A New Perspective on the Average Mixing Matrix

  • Gabriel Coutinho Universidade Federal de Minas Gerais
  • Chris Godsil University of Waterloo
  • Krystal Guo Universit√© libre de Bruxelles
  • Hanmeng Zhan University of Waterloo
Keywords: Algebraic graph theory, Quantum walks, Average mixing matrix


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.

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