dorsal/arxiv
View SchemaA note on graphs resistant to quantum uniform mixing
| Authors | William Adamczak, Kevin Andrew, Peter Hernberg, Christino Tamon |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0308073 |
| URL | https://arxiv.org/abs/quant-ph/0308073 |
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 $G \rtimes \Int_{2}$, 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.
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"abstract": "Continuous-time quantum walks on graphs is a generalization of\ncontinuous-time Markov chains on discrete structures. Moore and Russell proved\nthat the continuous-time quantum walk on the $n$-cube is instantaneous exactly\nuniform mixing but has no average mixing property. On complete (circulant)\ngraphs $K_{n}$, the continuous-time quantum walk is neither instantaneous\n(except for $n=2,3,4$) nor average uniform mixing (except for $n=2$). We\nexplore two natural {\\em group-theoretic} generalizations of the $n$-cube as a\n$G$-circulant and as a bunkbed $G \\rtimes \\Int_{2}$, where $G$ is a finite\ngroup. Analyses of these classes suggest that the $n$-cube might be special in\nhaving instantaneous uniform mixing and that non-uniform average mixing is\npervasive, i.e., no memoryless property for the average limiting distribution;\nan implication of these graphs having zero spectral gap. But on the bunkbeds,\nwe note a memoryless property with respect to the two partitions. We also\nanalyze average mixing on complete paths, where the spectral gaps are nonzero.",
"arxiv_id": "quant-ph/0308073",
"authors": [
"William Adamczak",
"Kevin Andrew",
"Peter Hernberg",
"Christino Tamon"
],
"categories": [
"quant-ph"
],
"title": "A note on graphs resistant to quantum uniform mixing",
"url": "https://arxiv.org/abs/quant-ph/0308073"
},
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