Quantum Speedup and Limitations on Matroid Property Problems

TL;DR

This paper explores quantum algorithms for matroid property problems, demonstrating quadratic speedups and establishing bounds on query complexity, with some problems achieving optimal or near-optimal quantum algorithms.

Contribution

It provides new quantum algorithms and lower bounds for various matroid property problems, advancing understanding of quantum speedups in combinatorial structures.

Findings

Quadratic quantum speedup for girth and circuit number calculations

Optimal quantum algorithm for uniform matroid decision problem

Lower bounds on query complexity for paving matroid decision problem

Abstract

This paper initiates the study of quantum algorithms for matroid property problems. It is shown 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 for 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, an asymptotically optimal quantum algorithm is proposed which achieves the lower bound, and for the girth problem, an almost optimal quantum algorithm is given with query complexity $O(\log n\sqrt{\binom{n}{\lfloor n/2\rfloor}})$. In addition, for the paving matroid decision problem, a lower bound $\Omega(\sqrt{\binom{n}{\lfloor n/2\rfloor}/n})$ on the query complexity is obtained, and an $O(\sqrt{\binom{n}{\lfloor n/2\rfloor}})$ quantum algorithm is presented.