Graph properties and circular functions: how low can quantum query complexity go?

In decision tree models, considerable attention has been paid on the effect of symmetry on computational complexity. That is, for a permutation group /spl Gamma/, how low can the complexity be for any Boolean function invariant under /spl Gamma/? In this paper, we investigate this question for quantum decision trees for graph properties, directed graph properties, and circular functions. In particular, we prove that the n-vertex Scorpion graph property has quantum query complexity /spl Theta//sup /spl tilde// (n/sup 1/2/), which implies that the minimum quantum complexity for graph properties is strictly less than that for monotone graph properties (known to be /spl Omega/(n/sup 2/3/)). A directed graph property, SINK, is also shown to have the /spl Theta//sup /spl tilde//(n/sup 1/2/) quantum query complexity. Furthermore, we give an N-ary circular function which has the quantum query complexity /spl Theta/ /sup /spl tilde//(N/sup 1/4/). Finally, we show that for any permutation group /spl Gamma/, as long as /spl Gamma/ is transitive, the quantum query complexity of any function invariant to /spl Gamma/ is at least /spl Omega/(N/sup 1/4/), which implies that our examples are (almost) the best ones in the sense of pinning down the complexity for the corresponding permutation group.