- 220

- 1

**1. Homework Statement**

One of the operators used in the Grover iteration is:

[tex] \hat{O_{\psi} }= 2 \mid \psi \rangle \langle \psi \mid - I[/tex]

where [tex] \mid \psi \rangle = \frac{1}{\sqrt{N}} \Sigma^{N-1}_{x=0} \mid x \rangle [/tex]

Show that the operator:

[tex] \hat{O_{0} }= 2 \mid 000...0 \rangle \langle 000...0 \mid - I[/tex]

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

**3. The Attempt at a Solution**

Now I know that the answer is :

[tex] \hat{O_{\psi} }=\hat{H}^{\otimes N}\hat{O_{0} }\hat{H}^{\otimes N}[/tex]

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

[tex]\hat{H}\hat{O_{0} }\hat{H}=

\left( \begin{array}{c c}

1 & 1 \\

1 & 1

\end{array} \right) - I

[/tex]

And the array is the sum

[tex] \mid 0 \rangle \langle 0\mid + \mid 0 \rangle \langle 1\mid + \mid 1 \rangle \langle 0\mid + \mid 1 \rangle \langle 1 \mid[/tex]

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