Title: Complex Networks: from Classical to Quantum,
Abstract: Recent progress in applying complex network theory to problems faced in
quantum information and computation has resulted in a beneficial crossover
between two fields. Complex network methods have successfully been used to
characterize quantum walk and transport models, entangled communication
networks, graph-theoretic models of emergent space-time and in detecting
mesoscale structure in quantum systems. Information physics is setting the
stage for a theory of complex and networked systems with quantum
information-inspired methods appearing in complex network science, including
information-theoretic distance and correlation measures for network
characterization. Novel quantum induced effects have been predicted in random
graphs---where edges represent entangled links---and quantum computer
algorithms have recently been proposed to offer super-polynomial enhancement
for several network and graph theoretic problems. Here we review the results at
the cutting edge, pinpointing the similarities and reconciling the differences
found in the series of results at the intersection of these two fields.