  A coinless, discrete-time quantum walk possesses a Hilbert space whose
dimension is smaller compared to the widely-studied coined walk. Coined walks
require the direct product of the site basis with the coin space, coinless
walks operate purely in the site basis, which is clearly minimal. These
coinless quantum walks have received considerable attention recently because
they have evolution operators that can be obtained by a graphical method based
on lattice tessellations and they have been shown to be as efficient as the
best known coined walks when used as a quantum search algorithm. We argue that
both formulations in their most general form are equivalent. In particular, we
demonstrate how to transform the one-dimensional version of the coinless
quantum walk into an equivalent extended coined version for a specific family
of evolution operators. We present some of its basic, asymptotic features for
the one-dimensional lattice with some examples of tessellations, and analyze
the mixing time and limiting probability distributions on cycles.
