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Faster search of clustered marked states with lackadaisical quantum walks

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Abstract

The nature of discrete-time quantum walks in the presence of multiple marked states can be found in the literature. An exceptional configuration of clustered marked states, which is a variant of multiple marked states, may be defined as a cluster of k marked states arranged in a \(\sqrt{k} \times \sqrt{k}\) array within a \(\sqrt{N} \times \sqrt{N}\) grid, where \(k=n^{2}\) and n an odd integer. In this article, we establish through numerical simulation that for lackadaisical quantum walks, which is the analogue of a three-state discrete-time quantum walks on a line, the success probability to find a vertex in the marked region of this exceptional configuration is nearly 1 with smaller run-time. We also show that the weights of the self-loop suggested for multiple marked states in the state-of-the-art works are not optimal for this exceptional configuration of clustered marked states. We propose a weight of the self-loop which gives the desired result for this configuration.

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Acknowledgements

The first author gratefully acknowledges fruitful discussion regarding quantum walks with Prof. Andris Ambainis, University of Latvia. The first author is also thankful to Dr. Alexander Rivosh, Dr. Nikolajs Nahimovs and Dr. Raqueline Azevedo Medeiros Santos for the discussion on exceptional configurations of quantum walks. The authors have no conflicts of interest to declare that are relevant to the content of this article.

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Authors and Affiliations

  1. A. K. Choudhury School of Information Technology, University of Calcutta, Kolkata, India

    Amit Saha, Debasri Saha & Amlan Chakrabarti

  2. Atos, Pune, India

    Amit Saha

  3. Advanced Computing and Microelectronics Unit, Indian Statistical Institute, Kolkata, India

    Ritajit Majumdar & Susmita Sur-Kolay

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  1. Amit Saha
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Saha, A., Majumdar, R., Saha, D. et al. Faster search of clustered marked states with lackadaisical quantum walks. Quantum Inf Process 21, 275 (2022). https://doi.org/10.1007/s11128-022-03606-6

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