  We investigate the ground states of spin models defined on networks that we
imprint (e.g. non-complex random networks like Erdos-Renyi or complex networks
like Watts-Strogatz, and Barabasi-Albert), and their response to decohering
processes which we model with network attacks. We quantify the complexity of
these ground states, and their response to the attacks, by calculating
distributions of network measures of an emergent network whose link weights are
the pairwise mutual information between spins. We focus on attacks which
projectively measure spins. We find that the emergent networks in the ground
state do not satisfy the usual criteria for complexity, and their average
properties are captured well by a single dimensionless parameter in the
Hamiltonian. While the response of classical networks to attacks is
well-studied, where classical complex networks are known to be more robust to
random attacks than random networks, we find counterintuitive results for our
quantum networks. We find that the ground states for Hamiltonians defined on
different classes of imprinted networks respond similarly to all our attacks,
and the attacks rescale the average properties of the emergent network by a
constant factor. Mean field theory explains these results for relatively dense
networks, but we also find the simple rescaling behavior away from the regime
of validity of mean field theory. Our calculations indicate that complex spin
networks are not more robust to projective measurement attacks, and presumably
also other quantum attacks, than non-complex spin networks, in contrast to the
classical case. Understanding the response of the spin networks to decoherence
and attacks will have applications in understanding the physics of open quantum
systems, and in designing robust complex quantum systems, possibly even a
robust quantum Internet in the long run, that is maximally resistant to
decoherence.
