Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science.The mixing property is necessary for Markov chains to approach stationary distribution...Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science.The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks.Quantum walks show promise for faster mixing times than classical methods but lack universal proof,especially in finite group settings.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 typically takes at least orderΩ(n^(2)log(1/2∈)),the continuous-time quantum walk mixing time on D_(2n)is of order O(n(log n)^(5)log(1/∈)),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 a class of sampling problems based on non-abelian groups.展开更多
基金supported by the Key-Area Research and Development Program of Guang-Dong Province(Grant No.2018B030326001)the National Natural Science Foundation of China(U1801661)Shenzhen Science and Technology Program(KQTD20200820113010023)。
文摘Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science.The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks.Quantum walks show promise for faster mixing times than classical methods but lack universal proof,especially in finite group settings.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 typically takes at least orderΩ(n^(2)log(1/2∈)),the continuous-time quantum walk mixing time on D_(2n)is of order O(n(log n)^(5)log(1/∈)),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 a class of sampling problems based on non-abelian groups.
