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While completely self-avoiding quantum walks have the distinct property ofleading to a trivial unidirectional transport of a quantum state, aninteresting and non-trivial dynamics can be constructed by restricting theself-avoidance to a subspace of the complete Hilbert space. Here, we present acomprehensive study of three two-dimensional quantum walks, which areself-avoiding in coin space, in position space and in both, coin and positionspace. We discuss the properties of these walks and show that all result indelocalisation of the probability distribution for initial states which arestrongly localised for a walk with a standard Grover coin operation. We alsopresent analytical results for the evolution in the form of limit distributionsfor the self-avoiding walks in coin space and in both, coin and position space.

Self-avoiding quantum walks: realisations in subspaces and limit theorems