  Quantum walks exhibit many unique characteristics compared to classical
random walks. In the classical setting, self-avoiding random walks have been
studied as a variation on the usual classical random walk. Classical
self-avoiding random walks have found numerous applications, most notably in
the modeling of protein folding. We consider the analogous problem in the
quantum setting. We complement a quantum walk with a memory register that
records where the walker has previously resided. The walker is then able to
avoid returning back to previously visited sites. We parameterise the strength
of the memory recording and the strength of the memory back-action on the
walker's motion, and investigate their effect on the dynamics of the walk. We
find that by manipulating these parameters the walk can be made to reproduce
ideal quantum or classical random walk statistics, or a plethora of more
elaborate diffusive phenomena. In some parameter regimes we observe a close
correspondence between classical self-avoiding random walks and the quantum
self-avoiding walk.
