2011/03/14 by Chris Godsil, Godsil, Chris · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Adjacency matrix #Combinatorics #Combinatorics (math.CO) #Discrete mathematics #FOS: Mathematics #FOS: Physical sciences #Graph #Limit (mathematics) #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix multiplication #Mixing (physics) #Operator (biology) #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Quantum and electron transport phenomena #Quantum computer #Quantum mechanics #Quantum walk #math.CO #quant-ph
paper · pdf · doi:10.48550/arxiv.1103.2578
published in arXiv (Cornell University) (Cornell University) · 20 pages, minor fixes, added section on discrete walks; fixed typos
openalex publication_date 2011/03/14 · arxiv created 2011/10/01 · arxiv updated 2011/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
If X is a graph with adjacency matrix A, then we define H(t) to be the operator exp(itA). The Schur (or entrywise) product H(t)∘ H(-t) is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the average mixing matrix. This can be defined as the limit of C-1 ∫0C H(t)∘ H(-t)\dt as C→∞. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of I, J (the all-ones matrix), and a permutation matrix.