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Unformatted text preview: t is signiﬁcant.)
Remember this formula! D. QUANTUM ALGORITHMS 137 ¶14. Combining this and the result in ¶10,
⌦ I )|
3 i = (H 2i = XX 1
( ) x · z + f ( x ) | zi | i .
z 22 n x 2 2 n ¶15. Measurement: Consider the ﬁrst n qubits and the amplitude of one
particular basis state, z = |0i⌦n .
Its amplitude is x22n 21 ( )f (x) .
n ¶16. Constant function: If the function is constant, then all the exponents
of 1 will be the same (either all 0 or all 1), and so the amplitude will
Therefore all the other amplitudes are 0 and any measurement must
yield 0 for all the bits (since only |0i⌦n has nonzero amplitude).
¶17. Balanced function: If the function is not constant then (ex hypothesi)
it is balanced.
But more speciﬁcally, if it is balanced, then there must be an equal
number of +1 and 1 contributions to the amplitude of |0i⌦n , so its
amplitude is 0.
Therefore, when we measure the state, at least one qubit must be
nonzero (since the all-0s state has amplitude 0).
¶18. Good and bad news: The good news is that with one quantum
function evaluation we have got a result that would require between 2
and O(2n 1 ) classical function evaluations (exponential speedup).
The bad news is that the algorithm has no known applications!
¶19. Even if it were useful, the problem could be solved probabilistically on
a classical computer with only a few evaluations of f .
¶20. However, it illustrates principles of quantum computing that can be
used in more useful algorithms....
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- Fall '13