  Quantum Random Walks (QRW) were first defined as one-particle sectors of
Quantum Lattice Gas Automata (QLGA). Recently, they have been generalized to
include history dependence, either on previous coin (internal, i.e., spin or
velocity) states or on previous position states. These models have the goal of
studying the transition to classicality, or more generally, changes in the
performance of quantum walks in algorithmic applications. We show that several
history dependent QRW can be identified as one-particle sectors of QLGA. This
provides a unifying conceptual framework for these models in which the extra
degrees of freedom required to store the history information arise naturally as
geometrical degrees of freedom on the lattice.
