4.7 Article

An index theorem for one-dimensional gapless non-unitary quantum walks

期刊

QUANTUM INFORMATION PROCESSING
卷 20, 期 9, 页码 -

出版社

SPRINGER
DOI: 10.1007/s11128-021-03212-y

关键词

Chiral symmetry; Non-unitary quantum walks; Gapless quantum walks; Supersymmetry; Witten index

资金

  1. JSPS KAKENHI [20J22684]
  2. Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University
  3. Grants-in-Aid for Scientific Research [20J22684] Funding Source: KAKEN

向作者/读者索取更多资源

Recent developments in the index theory of discrete-time quantum walks have allowed for the application of the theory to non-unitary time evolutions, showing that unitarity is not necessary for certain calculations in this context.
Recent developments in the index theory of discrete-time quantum walks allow us to assign a certain well-defined supersymmetric index to a unitary time evolution U and a Z(2)-grading operator Gamma satisfying the chiral symmetry condition, U* = Gamma U Gamma. In the present article, we extend this index theory to encompass non-unitary time evolutions U. The existing literature for unitary U assumes that U is essentially gapped to define the associated index, that is, the essential spectrum of U contains neither -1 nor +1. We show this assumption is not necessary if U fails to be unitary. As a concrete example, we consider the well-known non-unitary quantum walk model on the integer lattice Z introduced by Mochizuki-Kim-Obuse.

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