On the power of Ambainis lower bounds

The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb’s. We first use known Alb’s to derive (n1.5) lower bounds for BIPARTITENESS, BIPARTITENESS MATCHING and GRAPH MATCHING, in which the lower bound for BIPARTITENESS improves the previous (n) one.We then show that all the three known Ambainis lower bounds have a limitation √ N min{C0(f ), C1(f )}, where C0(f ) and C1(f ) are the 0and 1-certificate complexities, respectively. This implies that for many problems such asTRIANGLE, k-CLIQUE, BIPARTITENESS andBIPARTITE/GRAPHMATCHING which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis techniques. Another consequence is that all theAmbainis lower bounds are not tight. For total functions, this upper bound for Alb’s can be further improved to min{C0(f )C1(f ), √ N · CI(f )}, where CI(f ) is the size of max intersection of a 0and a 1-certificate set. Again this implies that Alb’s cannot improve the best known lower bound for some specific problems such as AND-OR TREE, whose precise quantum query complexity is still open. Finally, we generalize the three known Alb’s and give a new Alb style lower bound method, which may be easier to use for some problems. © 2005 Published by Elsevier B.V.