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arXiv:1901.09119 · Quantum Information Processing

Eigenvalues of quantum walk induced by recurrence properties of the underlying birth and death process: application to computation of an edge state

Yusuke Ide, Norio Konno, Etsuo Segawa

Published May 1, 2021• 2 citations
arXivHTMLPDF
DOI
10.1007/s11128-021-02999-0

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This paper underscores the intimate connection between the quantum walks generated by certain semi-infinite spin chain Hamiltonians and classical birth and death processes. It is observed that transition amplitudes between single excitation states of the spin chains have an expression in terms of orthogonal polynomials which is analogous to the Karlin-McGregor representation formula of the transition probability functions for classes of birth and death processes. As an application, we present a characterization of spin systems for which the probability to return to the point of origin at some time is 1 or almost 1.
In this paper, we consider a spectral analysis of the Correlated Random Walk (CRW) on the path. We apply an analytical method for the Quantum Walk to CRW. For the isospectral coin cases, we obtain all of the eigenvalues and the corresponding eigenvectors of the time evolution operator of CRW. We show that most of them are described by the eigenvalues and the eigenvectors of the Jacobi matrix related to the random walks on the path (the birth and death chains). The remaining eigenvector corresponding to an eigenvalue 1, is characterized by the stationary measure of the same birth and death chain. As a consequence, we derive the limiting distribution.
A new model maps a quantum random walk described by a Hadamard operator to a particular case of a birth and death process. The model is represented by a 2D Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries (this is possible by increasing the dimensionality of the system). The probability distribution of the walker population is preserved using the Markovian property. By applying a proper transformation to the population distribution of the random walk, the probability distributions of the quantum states |0>, 1> are revealed. Thus, the new model has two unique properties: it reveals the probability distribution of the quantum states as a unitary system and preserves the population distribution of the random walker as a Markovian system.