On the Cayley digraphs that are patterns of unitary matrices

This is a short note on some properties of a family of Cayley digraphs. A digraph D is the pattern of a matrix M when D has an arc ij if and only if the ij-th entry of M is nonzero. Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology and quantum computation. In this note, I prove that if a Cayley digraph is a line digraph then it is the pattern of a unitary matrix. I prove that for any finite group with two generators there exists a set of generators such that the Cayley digraph with respect to such a set is a line digraph and hence the pattern of a unitary matrix. Investigating this subject might be useful in the perspective of design quantum algorithms for groups (e.g. for hidden subgroup problems, et simil.).