%0 Journal Article
%J Proc. RANDOM
%D 2010
%T Quantum property testing for bounded-degree graphs
%A Andris Ambainis
%A Andrew M. Childs
%A Yi-Kai Liu
%X 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.
%B Proc. RANDOM
%P 365-376
%8 2010/12/14
%G eng
%U http://arxiv.org/abs/1012.3174v3
%! Proceedings of RANDOM 2011
%R 10.1007/978-3-642-22935-0_31
%0 Journal Article
%D 2009
%T The quantum query complexity of certification
%A Andris Ambainis
%A Andrew M. Childs
%A François Le Gall
%A Seiichiro Tani
%X 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.
%8 2009/03/06
%G eng
%U http://arxiv.org/abs/0903.1291v2
%! Quantum Information and Computation 10