Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)

Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from the known Poisson distribution of cycle counts in regular graphs, in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely. The result is analogous to the so called "diagonal approximation" in Quantum Chaos. We also show that by assuming that the spectral correlations are given by RMT to all orders, we can compute the leading deviations from the Poisson distribution for cycle counts. We provide numerical evidence which supports this conjecture.Comment: 15 pages, 5 figure

From quantum graphs to quantum random walks

We give a short overview over recent developments on quantum graphs and outline the connection between general quantum graphs and so-called quantum random walks.Comment: 14 pages, 6 figure

Decoherence in quantum walks - a review

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work since first posted and corrections from comments received; some non-trivial typos fixed. Comments now limited to changes that can be applied at proof stag