Document type
Book partVersion
Published versionPublication date
Publication license
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/178245
Invariance in Quantum Walks
Journal Title
Authors
Director/Tutor
Journal ISSN
Volume Title
Related resource
Abstract
In this Chapter, we present some interesting properties of quantum walks on the line. We
concentrate our attention in the emergence of invariance and provide some insights into
the ultimate origin of the observed behavior. In the first part of the Chapter, we review
the building blocks of the quantum-mechanical version of the standard random walk in
one dimension. The most distinctive difference between random and quantum walks is
the replacement of the random coin in the former by the action of a unitary operator upon
some internal property of the later. We provide explicit expressions for the solution to the
problem when the most general form for the homogeneous unitary operator is considered,
and we analyze several key features of the system as the presence of symmetries or
stationary limits. After that, we analyze the consequences of letting the properties of the
coin operator change from site to site, and from time step to time step. In spite of this lack
of homogeneity, the probabilistic properties of the motion of the walker can remain
unaltered if the coin variability is chosen adequately. Finally, we show how this invariance
can be connected to the gauge freedom of electromagnetism.
Subject
Subject (English)
Citation
Citation
MONTERO TORRALBO, Miquel. Invariance in Quantum Walks. Chapter 1 in: Bracken. Paul. 2016. Research Advances in Quantum Dynamics. IntechOpen. ISBN: 978-953-51-5072-5. DOI: 10.5772/61561. pp: 3-25.. [consulted: 8 of June of 2026]. Available at: https://hdl.handle.net/2445/178245