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Shor Algorithm Anuj Dawar

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Finding the eigenvalue is the same as finding its phase

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Target register Index register to store the eigenvalue Unitary operator for which we calculate phase of eigenvalue using phase kickback we measure the phase We initialize to all states

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Formulas for the phase estimation algorithm This is the input to QFT ()

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Because e j is an eigenvallue of U

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We found phase Assume k an integer Number of bits

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Concluding, we can calculate phase

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Easy to multiply but difficult to factor big integers.

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Assume: We want to find the smallest r such that the above is true

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Greatest common denominator

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So now what remains is to be able to find period, but this is something well done with spectral transforms.

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Phase is 1/r

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Reductions Solve RSA Factor big integers Find period Estimate Phase Fourier Transform

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