  In this paper we discuss the criticality of a quantum Ising spin chain with
competing random ferromagnetic and antiferromagnetic couplings. Quantum
fluctuations are introduced via random local transverse fields. First we
consider the chain with couplings between first and second neighbors only and
then generalize the study to a quantum analog of the Viana-Bray model, defined
on a small world random lattice. We use the Dasgupta-Ma decimation technique,
both analytically and numerically, and focus on the scaling of the lattice
topology, whose determination is necessary to define any infinite disorder
transition beyond the chain. In the first case, at the transition the model
renormalizes towards the chain, with the infinite disorder fixed point
described by Fisher. This corresponds to the irrelevance of the competition
induced by the second neighbors couplings. As opposed to this case, this
infinite disorder transition is found to be unstable towards the introduction
of an arbitrary small density of long range couplings in the small world
models.
