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

01-01-2017

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

Authors: Pascal Philipp, Renato Portugal

Published in: Quantum Information Processing | Issue 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.
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Footnotes
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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Metadata
Title
Exact simulation of coined quantum walks with the continuous-time model
Authors
Pascal Philipp
Renato Portugal
Publication date
01-01-2017
Publisher
Springer US
Published in
Quantum Information Processing / Issue 1/2017
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-016-1475-9

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