  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.
