# Homework Help: Hint for Quantum Computing Question Regarding QFT's and Eigenvalue Estimation

1. Feb 26, 2007

### ArjSiv

1. The problem statement, all variables and given/known data
I've pasted the actual question below:
http://www.zeta-psi.com/aj/qip5b.png [Broken]

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" [Broken]
http://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysis" [Broken]
$$M=\sum_i m_i P_i$$, I'm assuming $$P_i^n = P_i$$ where $$n >= 1$$ is.
Also, I'm assuming that $$M^2 \neq I$$, in that case I could have used the identity $$e^{iAx} = cos(x)I + isin(x)A$$ but I can't.

3. The attempt at a solution
I've tried letting $$|x> = \sum_{i=0}^{N-1} |i>$$, then from there, I can perform an inverse QFT on $$|x>U^x|\psi_j>$$ where $$|\psi_j>$$ is some eigenvalue of $$U$$ (and thus also $$U^x$$) to get me $$|\omega>$$ 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 $$M|\psi_j>$$, which I can then use to find $$M|\psi>$$ (since spectral decomposition lets me write this as the sum of eigenvectors).

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

I've also tried expanding $$e^{2\pi i M/N}$$ and I was able to (partially) factor out the $$M$$, but I wasn't sure where to go from there.

Last edited by a moderator: May 2, 2017