R_x(θ)R_y(θ)R_z(θ)

$R_x(\theta )=\exp (-iX\theta /2)=\left[\begin{array}{cc}\cos \left(\theta /2\right)& -i\sin \left(\theta /2\right)\\ -i\sin \left(\theta /2\right)& \cos \left(\theta /2\right)\end{array}\right]$

$R_y(\theta )=\exp (-iY\theta /2)=\left[\begin{array}{cc}\cos \left(\theta /2\right)& -\sin \left(\theta /2\right)\\ \sin \left(\theta /2\right)& \cos \left(\theta /2\right)\end{array}\right]$

$R_z(\theta )=\exp (-iZ\theta /2)=\left[\begin{array}{cc}\exp (-i\theta /2)& 0\\ 0& \exp \left(i\theta /2\right)\end{array}\right]$

としたとき、$U=R_x(\theta )R_y(\theta )R_z(\theta )$ について調べよう。

$U=\left[\begin{array}{cc}\cos \left(\theta /2\right)& -i\sin \left(\theta /2\right)\\ -i\sin \left(\theta /2\right)& \cos \left(\theta /2\right)\end{array}\right]\left[\begin{array}{cc}\cos \left(\theta /2\right)& -\sin \left(\theta /2\right)\\ \sin \left(\theta /2\right)& \cos \left(\theta /2\right)\end{array}\right]\left[\begin{array}{cc}\exp (-i\theta /2)& 0\\ 0& \exp \left(i\theta /2\right)\end{array}\right]$

$U=\left[\begin{array}{cc}\cos \left(\theta /2\right)& -i\sin \left(\theta /2\right)\\ -i\sin \left(\theta /2\right)& \cos \left(\theta /2\right)\end{array}\right]\left[\begin{array}{cc}\exp (-i\theta /2)\cos \left(\theta /2\right)& -\sin \left(\theta /2\right)\exp \left(i\theta /2\right)\\ \exp (-i\theta /2)\sin \left(\theta /2\right)& \cos \left(\theta /2\right)\exp \left(i\theta /2\right)\end{array}\right]$

$U=\left[\begin{array}{cc}({\cos }^2(\theta /2)-i{\sin }^2(\theta /2))\exp (-i\theta /2)& -(1+i)\sin \left(\theta /2\right)\cos \left(\theta /2\right)\exp \left(i\theta /2\right)\\ (1-i)\sin \left(\theta /2\right)\cos \left(\theta /2\right)\exp (-i\theta /2)& ({\cos }^2(\theta /2)+i{\sin }^2(\theta /2))\exp \left(i\theta /2\right)\end{array}\right]$

$c=\cos \left(\theta /2\right)$, $s=\sin \left(\theta /2\right)$ と置いて

$U=\left[\begin{array}{cc}(c^2-i{s}^2)(c-is)& -(1+i)sc(c+is)\\ (1-i)sc(c-is)& (c^2+i{s}^2)(c+is)\end{array}\right]=\left[\begin{array}{cc}{c}^3-s^3-i(c{s}^2+c^{2}s)& -s{c}^2+s^{2}c-i(s{c}^2+s^{2}c)\\ s{c}^2-s^{2}c-i(s{c}^2+s^{2}c)& c^3-s^3+i(c{s}^2+c^{2}s)\end{array}\right]$

$U=(c^3-s^3)I-isc(c+s)X-isc(c-s)Y-isc(s+c)Z$

えー、${(c^3-s^3)}^2+s^2{c}^2(2{(c+s)}^2+{(c-s)}^2)=1$ となるようで、上手くいってそうですね。

ということで、$U$ というのは、$(\cos \left(\theta /2\right)+\sin \left(\theta /2\right),\cos \left(\theta /2\right)-\sin \left(\theta /2\right),\cos \left(\theta /2\right)+\sin \left(\theta /2\right))$ 方向を軸として $\mathrm{\Theta }=2{\cos }^{-1}({\cos }^3(\theta /2)-{\sin }^3(\theta /2))$ だけ回すようなものとなる。この最後の $\mathrm{\Theta }$ は $0\le \theta \le \pi$ の範囲内で $\mathrm{\Theta }\approx \theta -0.03804\sin 2\theta$ としてよく近似できるようだ。