01146nas a2200145 4500008004100000245005500041210005400096260001500150300001200165520072600177100002100903700002300924700001600947856003700963 2010 eng d00aQuantum property testing for bounded-degree graphs0 aQuantum property testing for boundeddegree graphs c2010/12/14 a365-3763 a We study quantum algorithms for testing bipartiteness and expansion of
bounded-degree graphs. We give quantum algorithms that solve these problems in
time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing
expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus
ruling out the possibility of an exponential quantum speedup. Our quantum
algorithms follow from a combination of classical property testing techniques
due to Goldreich and Ron, derandomization, and the quantum algorithm for
element distinctness. The quantum lower bound is obtained by the polynomial
method, using novel algebraic techniques and combinatorial analysis to
accommodate the graph structure.
1 aAmbainis, Andris1 aChilds, Andrew, M.1 aLiu, Yi-Kai uhttp://arxiv.org/abs/1012.3174v300937nas a2200145 4500008004100000245005000041210004600091260001500137520051500152100002100667700002300688700002300711700002000734856003700754 2009 eng d00aThe quantum query complexity of certification0 aquantum query complexity of certification c2009/03/063 a We study the quantum query complexity of finding a certificate for a
d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show
that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and
Theta(d^{k/2}) for 1-certificates. In particular, this shows that the
zero-error quantum query complexity of evaluating such formulas is
O(d^{(k+1)/2}) (again neglecting a logarithmic factor). Our lower bound relies
on the fact that the quantum adversary method obeys a direct sum theorem.
1 aAmbainis, Andris1 aChilds, Andrew, M.1 aLe Gall, François1 aTani, Seiichiro uhttp://arxiv.org/abs/0903.1291v2