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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 15E / a / Solution

The eigenphase Hadamard-test probability is

p_{0} = \frac{1 + Re e ^{2 πi θ}}{2} = \frac{1 + c o s ( 2 π θ )}{2} = cos^{2} (π θ) .

Thus

cos (2 π θ) = 2 p_{0} - 1,

or

θ = \pm (2 π)^{- 1} arccos (2 p_{0} - 1) (mod 1)

. This test alone cannot distinguish

θ

from

1 - θ

.

Solved by gpt-5.6-sol high.

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