We present experimental evidence for a quantum phase transition in the easy-axis
S=3/2 anisotropic quasi-one-dimensional ferromagnet ${\mathrm{CoCl}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{D}}_{2}\mathrm{O}$ in a transverse field. Elastic neutron scattering shows that the magnetic order parameter vanishes at a transverse critical field ${\ensuremath{\mu}}_{0}{H}_{c}=16.05$(4) T, while inelastic neutron scattering shows that the gap in the magnetic excitation spectrum vanishes at the same field value, and reopens for
H>Hc. The field dependence of the order parameter and the gap are well described by critical exponents $\ensuremath{\beta}=0.45\ifmmode\pm\else\textpm\fi{}0.09$ and $z\ensuremath{\nu}$ close to
1/2, implying that the quantum phase transition in ${\mathrm{CoCl}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{D}}_{2}\mathrm{O}$ differs significantly from the textbook version of a
S=1/2 Ising chain in a transverse field. We attribute the difference to weak but finite three-dimensionality of the magnetic interactions.
We analytically investigated the phase diagrams of one-dimensional Ising and XY models. A second-order phase transition occurs in both models only with the transverse magnetic field. In this study, the fully connected ferromagnetic and anti-ferromagnetic XX interactions are applied in addition to the transverse magnetic field. The ferromagnetic XX interaction causes a first-order phase transition in both of the one-dimensional Ising and XY models. In contrast, by applying the anti-ferromagnetic one, the second-order phase transition can be avoided in the one-dimensional Ising model, and the critical amplitude becomes smaller in the one-dimensional XY model even though the second-order phase transition remains and the critical exponent does not change. Because the anti-ferromagnetic XX interaction works to destroy the order of the transverse magnetization, it begins to decrease at a larger strength of the transverse magnetic field. As a result, the susceptibility of the transverse magnetization with respect to the strength of the magnetic field becomes small. It is conjectured that the anti-ferromagnetic XX interaction is generally preferable to the ferromagnetic interaction to improve the performance of quantum annealing.