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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Return Probability of Quantum and Correlated Random Walks.

Chusei Kiumi1, Norio Konno2, Shunya Tamura1

  • 1Graduate School of Science and Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan.

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Summary

This study analyzes return probabilities for quantum and correlated random walks on a 1D lattice. Both walk types exhibit return probabilities expressible via Legendre polynomials, with quantum walks also linked to elliptic integrals.

Keywords:
Legendre polynomialcorrelated random walkelliptic integralquantum walkreturn probability

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Area of Science:

  • Quantum mechanics
  • Statistical physics
  • Probability theory

Background:

  • Return probability is a fundamental concept in classical random walk analysis.
  • Understanding return probabilities is crucial for characterizing the long-term behavior of random processes.

Purpose of the Study:

  • To investigate the return probability of quantum random walks.
  • To analyze the return probability of correlated random walks.
  • To compare these probabilities on a one-dimensional integer lattice.

Main Methods:

  • Employed the path counting method for analysis.
  • Utilized mathematical expressions involving Legendre polynomials.
  • Derived generating functions for quantum walk return probabilities.

Main Results:

  • Demonstrated that return probabilities for both quantum and correlated random walks can be expressed using Legendre polynomials.
  • Showed that the generating function for quantum walk return probability is related to elliptic integrals of the first and second kinds.

Conclusions:

  • Established a novel connection between quantum/correlated random walks and special functions (Legendre polynomials, elliptic integrals).
  • Provides a new analytical framework for studying quantum and correlated random walks on lattices.