  We have developed a very efficient numerical algorithm of the strong disorder
renormalization group method to study the critical behaviour of the random
transverse-field Ising model, which is a prototype of random quantum magnets.
With this algorithm we can renormalize an N-site cluster within a time
N*log(N), independently of the topology of the graph and we went up to
N~4*10^6. We have studied regular lattices with dimension D<=4 as well as
Erdos-Renyi random graphs, which are infinite dimensional objects. In all cases
the quantum critical behaviour is found to be controlled by an infinite
disorder fixed point, in which disorder plays a dominant role over quantum
fluctuations. As a consequence the renormalization procedure as well as the
obtained critical properties are asymptotically exact for large systems. We
have also studied Griffiths singularities in the paramagnetic and the
ferromagnetic phases and generalized the numerical algorithm for another random
quantum systems.
