Quantum speedup and limitations on matroid property problems

Matroid theory has been developed to be a mature branch of mathematics and has extensive applications in combinatorial optimization, algorithm design and so on. On the other hand, quantum computing has attracted much attention and has been shown to surpass classical computing on solving some computational problems. Surprisingly, crossover studies of the two fields seem to be missing in the literature. 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 \(\Omega \left( {\sqrt {\left( {\matrix{n \cr {\left\lfloor {n/2} \right\rfloor } \cr } } \right)} } \right)\) on the query complexity of 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 \(O\left( {\log n\sqrt {\left( {\matrix{n \cr {\left\lfloor {n/2} \right\rfloor } \cr } } \right)} } \right)\). In addition, for the paving matroid decision problem, a lower bound \(\Omega \left( {\sqrt {\left( {\matrix{n \cr {\left\lfloor {n/2} \right\rfloor } \cr } } \right)/n} } \right)\) on the query complexity is obtained, and an \(O\left( {\sqrt {\left( {\matrix{n \cr {\left\lfloor {n/2} \right\rfloor } \cr } } \right)} } \right)\) quantum algorithm is presented.