A walk on max-plus algebra

Sennosuke Watanabe; Akiko Fukuda; Etsuo Segawa; Iwao Sato

doi:10.1016/j.laa.2020.03.025

A walk on max-plus algebra

Sennosuke Watanabe, Akiko Fukuda, Etsuo Segawa, Iwao Sato

研究成果: Article › 査読

抄録

Max-plus algebra is a kind of idempotent semiring over R_max:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ²-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.

本文言語	English
ページ（範囲）	29-48
ページ数	20
ジャーナル	Linear Algebra and Its Applications
巻	598
DOI	https://doi.org/10.1016/j.laa.2020.03.025
出版ステータス	Published - 2020 8月 1

ASJC Scopus subject areas

代数と数論
数値解析
幾何学とトポロジー
離散数学と組合せ数学

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10.1016/j.laa.2020.03.025

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abstract = "Max-plus algebra is a kind of idempotent semiring over Rmax:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ2-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.",

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AU - Segawa, Etsuo

AU - Sato, Iwao

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N2 - Max-plus algebra is a kind of idempotent semiring over Rmax:=R∪{−∞} with two operations ⊕:=max and ⊗:=+. In this paper, we introduce a new model of a walk on one dimensional lattice on Z, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the ℓ2-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk. Moreover, spectral analysis on the total time evolution operator is also given.

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A walk on max-plus algebra

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