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 Γ, how low can the complexity be for any boolean function invariant under Γ? 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 nvertex Scorpion graph property has quantum query complexity ˜Θ(n1/2 ), which implies that the minimum quantum complexity for graph properties is strictly less than that for monotone graph properties (known to be Ω(n2/3 )). A directed graph property, SINK, is also shown to have the ˜Θ(n1/2 ) quantum query complexity. Furthermore, we give an N-ary circular function which has the quantum query complexity ˜Θ(N1/4 ). Finally, we show that for any permutation group Γ, as long as Γ is transitive, the quantum query complexity of any function invariant to Γ is at least Ω(N1/4 ), whi...

• Type: Conference
• Year: 2004
• Where: COCO
• Authors: Xiaoming Sun, Andrew Chi-Chih Yao, Shengyu Zhang
• Keywords: COCO 2004, Graph Properties, Permutation Group Γ, Quantum Query Complexity