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International Conference on Quantum Information

Connectivity effects in the coined quantum walk search algorithm

Neil B. Lovett, Viv M. Kendon

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We show numerically how the quantum walk search algorithm depends on both the spatial dimension and connectivity of the structure being searched. We investigate this using a simple form of tunnelling to interpolate between lattices.
DOI
10.1364/icqi.2011.qmi16

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We carry out a numerical study of the quantum walk search algorithm of Shenvi, Kempe and Whaley Shenvi et al. (2003) and the factors that affect its efficiency in finding an individual state from an unsorted set. Previous work has focused purely on the effects of the dimensionality of the dataset to be searched. In the current paper we consider the effects of interpolating between dimensions, the connectivity of the dataset and the possibility of disorder in the underlying substrate: all these factors affect the efficiency of the search algorithm. We show that in addition to the strong dependence on the spatial dimension of the structure to be searched, there are also secondary dependencies on the connectivity and symmetry of the lattice, with greater connectivity providing a more efficient algorithm. We also show that the algorithm can tolerate a non-trivial level of disorder in the underlying substrate.
We carry out a numerical study of the quantum walk search algorithm of Shenvi, Kempe and Whaley Shenvi et al. (2003) and the factors that affect its efficiency in finding an individual state from an unsorted set. Previous work has focused purely on the effects of the dimensionality of the dataset to be searched. In the current paper we consider the effects of interpolating between dimensions, the connectivity of the dataset and the possibility of disorder in the underlying substrate: all these factors affect the efficiency of the search algorithm. We show that in addition to the strong dependence on the spatial dimension of the structure to be searched, there are also secondary dependencies on the connectivity and symmetry of the lattice, with greater connectivity providing a more efficient algorithm. We also show that the algorithm can tolerate a non-trivial level of disorder in the underlying substrate.
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One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.