Abstract
We study tensor network states defined on an underlying graph which is
sparsely connected. Generic sparse graphs are expander graphs with a high
probability, and one can represent volume law entangled states efficiently with
only polynomial resources. We find that message-passing inference algorithms
such as belief propagation can lead to efficient computation of local
expectation values for a class of tensor network states defined on sparse
graphs. As applications, we study local properties of square root states, graph
states, and also employ this method to variationally prepare ground states of
gapped Hamiltonians defined on generic graphs. Using the variational method we
study the phase diagram of the transverse field quantum Ising model defined on
sparse expander graphs.