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- W2609471214 abstract "We present a detailed analysis of the real-space renormalization group (RG) for discrete-time quantum walks on fractal networks. The RG-flow for such a walk on a dual Sierpinski gasket is obtained explicitly after transforming the unitary evolution equation into Laplace space. Unlike for classical random walks, we find that the long-time asymptotics of the quantum walk requires consideration of a diverging number of Laplace-poles, which we demonstrate exactly for the closed form solution available for the walk on a 1d-loop. In particular, we calculate the probability of the walk to overlap with its starting position, which oscillates with a period that scales as $N^{d^Q_w/d_f}$ with system size $N$, consistent with our simulations. While the largest Jacobian eigenvalue $lambda_1$ of the RG-flow merely reproduces the fractal dimension, $d_f = log_2 lambda_1$, the asymptotic analysis shows that the second eigenvalue $lambda_2$ becomes essential to determine the dimension of the quantum walk via $d^Q_w = log_2 sqrt{lambda_1lambda_2}$. We trace this fact to a delicate cancellation caused by unitarity. We obtain identical relations for other networks, although the details of the RG-analysis may exhibit surprisingly distinct features. Thus, our conclusions - which trivially extend to regular lattices - appear to be quite general and likely apply to networks beyond those studied here." @default.
- W2609471214 created "2017-05-05" @default.
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- W2609471214 date "2017-04-15" @default.
- W2609471214 modified "2023-10-14" @default.
- W2609471214 title "Asymptotic Analysis of Coined Quantum Walks on Fractal Networks" @default.
- W2609471214 hasPublicationYear "2017" @default.
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