# Publications by Tomi Johnson

## Degree Distribution in Quantum Walks on Complex Networks

Physical Review X **3** (2013) 041007

We relate the average long time probability of finding a quantum walker at a node to that for a corresponding classical walk. The latter is proportional to the node degree. We replace the state dependence of quantum evolution by a partial order, bounding the quantumness of the walker in terms of the energy of the initial state and the spectral gap of the complex network. For a uniform initial state, we identify complex network classes and regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate. The importance of quantum effects, or the quantumness of a complex network, is given by the R\'enyi entropy of order 1/2 of the normalized weighted degrees, which can be upper bounded by the Shannon entropy.

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