Perfect state transfer on signed graphs

  • Authors:

  • John Brown;Chris Godsil;Devlin Mallory;Abigail Raz;Christino Tamon

  • Affiliations:

  • Department of Mathematics, SUNY Potsdam, Potsdam, NY;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada;Department of Mathematics, University of California at Berkeley, Berkeley, CA;Department of Mathematics, Wellesley College, Wellesley, MA;Department of Computer Science, Clarkson University, Potsdam, NY

  • Venue:
  • Quantum Information & Computation
  • Year:
  • 2013

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Abstract

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative 2-clique with any positive (n, 3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the 2-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic. Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.