Coined Quantum Walks on Graphs

Abstract

In the previous chapters, we have addressed coined quantum walks on specific graphs of wide interest, such as lattices and hypercubes. In this chapter, we define coined quantum walks on graphs . The concepts of graph theory reviewed in Appendix B are required here for a full understanding of the definition of the coined quantum walk. We split the presentation into class 1 and class 2 graphs. Class 1 comprises graphs whose maximum degree coincides with the edge-chromatic number , and class 2 comprises the remaining ones. For graphs in class 1, we can use the standard coin-position or position-coin notation , and we can give the standard interpretation that the vertices are the positions and the edges are the directions. For graphs in class 2, on the other hand, we can use neither the coin-position nor position-coin notation; we have to use the arc notation and replace the simple graph by an associated symmetric digraph , whose underlying graph is the original graph. In this case, the walker steps on the arcs of the digraph. After those considerations, we are able to define formally coined quantum walks.