Impacts of Multiple Solutions on the Lackadaisical Quantum Walk Search Algorithm

Carvalho, Jonathan H. A. de; Souza, Luciana Sandra Bastos de; Neto, Fernando M. de Paula; Ferreira, Tiago A. E.

doi:10.1007/978-3-030-61377-8_9

Cited by 5 publications

(9 citation statements)

References 25 publications

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“…In this section, we present the extension of the work developed in [33]. The novelties deal with the application of the LQW algorithm on grids of arbitrary dimensions with multiple solutions.…”

Section: Lackadaisical Quantum Walk On D-dimensional Gridsmentioning

confidence: 99%

“…Finally, the system begins in the uniform distribution between each of the N vertices of the d-dimensional grid with the weighted superposition of coin states generalized in Equation 7. The step where the simulation stops depends on a stopping condition, but the more appropriate one is to monitor the probability's evolution about the m marked vertices until achieving its maximum [33]. Now, the LQW algorithm can be simulated on higher-than-two-dimensional grids with multiple solutions.…”

Section: Generalization To D-dimensional Gridsmentioning

confidence: 99%

“…In real applications, the solutions could be at any grid point. However, for methodological reasons, the experiment setup is controlled in terms of solutions' positions [26,31,33]. It is not different here.…”

Section: Application On D-dimensional Gridsmentioning

confidence: 99%

“…After that, Giri and Korepin [32] showed that one of these m solutions can be obtained with sufficiently high probability in O( N m log N m ) steps. This paper is an extension of the conference paper presented in [33]. The work as a whole is an extensive experimental analysis of the LQW algorithm searching for multiple solutions on grids.…”

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confidence: 99%

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On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids

Carvalho¹,

Souza²,

Neto³

et al. 2021

Preprint

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Quantum computing holds the promise of improving the information processing power to levels unreachable by classical computation. Quantum walks are heading the development of quantum algorithms for searching information on graphs more efficiently than their classical counterparts. A quantum-walk-based algorithm that is standing out in the literature is the lackadaisical quantum walk. The lackadaisical quantum walk is an algorithm developed to search two-dimensional grids whose vertices have a self-loop of weight l. In this paper, we address several issues related to the application of the lackadaisical quantum walk to successfully search for multiple solutions on grids. Firstly, we show that only one of the two stopping conditions found in the literature is suitable for simulations. We also demonstrate that the final success probability depends on the space density of solutions and the relative distance between solutions. Furthermore, this work generalizes the lackadaisical quantum walk to search for multiple solutions on grids of arbitrary dimensions. In addition, we propose an optimal adjustment of the self-loop weight l for such scenarios of arbitrary dimensions. It turns out the other fits of l found in the literature are particular cases. Finally, we observe a two-to-one relation between the steps of the lackadaisical quantum walk and the ones of Grover's algorithm, which requires modifications in the stopping condition. In conclusion, this work deals with practical issues one should consider when applying the lackadaisical quantum walk, besides expanding the technique to a wider range of search problems.

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Section: Lackadaisical Quantum Walk On D-dimensional Gridsmentioning

confidence: 99%

Section: Generalization To D-dimensional Gridsmentioning

confidence: 99%

Section: Application On D-dimensional Gridsmentioning

confidence: 99%

mentioning

confidence: 99%

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On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids

Carvalho¹,

Souza²,

Neto³

et al. 2021

Preprint

Self Cite

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“…Later, the unweighted self-loops at each vertex were replaced by a single self-loop of real-valued weight at each vertex, such that if is an integer, it is equivalent to the original definition of integer self-loops per vertex [10]. This generalization to real-valued weights led to speedups for spatial search on a variety of graphs, including the discrete torus with one marked vertex [11] and multiple marked vertices [12][13][14][15][16], periodic square lattices of arbitrary dimension [17,18], strongly regular graphs [18], Johnson graphs [18], the hypercube [18], regular locally arc-transitive graphs [19], the triangular lattice [20], and the honeycomb lattice [20]. All of these graphs are vertex transitive, meaning they have symmetries such that each vertex has the same structure.…”

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confidence: 99%

Search by Lackadaisical Quantum Walk with Symmetry Breaking

Rapoza,

Wong

2021

Preprint

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The lackadaisical quantum walk is a lazy version of a discrete-time, coined quantum walk, where each vertex has a weighted self-loop that permits the walker to stay put. They have been used to speed up spatial search on a variety of graphs, including periodic lattices, strongly regular graphs, Johnson graphs, and the hypercube. In these prior works, the weights of the self-loops preserved the symmetries of the graphs. In this paper, we show that the self-loops can break all the symmetries of vertex-transitive graphs while providing the same computational speedups. For vertex-transitive graphs with large numbers of vertices, only the weight of the self-loop at the marked vertex matters, and the remaining self-loop weights can be chosen randomly.

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Lackadaisical Quantum Walk in the Hypercube to Search for Multiple Marked Vertices

Souza¹,

Carvalho²,

Ferreira³

2021

Lecture Notes in Computer Science

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Impacts of Multiple Solutions on the Lackadaisical Quantum Walk Search Algorithm

Cited by 5 publications

References 25 publications

On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids

On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids

Search by Lackadaisical Quantum Walk with Symmetry Breaking

Lackadaisical Quantum Walk in the Hypercube to Search for Multiple Marked Vertices

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