Mixing of Quantum Walks on Generalized Hypercubes

We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the hypercube with an additive matching x 7→ x⊕η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes Moore-Russell’s result on the hypercube [11]. • The class of Hamming graphs H(n, q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n, q) with q < 5 was known. • The bunkbed graph Bn(Af ) whose adjacency matrix is I⊗Qn+X⊗Af , where Af is a Z 2 -circulant matrix defined by a Boolean function f , is not uniform mixing if the Fourier transform of f has support of size smaller than 2. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.