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arXiv:1508.05270 · Quantum Information Processing

Mimicking the Hadamard discrete-time quantum walk with a time-independent Hamiltonian

Jalil Khatibi Moqadam, M. C. de Oliveira

Published Mar 28, 2019• 1 citation
arXivHTMLPDF
DOI
10.1007/s11128-019-2262-1

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We consider a one-dimensional space-inhomogeneous discrete time quantum walk. This model is the Hadamard walk with one defect at the origin which is different from the model introduced by Wojcik et al. [14]. We obtain a stationary measure of the model by solving the eigenvalue problem and an asymptotic behaviour of the return probability by the path counting approach. Moreover, we get the time-averaged limit measure using the space-time generating function method. The measure is symmetric for the origin and independent of the initial coin state at the starting point. So localization depends only on the parameter which determines the model.
We study a one-parameter family of discrete-time quantum walk models on Z\mathbb{Z}Z and Z2{\mathbb{Z}}^{2}Z2 associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on Z\mathbb{Z}Z and Z2{\mathbb{Z}}^{2}Z2 is proved. Symmetrization on Z\mathbb{Z}Z and Z2{\mathbb{Z}}^{2}Z2 is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on Z\mathbb{Z}Z. A necessary condition for the existence of a phenomenon known as localization is given.