Quantum Circuit for Calculating Symmetrized Functions Via Grover-like Algorithm

Abstract

In this paper, we give a quantum circuit that calculates symmetrized functions. Our algorithm applies the original Grover's algorithm or a variant thereof such as AFGA (adaptive fixed point Grover's algorithm). Our algorithm uses AFGA in conjunction with two new techniques we call "targeting two hypotheses" and "blind targeting". Suppose AFGA drives the starting state $|s\rangle$ to the target state $|t\rangle$. When targeting two hypotheses, $|t\rangle$ is a superposition $a_0|0\rangle + a_1|1\rangle$ of two orthonormal states or hypotheses $|0\rangle$ and $|1\rangle$. When targeting blindly, the value of $\langle t| s\rangle$ is not known a priori.