Grover iteration

1. Apr 19, 2008

neu

1. The problem statement, all variables and given/known data
One of the operators used in the Grover iteration is:

$$\hat{O_{\psi} }= 2 \mid \psi \rangle \langle \psi \mid - I$$

where $$\mid \psi \rangle = \frac{1}{\sqrt{N}} \Sigma^{N-1}_{x=0} \mid x \rangle$$

Show that the operator:

$$\hat{O_{0} }= 2 \mid 000...0 \rangle \langle 000...0 \mid - I$$

acting on a register of $$log_{2} N$$ qubits, the operator $$\hat{O_{\psi} }$$ can be realised with the use of hadamard operators

3. The attempt at a solution

Now I know that the answer is :

$$\hat{O_{\psi} }=\hat{H}^{\otimes N}\hat{O_{0} }\hat{H}^{\otimes N}$$

and i tried to evaluate this explicity. for example, in the case N =1

$$\hat{H}\hat{O_{0} }\hat{H}= \left( \begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array} \right) - I$$

And the array is the sum

$$\mid 0 \rangle \langle 0\mid + \mid 0 \rangle \langle 1\mid + \mid 1 \rangle \langle 0\mid + \mid 1 \rangle \langle 1 \mid$$

but This is reverse engineering which I'm not happy with. Does anybody know how to derive the relationship $$\hat{O_{\psi} }=\hat{H}^{\otimes N}\hat{O_{0} }\hat{H}^{\otimes N}$$??