  In this paper I propose a new method of encoding discrete variables into
Ising model qubits for quantum optimization. The new method is based on the
physics of domain walls in one dimensional Ising spin chains. I find that these
encodings and the encoding of arbitrary two variable interactions is possible
with only two body Ising terms. Following on from similar results for the `one
hot' method of encoding discrete variables [Hadfield et. al. Algorithms 12.2
(2019): 34] I also demonstrate that it is possible to construct two body mixer
terms which do not leave the logical subspace, an important consideration for
optimising using the quantum alternating operator ansatz (QAOA). I additionally
discuss how, since the couplings in the domain wall encoding only need to be
ferromagnetic and therefore could in principle be much stronger than
anti-ferromagnetic couplers, application specific quantum annealers for
discrete problems based on this construction may be beneficial. Finally, I
compare embedding for synthetic scheduling and colouring problems with the
domain wall and one hot encodings on two graphs which are relevant for quantum
annealing, the chimera graph and the Pegasus graph. For every case I examine I
find a similar or better performance from the domain wall encoding as compared
to one hot, but this advantage is highly dependent on the structure of the
problem. For encoding some problems, I find an advantage similar to the one
found by embedding in a Pegasus graph compared to embedding in a chimera graph.
