Quantum Circuit for Calculating Mobius-like Transforms Via Grover-like Algorithm

In this paper, we give quantum circuits for calculating two closely related linear transforms that we refer to jointly as Mobius-like transforms. The first is the Mobius transform of a function $f^{-}(S^-)\in \mathbb{C}$, where $S^-\subset \{0,1,\ldots,n-1\}$. The second is a marginal of a probability distribution $P(y^n)$, where $y^n\in Bool^n$. Known classical algorithms for calculating these Mobius-like transforms take ${\cal O}(2^n)$ steps. Our quantum algorithm is based on a Grover-like algorithm and it takes ${\cal O}(\sqrt{2^n})$ steps.