# Factorizing rotation operator into finite products

1. Feb 21, 2014

### cattlecattle

I know that any unitary operator on the spin-1/2 Hilbert space can be identified as a fixed-axis rotation operator up to a global phase, i.e.,
$$U=e^{i\alpha}R_{\hat{n}}(\theta)=e^{i\alpha}e^{-i\theta\mathbf{\hat{n}}\cdot\mathbf{\sigma}/2\hbar}$$

And it's also a known result that any fixed-axis rotation operator can be factorized into 3 Euler rotations z-y-z
$$R_{\hat{n}}(\theta)=R_{\hat{z}}(\cdot)R_{\hat{y}}(\cdot)R_{\hat{z}}(\\cdot)$$

Now there is a more general claim that as long as $\mathbf{\hat{u}}$ and $\mathbf{\hat{v}}$ are not parallel, arbitrary $R_{\hat{n}}$ can be factorized into a finite series of alternate rotations around $\mathbf{\hat{u}}$ and $\mathbf{\hat{v}}$
i.e.,
$$R_{\hat{n}}(\theta)=R_{\hat{u}}(\beta_1)R_{\hat{v}}(\gamma_1)\cdots R_{\hat{u}}(\beta_k)R_{\hat{v}}(\gamma_k)$$
for some fixed k (which only depends on $\mathbf{\hat{u}}$ and $\mathbf{\hat{v}}$) and $\beta_i$ and $\gamma_i$

I attempted to prove this general claim but to no avail. Apparently this is a pretty important claim that's was used in Nielsen and Chuang's Quantum Computation and Quantum Information (in section 4.5 where they try to prove that H, S, pi/8 and CNOT can approximate any single-qubit gate to arbitrary accuracy). I've also searched online for this result, but didn't find any reference to it. Any pointers will be greatly appreciated