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Quantum Information Processing
Erschienen in: Quantum Information Processing 1/2017

01.01.2017

Exact simulation of coined quantum walks with the continuous-time model

verfasst von: Pascal Philipp, Renato Portugal

Erschienen in: Quantum Information Processing | Ausgabe 1/2017

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Abstract

The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.
Nächster Artikel Site recurrence of open and unitary quantum walks on the line

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Fußnoten
1
Let \(n_t\) be a Poisson random variable and denote the expected value by \(\langle \,\cdot \,\rangle \). For the continuous-time transition matrix \(M_{\mathrm{CT}}(t)\) and the discrete-time transition matrix \(M_{\mathrm{DT}}(k)\) of the RW, we have
$$\begin{aligned} M_{\mathrm{CT}}(t) = \langle M_{\mathrm{DT}}(n_t) \rangle = \sum _{k=0}^\infty P(n_t=k)M_{\mathrm{DT}}(k) = \text {exp}[t(M_{\mathrm{DT}}-I)] . \end{aligned}$$
.
 
2
In order to write down a matrix representation of C or S, one first has to decide how to list the basis elements \(|v,j\rangle \). Arranging them with respect to v yields a matrix representation of C that is in block diagonal form. The representation of S can be made block diagonal by ordering in the j-component.
 
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Metadaten
Titel
Exact simulation of coined quantum walks with the continuous-time model
verfasst von
Pascal Philipp
Renato Portugal
Publikationsdatum
01.01.2017
Verlag
Springer US
Erschienen in
Quantum Information Processing / Ausgabe 1/2017
Print ISSN: 1570-0755
Elektronische ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-016-1475-9

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