A block-encoding approach to the Segawa-Suzuki spectral mapping theorem
Abstract
We revisit the Segawa--Suzuki spectral mapping theorem for abstract Szegedy-type quantum walks from a block-encoding viewpoint. Retaining both eigenspaces of the involution underlying the chiral symmetry yields a natural two-by-two block representation in which the usual discriminant operator and its counterpart form the diagonal blocks. The non-endpoint spectral mapping theorem then follows from an elementary Schur decomposition, substantially simplifying the original proof by avoiding its invariant-subspace decomposition and approximate-eigenvector arguments. The same framework establishes the result for the spectrum, point spectrum, and Fredholm essential spectrum and gives a direct block-theoretic description of the endpoint inherited and birth eigenspaces. We illustrate the construction with a one-dimensional split-step quantum walk.
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BibTeX
@misc{tanaka2026blockencoding,
title = {{A block-encoding approach to the Segawa-Suzuki spectral mapping theorem}},
author = {Yohei Tanaka},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00498},
eprint = {2611.00498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00498},
}
RIS
TY - UNPB
TI - A block-encoding approach to the Segawa-Suzuki spectral mapping theorem
AU - Yohei Tanaka
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00498
UR - https://arxiv.org/abs/2611.00498
AB - We revisit the Segawa--Suzuki spectral mapping theorem for abstract Szegedy-type quantum walks from a block-encoding viewpoint. Retaining both eigenspaces of the involution underlying the chiral symmetry yields a natural two-by-two block representation in which the usual discriminant operator and its counterpart form the diagonal blocks. The non-endpoint spectral mapping theorem then follows from an elementary Schur decomposition, substantially simplifying the original proof by avoiding its invariant-subspace decomposition and approximate-eigenvector arguments. The same framework establishes the result for the spectrum, point spectrum, and Fredholm essential spectrum and gives a direct block-theoretic description of the endpoint inherited and birth eigenspaces. We illustrate the construction with a one-dimensional split-step quantum walk.
ER -