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B... B... & 38 ..... ... " " ... " ... "   "  ... " ...  " .. X ..... bXX X b XX XX b B B... b ........ b b . . b . .....  In the left case, all vectors point in different directions, and they tend to cancel each other. This will cause destructive interference, which will cause the amplitude of such k s to be small. In the right case, all vectors point almost to the same direction. In this case there will be constructive interference of all the vectors. This happens when e2πikr/Q is close to one, or when kr mod Q is close to zero.

It is left to show how the above computation can be made unitary. The idea is that it is not necessary to measure each set of qubits, in order to count the number of 1 s. Instead of measuring these bits, we will apply a unitary transformation that counts the portion of 1’s out of m and writes this portion down on a counting register. If we denote by w(i) the number of 1 s in a string i, or the weight of the string, then this transformation will be: |i |0 −→ |i |w(i)/m . (32) The resulting state will look something like: |Ψ ⊗ pw (i)(1 − p)m−w(i) |i |w(i) (33) i with perhaps extra phases.

We start with |α and rotate by 2θ 21 times. The angle between our vector and |α is η/ . We can now project on |α (by rotating |α to |0 and projecting on |0 ). The result is distributed like a coin flip with bias cos2(η/ ). We can repeat this experiment poly( 1 ) number of times. This will allow us to estimate the bias cos2 (η/ ) and from it |η|/ , up to a 1/4, with exponentially small error probability. Thus we can estimate |η| up to /4 in O( 1 ) time. Estimating the mean to a precision . 5], where N is assumed to be very large.

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