# Spatial search by quantum walk

@article{Childs2004SpatialSB,
title={Spatial search by quantum walk},
author={Andrew M. Childs and Jeffrey Goldstone},
journal={Physical Review A},
year={2004},
volume={70},
pages={022314}
}
• Published 6 June 2003
• Computer Science
• Physical Review A
Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of N items laid out in d spatial dimensions can be searched in time of order {radical}(N) for d>2, and in time of order {radical}(N) poly(log N) for d=2. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the…
460 Citations
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• Computer Science
• 2003
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It is shown that spatial search on Johnson graphs by continuous-time quantum walk achieves the Grover lower bound with success probability 1 asymptotically for every fixed diameter, where N is the number of vertices.
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• 2021
This article provides a new continuous-time quantum walk search algorithm that can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks.
Quantum search of spatial regions
• Computer Science
44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
• 2003
An 0(/spl radic/n)-qubit communication protocol for the disjointness problem is given, which improves an upper bound of Hoyer and de Wolf and matches a lower bound of Razborov.
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ArXiv
• 2022
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• 2015
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• Mathematics
ICCCS
• 2018
The spatial search for a single marked vertex by continuous-time quantum walk (CTQW) is generalized to the search for multiple marked vertices and it is found that although the different Hamming distance lead to unequal search time, this search can be done in \(\mathrm{O}\left( {\sqrt{N} } \right) time for all two uniform marked Vertices.
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• 2020
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• Computer Science
• 2014
By constructing lattice Hamiltonians exhibiting Dirac points in their dispersion relations and exploiting the linear behavior near a Dirac point, this work develops algorithms that solve the problem of searching a general $d$-dimensional lattice of $N$ vertices for a single marked item using a continuous-time quantum walk in a time of $O(\sqrt{N})$.

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