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Randomizing Quantum Walk

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Abstract

The conditional shift operator in Discrete-time Quantum Walk (DTQW) shifts the position of the walker by unit shift-size depending on the coin state. This scenario can be generalized by choosing the shift-size different from the unit size. The first generalization made in this work is that shift-size is greater than unit size. The second variant is made out of allowing shift-size in positive and negative directions to be not equal to each other. The third type is developed by choosing the shift-size randomly at each step. We have calculated several parameters for these walks. The probability in each case evolves depending on the choice of the shift-sizes. All three walks spread faster than the standard DTQW. In the case of random shift-size, the probability becomes random but still follows some specific pattern. The increase in shift-size, on one hand, preserves the overall behavior but on other hand increases standard deviations σ for random choice of shift-size. For all these variants the Shannon entropy remains the same for the initial steps and then becomes small (after a few steps) than the Shannon entropy of standard DTQW. The Shannon entropy for random shift-size is of random behavior. The entanglement entropy remains the same as that of DTQW with small variations at higher steps. For the special case of the walk with a left-side shift-size greater than the right-side shift-size, the entropy reduces drastically after a few initial steps.

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Authors and Affiliations

  1. Department of Physics, Kohat University of Science and Technology, Kohat, 26000, Khyber-Pakhtunkhwa, Pakistan

    Anwar Zaman, Rashid Ahmad, Safia Bibi & Sajid Khan

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  1. Anwar Zaman
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  2. Rashid Ahmad
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  3. Safia Bibi
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Correspondence to Rashid Ahmad.

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Zaman, A., Ahmad, R., Bibi, S. et al. Randomizing Quantum Walk. Int J Theor Phys 61, 135 (2022). https://doi.org/10.1007/s10773-022-05113-x

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