2005/09/29 by Yuki Kelly Itakura, Itakura, Yuki Kelly · 1 citation
Computer Science · Physics and Astronomy · #Computability, Logic, AI Algorithms #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Quantum-Dot Cellular Automata #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0509206
70 pages, 8 figures, MMath Essay for Univerisity of Waterloo
arxiv created 2005/09/29 · openalex publication_date 2005/09/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle finding problem (quant-ph/0310134) and matrix verification problem (quant-ph/0409035) to compare the powers of the two different walks. We also present Ambainis's method of lower bounding technique (quant-ph/0002066) to obtain a lower bound for this problem. It turns out Szegedy's algorithm can be generalized to solve similar problems. Therefore we use Szegedy's theorem to analyze the problem of matrix set commutativity. We give an O(k4/5n9/5) algorithm as well as a lower bound of Omega(k1/2n). We generalize the technique used in coming up with the upper bound to solve a broader range of similar problems. This is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter, here, k and n.