Electric Circuit Induced by Quantum Walk
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Electric Circuit Induced by Quantum Walk Yusuke Higuchi1,2 · Mohamed Sabri3 · Etsuo Segawa4,5 Received: 20 March 2020 / Accepted: 13 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the ∞ -category initial state so that the internal graph receives time independent input from the tails, say α in , at every time step. We show that the response of the Szegedy walk to the input, which is the output, say β out , from the internal graph to the tails in the long time limit, is drastically changed depending on the reversibility of the underlying random walk. If the underlying random walk is reversible, we have β out = Sz(mδ E )α in , where the unitary matrix Sz(mδ E ) is the reflection matrix to the unit vector mδ E which is determined by the boundary of the internal graph δ E. Then the global dynamics so that the internal graph is regarded as one vertex recovers the local dynamics of the Szegedy walk in the long time limit. Moreover if the underlying random walk of the Szegedy walk is reversible, then we obtain that the stationary state is expressed by a linear combination of the reversible measure and the electric current on the electric circuit determined by the internal graph and the random walk’s reversible measure. On the other hand, if the underlying random walk is not reversible, then the unitary matrix is just a phase flip; that is, β out = −α in , and the stationary state is similar to the current flow but satisfies a different type of the Kirchhoff laws. Keywords Quantum walk · Scattering · Electric circuit · Reversibility
Communicated by Hal Tasaki.
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Etsuo Segawa [email protected] Yusuke Higuchi [email protected] Mohamed Sabri [email protected]
1
Mathematics Laboratories, College of Arts and Sciences, Showa University, Fujiyoshida, Yamanashi 403-0005, Japan
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Present Address: Department of Mathematics, Faculty of Sciences, Gakushuin University, Mejiro, Toshima-ku, Tokyo, Japan
3
Graduate School of Information Sciences, Tohoku University, Aoba, Sendai 980-0845, Japan
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Graduate School of Education Center, Yokohama National University, Yokohama, Japan
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Graduate School of Environmental Sciences, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan
123
Yu. Higuchi et al.
1 Introduction An irreducible random walk on a finite graph has a stationary state which is expressed by the eigenvector of the maximal eigenvalue 1. On the other hand, for a quantum walk on a finite graph, it is not so easy to obtain a stationarity of a quantum walk from a natural initial state. The difference between them arises from the following; the time evolution operator of a quantum walk is described by some unitary operator on Hilbert space and then its spectrum lies on the unit circle in the complex plain. However if the graph is infinite, e.g., d-dimensional lattic
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