Non-uniform Mixing of Quantum Walks on the Symmetric Group

Abstract: It is well-known that classical random walks on regular graphs converge to the uniform distribution. Quantum walks, in their various forms, are quantizations of their corresponding classical random walk processes. Gerhardt and Watrous (2003) demonstrated that continuous-time quantum walks do not converge to the uniform distribution on certain Cayley graphs of the Symmetric group, which by definition are all regular. In this paper, we demonstrate that discrete-time quantum walks, in the sense of quantized Markov chains as introduced by Szegedy (2004), also do not converge to the uniform distribution. We analyze the spectra of the Szegedy walk operators using the representation theory of the symmetric group. In the discrete setting, the analysis is complicated by the fact that we work within a Hilbert space of a higher dimension than the continuous case, spanned by pairs of vertices. Our techniques are general, and we believe they can be applied to derive similar analytical results for other non-commutative groups using the characters of their irreducible representation.