Lower bounds on quantum query complexity for read-once formulas with XOR and MUX operators

We introduce a complexity measure r for the class ℱ of read-once formulas over the basis {AND,OR,NOT,XOR, MUX} and show that for any Boolean formula F in the class ℱ r(F) is a lower bound on the quantum query complexity of the Boolean function that F represents. We also show that for any Boolean function f represented by a formula in ℱ the deterministic query complexity of f is only quadratically larger than the quantum query complexity of f . Thus, the paper gives further evidence for the conjecture that there is an only quadratic gap for all functions.