International Journal of Engineering and Applied Sciences (IJEAS)

Application of Segawa-Suzuki Spectral Mapping Theorem

Hong Ji, Caishi Wang, Nan Fan

Published Jul 31, 2021• 0 citations

Abstract

In this paper, we consider a class of unitary operators associated with the evolution operators of the quantum walk .By using Segawa and Suzuki's spectral mapping theorem, we obtain their spectra.

DOI: 10.31873/ijeas.8.7.10

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International Journal of Engineering and Applied Sciences (IJEAS)

Application of Segawa-Suzuki Spectral Mapping Theorem

Hong Ji, Caishi Wang, Nan Fan

Published Jul 31, 2021• 0 citations

Publisher

Abstract

In this paper, we consider a class of unitary operators associated with the evolution operators of the quantum walk .By using Segawa and Suzuki's spectral mapping theorem, we obtain their spectra.

DOI: 10.31873/ijeas.8.7.10

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Yusuke Higuchi, Etsuo Segawa, Akito Suzuki

Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$ , we introduce an abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on $\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a unitary involution on $\mathcal{H}$ . In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, we introduce an abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on $\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a unitary involution on $\mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.

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Given two Hilbert spaces,

\mathcal{H}

and

\mathcal{K}

, we introduce an abstract unitary operator

U

\mathcal{H}

and its discriminant

T

\mathcal{K}

induced by a coisometry from

\mathcal{H}

\mathcal{K}

and a unitary involution on

\mathcal{H}

. In a particular case, these operators

U

and

T

become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between

U

and

T

via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.