Quantum walks on infinite graphs
speaker: Etsuo Segawa (Tohoku University)abstract: In my talk, we consider the quantum walks on infinite graph \(G\) by abelian regular covering of finite graph \(G_o\). We show that the Grover walk on \(G\) is reduced to a twisted quantum walk on the fundamental finite graph with covering transformation group \(\Gamma\), \(G_o=\Gamma\backslash G\). At first we obtain the mapping theorem from a twisted walk on \(G^{(o)}\); the spectrum of the time evolution is decomposed into inherited part from the twisted walk and \(\pm 1\) with multiplicities \(b_1(G_o)-1+m_{\pm}\), respectively. Here \(b_1(G_o)\) is the first Betti number of \(G_o\) and \(m_{\pm 1}\) is the multiplicities of eigenvalues \(\pm 1\) of the transition matrix of the twisted walk on \(G_o\). Secondly as an application of the above mapping theory, we show that as far as the fundamental graph of infinite \(G\) satisfies \(b_1(G_o)\geq 2\), then the appropriate initial state provides localization of the Grover walk; this is due to the eigenspaces of \(\{\pm 1\}\) of the twisted quantum walk on \(G^{(o)}\). Their eigenvectors pulled back to the original graph \(G\) have finite supports associated with even closed paths in \(G\). Moreover we also discuss the contribution of the inherited part of the eivenvalues by the twisted walk to the linear spreading of the Grover walk. We partially obtain an abstractive form of the density function for the weak limit theorem in \(\mathbb{Z}^d\) case.
timetable:
Mon 11 Nov, 16:30 - 17:00, Sala Stemmi
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