dorsal/arxiv
View SchemaQuantum walks on graphs and quantum scattering theory
| Authors | Edgar Feldman, Mark Hillery |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0403066 |
| URL | https://arxiv.org/abs/quant-ph/0403066 |
| Journal | Coding Theory and Quantum Computing, edited by D. Evans, J. Holt, C. Jones, K. Klintworth, B. Parshall, O. Pfister, and H. Ward, Contemporary Mathematics, 381, 71 (2005). |
Abstract
We discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply advances one unit at each time step, so that its behavior there is analogous to free propagation We are interested in how many steps it will take the particle, starting on one tail and propagating through the graph (where its propagation is not free), to emerge onto the other tail. The probability to make such a walk in n steps and the hitting time for such a walk can be expressed in terms of the transmission amplitude for the graph, which is one element of its S matrix. Demonstrating this necessitates a study of the analyticity properties of the transmission and reflection amplitudes of a graph. We show that the graph can have bound states that cannot be accessed by a particle entering the graph from one of its tails. Time-reversal invariance of a quantum walk is defined and used to show that the transmission amplitudes for the particle entering the graph from different directions are the same if the walk is time-reversal invariant.
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"abstract": "We discuss a particular kind of quantum walk on a general graph. We affix two\nsemi-infinite lines to a general finite graph, which we call tails. On the\ntails, the particle making the walk simply advances one unit at each time step,\nso that its behavior there is analogous to free propagation We are interested\nin how many steps it will take the particle, starting on one tail and\npropagating through the graph (where its propagation is not free), to emerge\nonto the other tail. The probability to make such a walk in n steps and the\nhitting time for such a walk can be expressed in terms of the transmission\namplitude for the graph, which is one element of its S matrix. Demonstrating\nthis necessitates a study of the analyticity properties of the transmission and\nreflection amplitudes of a graph. We show that the graph can have bound states\nthat cannot be accessed by a particle entering the graph from one of its tails.\nTime-reversal invariance of a quantum walk is defined and used to show that the\ntransmission amplitudes for the particle entering the graph from different\ndirections are the same if the walk is time-reversal invariant.",
"arxiv_id": "quant-ph/0403066",
"authors": [
"Edgar Feldman",
"Mark Hillery"
],
"categories": [
"quant-ph"
],
"journal_ref": "Coding Theory and Quantum Computing, edited by D. Evans, J. Holt,\n C. Jones, K. Klintworth, B. Parshall, O. Pfister, and H. Ward, Contemporary\n Mathematics, 381, 71 (2005).",
"title": "Quantum walks on graphs and quantum scattering theory",
"url": "https://arxiv.org/abs/quant-ph/0403066"
},
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