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This work focuses on the quantum mixing time, which is crucial for efficientquantum sampling and algorithm performance. We extend Richter's previousanalysis of continuous time quantum walks on the periodic lattice$\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \dots \times \mathbb{Z}_{n_d}$,allowing for non-identical dimensions $n_i$. We present two quantum walks thatachieve faster mixing compared to classical random walks. The first is acoordinate-wise quantum walk with a mixing time of $O\left(\left(\sum{i=1}^{d}n_i \right) \log{(d/\epsilon)}\right)$ and $O(d \log(d/\epsilon))$measurements. The second is a continuous-time quantum walk with$O(\log(1/\epsilon))$ measurements, conjectured to have a mixing time of$O\left(\sum_{i=1}^d n_i(\log(n_1))^2 \log(1/\epsilon)\right)$. Our resultsdemonstrate a quadratic speedup over classical mixing times on the generalizedperiodic lattice. We provide analytical evidence and numerical simulationssupporting the conjectured faster mixing time. The ultimate goal is to provethe general conjecture for quantum walks on regular graphs.

Quantum walk mixing is faster than classical on periodic lattices