  We give a generalization to an infinite tree geometry of Vidal's infinite
time-evolving block decimation (iTEBD) algorithm for simulating an infinite
line of quantum spins. We numerically investigate the quantum Ising model in a
transverse field on the Bethe lattice using the Matrix Product State ansatz. We
observe a second order phase transition, with certain key differences from the
transverse field Ising model on an infinite spin chain. We also investigate a
transverse field Ising model with a specific longitudinal field. When the
transverse field is turned off, this model has a highly degenerate ground state
as opposed to the pure Ising model whose ground state is only doubly
degenerate.
