Lower Bounds on Quantum Query Complexity for Read-Once Formulas with XOR and MUX Operators

Abstract

We introduce a complexity measure r for the class $\mathcal{F}$ of read-once formulas over the basis {AND, OR, NOT, XOR, MUX} and show that for any Boolean formula F in the class $\mathcal{F}$, 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 $\mathcal{F}$, 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.