Quantum walks advantage on the dihedral group for uniform sampling problem

• URL: http://arxiv.org/abs/2312.15693v1
• Date: Mon, 25 Dec 2023 11:21:55 GMT
• Title: Quantum walks advantage on the dihedral group for uniform sampling problem
• Authors: Shyam Dhamapurkar, Yuhang Dang, Saniya Wagh, and Xiu-Hao Deng
• Abstract summary: Mixing through walks is the process for a Markov chain to approximate a stationary distribution for a group. Quantum walks have shown potential advantages in mixing time over the classical case but lack general proof in the finite group case. This work has potential applications in algorithms for sampling non-abelian groups, graph isomorphism tests, etc.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Random walk algorithms, including sampling and approximations, have played a significant role in statistical physics and theoretical computer science. Mixing through walks is the process for a Markov chain to approximate a stationary distribution for a group. Quantum walks have shown potential advantages in mixing time over the classical case but lack general proof in the finite group case. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group $D_{2n}$ for odd $n$, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which is typically of order $O(n^2 \log(2n/\epsilon))$, the continuous-time quantum walk mixing time on $D_{2n}$ is of order $O(n (\log n)^5 \log(1/\epsilon))$, achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on $D_{2n}$. This work has potential applications in algorithms for sampling non-abelian groups, graph isomorphism tests, etc.

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