@article {1243,
title = {Quantum property testing for bounded-degree graphs},
journal = {Proc. RANDOM},
year = {2010},
month = {2010/12/14},
pages = {365-376},
abstract = { 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.
},
doi = {10.1007/978-3-642-22935-0_31},
url = {http://arxiv.org/abs/1012.3174v3},
author = {Andris Ambainis and Andrew M. Childs and Yi-Kai Liu}
}
@article {1256,
title = {The quantum query complexity of certification},
year = {2009},
month = {2009/03/06},
abstract = { 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.
},
url = {http://arxiv.org/abs/0903.1291v2},
author = {Andris Ambainis and Andrew M. Childs and Fran{\c c}ois Le Gall and Seiichiro Tani}
}