# metadata.dc.description.abstract[].value
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.

# metadata.dc.rights[].value
cc by (c) Montero Torralbo, Miquel, 2016

# metadata.dc.source[].value
Llibres / Capítols de llibre (Física de la Matèria Condensada)