  This work focuses on the quantum mixing time, which is crucial for efficient
quantum sampling and algorithm performance. We extend Richter's previous
analysis 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 that
achieve faster mixing compared to classical random walks. The first is a
coordinate-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 results
demonstrate a quadratic speedup over classical mixing times on the generalized
periodic lattice. We provide analytical evidence and numerical simulations
supporting the conjectured faster mixing time. The ultimate goal is to prove
the general conjecture for quantum walks on regular graphs.
