Quantum walk mixing is faster than classical on periodic lattices

The quantum mixing time is a critical factor affecting the efficiency of quantum sampling and algorithm performance. It refers to the minimum time required for a quantum walk to approach its limiting distribution closely and has implications across the areas of quantum computation. This work focuses on the continuous time quantum walk mixing on a regular graph, evolving according to the unitary map U=eiĀt, where the Hamiltonian Ā is the normalized adjacency matrix of the graph. In [Physical Review A 76, 042306 (2007).], Richter previously showed that this walk mixes in time O(ndlog(d)log(1/ϵ)) with O(log(d)log(1/ϵ)) intermediate measurements when the graph is the d−dimensional periodic lattice Zn×Zn×⋯×Zn. We extend this analysis to the periodic lattice L=Zn1×Zn2×⋯×Znd, relaxing the assumption that ni are identical. We provide two quantum walks on periodic lattices that achieve faster mixing compared to classical random walks: 1. A coordinate-wise quantum walk that mixes in O((∑i=1dni)log(d/ϵ)) time with O(dlog(d/ϵ)) measurements. 2. A continuous-time quantum walk with O(log(1/ϵ)) measurements that conjecturally mixes in O(∑i=1dni(log(n1))2log(1/ϵ)) time. Our results demonstrate a quadratic speedup over the classical mixing time O(dn12log(d/ϵ)) on the generalized periodic lattice L. We have provided analytical evidence and numerical simulations to support the conjectured faster mixing time of the continuous-time quantum walk algorithm. Making progress towards proving the general conjecture that quantum walks on regular graphs mix in O(δ−1/2log(N)log(1/ϵ)) time, where δ is the spectral gap and N is the number of vertices.