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Grover iteration

  • Thread starter neu
  • Start date
neu
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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]??
 

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