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 $O(\varphi ,{\vec{k}}_1)\in SO(3)$ and $O(\varphi ,{\vec{k}}_2)\in SO(3)$ 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 ${\varphi }_1=\varphi ={\varphi }_2$ are rational multiple of $\pi$ and show that $\gamma$ is not a rational multiple of $\pi$ unless $\varphi \in \{\frac{k\pi }{2}:k\in \mathbb{Z}\}$. The proof presented in this paper boils down to finding all pairs $\gamma ,\varphi \in \{a\pi :a\in \mathbb{Q}\}$ that are solutions of $\cos \frac{\gamma }{2}={\cos }^2\frac{\varphi }{2}$.

12 pages

References in corpus (4)

• Silica-on-Silicon Waveguide Quantum Circuits
• Universality of single qudit gates
• Classical simulation of fermionic linear optics augmented with noisy ancillas
• Criteria for universality of quantum gates