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Hint for Quantum Computing Question Regarding QFT's and Eigenvalue Estimation

  1. Feb 26, 2007 #1
    1. The problem statement, all variables and given/known data
    I've pasted the actual question below:

    I don't think there are many quantum computing specific things here other than the circuit (which I can derive easily if I can figure out the algorithm)

    2. Relevant equations

    The Quantum Fourier Transform: http://en.wikipedia.org/wiki/Quantum_fourier_transform
    Helpful Identities:
    [tex]M=\sum_i m_i P_i[/tex], I'm assuming [tex]P_i^n = P_i[/tex] where [tex]n >= 1[/tex] is.
    Also, I'm assuming that [tex]M^2 \neq I[/tex], in that case I could have used the identity [tex]e^{iAx} = cos(x)I + isin(x)A[/tex] but I can't.

    3. The attempt at a solution
    I've tried letting [tex]|x> = \sum_{i=0}^{N-1} |i>[/tex], then from there, I can perform an inverse QFT on [tex]|x>U^x|\psi_j>[/tex] where [tex]|\psi_j>[/tex] is some eigenvalue of [tex]U[/tex] (and thus also [tex]U^x[/tex]) to get me [tex]|\omega>[/tex] which I could use in replacement of M in the definition of U.

    Assuming I'm not making a trivial mistake, I'm assuming the observable itself I want to find is [tex]M|\psi_j>[/tex], which I can then use to find [tex]M|\psi>[/tex] (since spectral decomposition lets me write this as the sum of eigenvectors).

    I think the key somehow revolves around writing the eigenvalues of [tex]M|\psi_j>[/tex] in terms of the eigenvalues of U for each [tex]|\psi>[/tex], assuming that they even have the same eigenvector bases.

    I've also tried expanding [tex]e^{2\pi i M/N}[/tex] and I was able to (partially) factor out the [tex]M[/tex], but I wasn't sure where to go from there.
    Last edited: Feb 26, 2007
  2. jcsd
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