Quantum Speedup and Limitations on Matroid Properties.
Year: 2021
Type: preprint
Abstract: In this paper, we study for the first time how well quantum algorithms perform on matroid properties. We show that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats, hyperplanes) of a matroid, and the decision problem of deciding whether a matroid is uniform or Eulerian, by giving a uniform lower bound $\Omega(\sqrt{\binom{n}{\lfloor n/2\rfloor}})$ on the query complexity for all these problems. On the other hand, for the uniform matroid decision problem, we present an asymptotically optimal quantum algorithm which achieves the lower bound, and for the girth problem, we give an almost optimal quantum algorithm with query complexity $O(\log n\sqrt{\binom{n}{\lfloor n/2\rfloor}})$. In addition, for the paving matroid decision problem, we prove a lower bound $\Omega(\sqrt{\binom{n}{\lfloor n/2\rfloor}/n})$ on the query complexity, and give an $O(\sqrt{\binom{n}{\lfloor n/2\rfloor}})$ quantum algorithm.
Source: arXiv (Cornell University)
Authors: Xiaowei Huang, Jingquan Luo, Lvzhou Li
Institution: Sun Yat-sen University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3215810853