Saniya Wagh | Searcharxiv

arXiv2023

Quantum walks advantage on the dihedral group for uniform sampling problem

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 order $\Omega(n^2 \log(1/2\epsilon))$, 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 a class of sampling problems based on non-abelian groups.

quant-ph↗

arXiv2023

Character Sheaves on Tori over Local Fields

Let $\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field ${k}$ and ring of integers $\breve{{O}}$. Let $T$ be a torus defined over $\breve{K}$. Let $L^+T$ denote the connected commutative pro-algebraic group over ${k}$ obtained by applying the Greenberg functor to the connected N\'eron model of $T$ over $\breve{{O}}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $\pi_1(L^+T)$. We then study multiplicative local systems (or character sheaves) on $L^+T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^+T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence.

math.RT↗