Abstract. In this note, we give a characterization of the adjacency matrix of the line digraph of a regular digraph and we discuss a generalization.On the light of the characterization given, we remark that the underlying digraph of a coined quantum random walk is the line digraph of a regular digraph. 1. A characterization of the adjacency matrix of the line digraph of a regular digraph 1.1. Set-up. The notion of line digraph has been introduced by Harary and Norman [HN60] in 1960. A classic survey on line graphs and digraphs is [HB78]; a recent one is [P95]. Line digraphs are used in the design and analysis of interconnection networks (see e.g. [FYA84]). Line digraphs are also used in algorithms for DIRECTED MAX-CUT [CE90] and TRAVELLING SALESMAN [GKWS98]. Definition 1 (Line digraph). The line digraph − → LD of a digraph ( D = (V, A) is theAbstract. In this note, we give a characterization of the adjacency matrix of the line digraph of a regular digraph and we discuss a generalization.On the light of the characterization given, we remark that the underlying digraph of a coined quantum random walk is the line digraph of a regular digraph. 1. A characterization of the adjacency matrix of the line digraph of a regular digraph 1.1. Set-up. The notion of line digraph has been introduced by Harary and Norman [HN60] in 1960. A classic survey on line graphs and digraphs is [HB78]; a recent one is [P95]. Line digraphs are used in the design and analysis of interconnection networks (see e.g. [FYA84]). Line digraphs are also used in algorithms for DIRECTED MAX-CUT [CE90] and TRAVELLING SALESMAN [GKWS98]. Definition 1 (Line digraph). The line digraph − → LD of a digraph ( D = (V, A) is the

Oct 8, 20023 citations

Quantum Walks on Finite Graphs

Renato Portugal

Springer eBooksJan 1, 20130 citations

Factoring Discrete Quantum Walks on Distance Regular Graphs into Continuous Quantum Walks

Hanmeng Zhan

We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$ . Given that $X$ has diameter $d$ and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of $X$ . We also obtain a similar factorization for any graph $X$ in a Bose Mesner algebra.We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph $X$. Given that $X$ has diameter $d$ and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on $X$ is a product of at most $d$ commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of $X$. We also obtain a similar factorization for any graph $X$ in a Bose Mesner algebra.

arXiv:2008.01224Aug 3, 20200 citations

Coined Quantum Walks on Graphs

Renato Portugal

Quantum science and technologyJan 1, 20181 citation

A New Perspective on the Average Mixing Matrix

Gabriel Coutinho, Chris D. Godsil, Krystal Guo +1 more

We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. We show that it is the matrix of transformation of the orthogonal projection onto the commutant algebra of the adjacency matrix, restricted to diagonal matrices. Using this formulation of the average mixing matrix, we find connections between its rank and automorphisms of the graph.

The Electronic Journal of CombinatoricsOct 19, 20185 citations

We consider a discrete-time quantum walk, called the Grover walk, on a distance regular graph

X

. Given that

X

has diameter

d

and invertible adjacency matrix, we show that the square of the transition matrix of the Grover walk on

X

is a product of at most

d

commuting transition matrices of continuous-time quantum walks, each on some distance digraph of the line digraph of

X

. We also obtain a similar factorization for any graph

X

in a Bose Mesner algebra.