# Writing Grover's Iterator in different computational phase [duplicate]

This question is from Nielson and Chuang's Quantum Computation and Quantum Information: Here $$|\alpha\rangle$$ is given by: $$\frac{1}{\sqrt{N-M}}\sum_{x} "|x\rangle$$ where $$\sum_{x}" |x\rangle$$ is the sum of states which are not the solution.

and $$|\beta\rangle$$ is given by: $$\frac{1}{\sqrt{M}}\sum_{x} ' |x\rangle$$ where it represents all $$M$$ solutions.

$$\theta$$ is defined by: $$\cos\Big(\frac{\theta}{2}\Big) = \sqrt{\frac {N-M}{N}}$$

I understand everything mentioned above as well as the further geometric visualization. However, I am having trouble trying to come up with a justified way to solve this exercise. The only thing that I understand is that $$G$$ must be a unitary Matrix made up of trigonometric functions. How can I approach this problem?

You can rewrite $$|\psi\rangle$$ in terms of $$\alpha$$ and $$\beta$$ i.e. $$|\psi\rangle = \cos(\theta/2)|\alpha\rangle + \sin(\theta/2)|\beta\rangle$$ Also, G rotates the state vector by $$\theta$$ towards $$\beta$$. $$G|\psi\rangle = \cos(3\theta/2)|\alpha\rangle + \sin(3\theta/2)|\beta\rangle$$ $$\begin{bmatrix} x_{11} & x_{12}\\ x_{21} & x_{22}\\ \end{bmatrix} \begin{bmatrix} \cos(\theta/2)\\ \sin(\theta/2)\\ \end{bmatrix} = \begin{bmatrix} \cos(3\theta/2)\\ \sin(3\theta/2)\\ \end{bmatrix} = \begin{bmatrix} \cos(\theta)\cos(\theta/2)-\sin(\theta)\sin(\theta/2)\\ \sin(\theta)\cos(\theta/2)+\cos(\theta)\sin(\theta/2)\\ \end{bmatrix}$$ By comparison, you can find the value of $$x_{11},x_{12},x_{21}$$ and $$x_{22}$$, and hence, G which comes out to be $$\begin{bmatrix} \cos(\theta) && -\sin(\theta)\\ \sin(\theta) && \cos(\theta)\\ \end{bmatrix}$$