Products of finite order rotations and quantum gates universality

Abstract

We consider a product of two finite order quantum $SU(2)$-gates $U_1$, $U_2$ and ask when $U_1\cdot U_2$ has an infinite order. Using the fact that $SU(2)$ is a double cover of $SO(3)$ we actually study the product $O(\gamma,\vec{k}_{12})$ of two rotations $O(\phi,\vec{k}_1)\in SO(3)$ and $O(\phi,\vec{k}_2)\in SO(3)$ about axes $\vec{k}_1$, $\vec{k}_2\in \mathbb{R}^3$. In particular we focus on the case when $\vec{k}_1\cdot\vec{k}_2=0$, and $\phi_1=\phi=\phi_2$ are rational multiple of $\pi$ and show that $\gamma$ is not a rational multiple of $\pi$ unless $\phi\in\{\frac{k\pi}{2}:k\in\mathbb{Z}\}$. The proof presented in this paper boils down to finding all pairs $\gamma,\phi\in \{a\pi : a\in\mathbb{Q}\}$ that are solutions of $\cos\frac{\gamma}{2}=\cos^2\frac{\phi}{2}$.