A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Lifting Markov chains to speed up mixing
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On the Digraph of a Unitary Matrix
SIAM Journal on Matrix Analysis and Applications
Quantum Walk Algorithm for Element Distinctness
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Coins make quantum walks faster
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the structure of the adjacency matrix of the line digraph of a regular digraph
Discrete Applied Mathematics
Quantum walks on directed graphs
Quantum Information & Computation
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A quantum walk, i.e., the quantum evolution of a particle on a graph, is termed scalar if the internal space of the moving particle (often called the coin) has dimension one. Here, we study the existence of scalar quantum walks on Cayley graphs, which are built from the generators of a group. After deriving a necessary condition on these generators for the existence of a scalar quantum walk, we present a general method to express the evolution operator of the walk, assuming homogeneity of the evolution. We use this necessary condition and the subsequent constructive method to investigate the existence of scalar quantum walks on Cayley graphs of groups presented with two or three generators. In this restricted framework, we classify all groups - in terms of relations between their generators - that admit scalar quantum walks, and we also derive the form of the most general evolution operator. Finally, we point out some interesting special cases, and extend our study to a few examples of Cayley graphs built with more than three generators.