $h_t^{(\infty)}$. This feature is completely analogous to a fast exponential saturation of the critical temperatures $T_c^{(p)}$ we had observed for the classical Ising model on the identical series of hyperbolic lattices in our earlier studies [@hctmrg-Ising-5-4; @hctmrg-Ising-p-4]. However, we have not found physical interpretation of this phenomenon yet. The quantum spin systems (as well as the classical ones) investigated on the hyperbolic lattices belong to the mean-field universality class, since infinite Hausdorff dimension of the hyperbolic lattice geometry exceeds the critical lattice dimensions $D_c = 3$ (for quantum models) and $D_c=4$ (for the classical ones). The linearity of the squared magnetization in the vicinity of the phase transition confirms the mean-field-like behavior, in which the associated magnetic exponents $\beta_p = \frac{1}{2}$.

Although the set of the calculated phase transition magnetic fields $h_t^{(p)}$ and the ground-state energies $E_0^{(p)}$ are restricted to $4 \leq p \leq 11$, which is far away from the asymptotics $p\to\infty$, the fast convergence and the exponential character of $h_t^{(p)}$ and $E_0^{(p)}$ with increasing $p$ enables to estimate 