  It has been shown that certain quantum walks give rise to relativistic wave
equations, such as the Dirac and Weyl equations, in their long-wavelength
limits. This intriguing result raises the question of whether something similar
can happen in the multi-particle case. We construct a one-dimensional quantum
cellular automaton (QCA) model which matches the quantum walk in the single
particle case, and which approaches the quantum field theory of free fermions
in the long-wavelength limit. However, we show that this class of constructions
does not generalize to higher spatial dimensions in any straightforward way,
and that no construction with similar properties is possible in two or more
spatial dimensions. This rules out the most common approaches based on QCAs. We
suggest possible methods to overcome this barrier while retaining locality.

Quantum Physics