Cluster states, algorithms and graphs

Abstract

The present paper is concerned with the concept of the one-way quantum computer, beyond binary-systems, and its relation to the concept of stabilizer quantum codes. This relation is exploited to analyze a particular class of quantum algorithms, called graph algorithms, which correspond in the binary case to the Clifford group part of a network and which can efficiently be implemented on a one-way quantum computer. These algorithms can ``completely be solved" in the sense that the manipulation of quantum states in each step can be computed explicitly. Graph algorithms are precisely those which implement encoding schemes for graph codes. Starting from a given initial graph, which represents the underlying resource of multipartite entanglement, each step of the algorithm is related to a explicit transformation on the graph.