Symmetric quantum walks on Hamming graphs and their limit distributions

We study a class of symmetric quantum walks on Hamming graphs, where the distance between vertices specifies the transition probability. A special model is the simple quantum walk on the hypercube, which has been discussed in the literature. Eigenvalues of the unitary operator of the quantum walks are zeros of certain self-reciprocal polynomials. We obtain a spectral representation of the wave vector, where our systematic treatment relies on the coin space isomorphic to the state space and the commutative association scheme. The Grover coin is extended to the reflection about a vector in an invariant subspace of the Terwilliger algebra. The limit distributions of several quantum walks are obtained.