1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Grover iteration

  1. Apr 19, 2008 #1

    neu

    User Avatar

    1. The problem statement, all variables and given/known data
    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]??
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted