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Article overview
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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 | Abstract: | Continuous-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 | Services: | Forum | Review | PDF | Favorites |
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