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29 March 2024
 
  » arxiv » quant-ph/0308073

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A note on graphs resistant to quantum uniform mixing
William Adamczak ; Kevin Andrew ; Peter Hernberg ; Christino Tamon ;
Date 13 Aug 2003
Subject quant-ph
AbstractContinuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the $n$-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs $K_{n}$, the continuous-time quantum walk is neither instantaneous (except for $n=2,3,4$) nor average uniform mixing (except for $n=2$). We explore two natural {em group-theoretic} generalizations of the $n$-cube as a $G$-circulant and as a bunkbed $G times Int_{2}$, where $G$ is a finite group. Analyses of these classes suggest that the $n$-cube might be special in having instantaneous uniform mixing and that non-uniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero.
Source arXiv, quant-ph/0308073
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