  The antiferromagnetic Ising chain in both transverse and longitudinal
magnetic fields is one of the paradigmatic models of a quantum phase
transition. The antiferromagnetic system exhibits a zero-temperature critical
line separating an antiferromagnetic phase and a paramagnetic phase; the
critical line connects an integrable quantum critical point at zero
longitudinal field and a classical first-order transition point at zero
transverse field. Using a strong-disorder renormalization group method
formulated as a tree tensor network, we study the zero-temperature phase of the
quantum Ising chain with bond randomness. We introduce a new matrix product
operator representation of high-order moments, which provides an efficient and
accurate tool for determining quantum phase transitions via the Binder cumulant
of the order parameter. Our results demonstrate an infinite-randomness quantum
critical point in zero longitudinal field accompanied by pronounced quantum
Griffiths singularities, arising from rare ordered regions with anomalously
slow fluctuations inside the paramagnetic phase. The strong Griffiths effects
are signaled by a large dynamical exponent $z>1$, which characterizes a
power-law density of low-energy states of the localized rare regions and
becomes infinite at the quantum critical point. Upon application of a
longitudinal field, the quantum phase transition between the paramagnetic phase
and the antiferromagnetic phase is completely destroyed. Furthermore, quantum
Griffiths effects are suppressed, showing $z<1$, when the dynamics of the rare
regions is hampered by the longitudinal field.
