Recent progress in applying complex network theory to problems in quantum information has resulted in a beneficial cross-over. Complex network methods have successfully been applied to transport and entanglement models while information physics is setting the stage for a theory of complex systems with quantum information-inspired methods. Novel quantum induced effects have been predicted in random graphs—where edges represent entangled links—and quantum computer algorithms have been proposed to offer enhancement for several network problems. Here we review the results at the cutting edge, pinpointing the similarities and the differences found at the intersection of these two fields.

Quantum mechanics still provides new unexpected effects when considering the transport of energy and information. Models of continuous time quantum walks, which implicitly use time-reversal symmetric Hamiltonians, have been intensely used to investigate the effectiveness of transport. Here we show how breaking time-reversal symmetry of the unitary dynamics in this model can enable directional control, enhancement and suppression of quantum transport. Examples ranging from exciton transport to complex networks are presented. This opens new prospects for more efficient methods to transport energy and information.

Given its importance to many other areas of physics, from condensed-matter physics to thermodynamics, time-reversal symmetry has had relatively little influence on quantum information science. Here we develop a network-based picture of time-reversal theory, classifying Hamiltonians and quantum circuits as time symmetric or not in terms of the elements and geometries of their underlying networks. Many of the typical circuits of quantum information science are found to exhibit time asymmetry. Moreover, we show that time asymmetry in circuits can be controlled using local gates only and can simulate time asymmetry in Hamiltonian evolution. We experimentally implement a fundamental example in which controlled time-reversal asymmetry in a palindromic quantum circuit leads to near-perfect transport. Our results pave the way for using time-symmetry breaking to control coherent transport and imply that time asymmetry represents an omnipresent yet poorly understood effect in quantum information science.

In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long-time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks, including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy, and at higher energies, the difference, the so-called quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Rényi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.

Determining community structure is a central topic in the study of complex networks, be it technological, social, biological or chemical, static or in interacting systems. In this paper, we extend the concept of community detection from classical to quantum systems—a crucial missing component of a theory of complex networks based on quantum mechanics. We demonstrate that certain quantum mechanical effects cannot be captured using current classical complex network tools and provide new methods that overcome these problems. Our approaches are based on defining closeness measures between nodes, and then maximizing modularity with hierarchical clustering. Our closeness functions are based on quantum transport probability and state fidelity, two important quantities in quantum information theory. To illustrate the effectiveness of our approach in detecting community structure in quantum systems, we provide several examples, including a naturally occurring light-harvesting complex, LHCII. The prediction of our simplest algorithm, semiclassical in nature, mostly agrees with a proposed partitioning for the LHCII found in quantum chemistry literature, whereas our fully quantum treatment of the problem uncovers a new, consistent, and appropriately quantum community structure.