dorsal/arxiv
View SchemaEntanglement in coined quantum walks on regular graphs
| Authors | Ivens Carneiro, Meng Loo, Xibai Xu, Mathieu Girerd, Viv Kendon, Peter L. Knight |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0504042 |
| URL | https://arxiv.org/abs/quant-ph/0504042 |
| DOI | 10.1088/1367-2630/7/1/156 |
| Journal | New J. Phys. 7 (2005) 156 |
Abstract
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC, pp 5968) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.
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"abstract": "Quantum walks, both discrete (coined) and continuous time, form the basis of\nseveral recent quantum algorithms. Here we use numerical simulations to study\nthe properties of discrete, coined quantum walks. We investigate the variation\nin the entanglement between the coin and the position of the particle by\ncalculating the entropy of the reduced density matrix of the coin. We consider\nboth dynamical evolution and asymptotic limits for coins of dimensions from two\nto eight on regular graphs. For low coin dimensions, quantum walks which spread\nfaster (as measured by the mean square deviation of their distribution from\nuniform) also exhibit faster convergence towards the asymptotic value of the\nentanglement between the coin and particle\u0027s position. For high-dimensional\ncoins, the DFT coin operator is more efficient at spreading than the Grover\ncoin. We study the entanglement of the coin on regular finite graphs such as\ncycles, and also show that on complete bipartite graphs, a quantum walk with a\nGrover coin is always periodic with period four. We generalize the \u0027glued\ntrees\u0027 graph used by Childs et al (2003 Proc. STOC, pp 5968) to higher\nbranching rate (fan out) and verify that the scaling with branching rate and\nwith tree depth is polynomial.",
"arxiv_id": "quant-ph/0504042",
"authors": [
"Ivens Carneiro",
"Meng Loo",
"Xibai Xu",
"Mathieu Girerd",
"Viv Kendon",
"Peter L. Knight"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/1367-2630/7/1/156",
"journal_ref": "New J. Phys. 7 (2005) 156",
"title": "Entanglement in coined quantum walks on regular graphs",
"url": "https://arxiv.org/abs/quant-ph/0504042"
},
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