Lecture Notes 3.D

2n z 22 n x 2 2 n 15 measurement consider the rst n

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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, | ⌦n ⌦ I )| 3 i = (H 2i = XX 1 ( ) x · z + f ( x ) | zi | i . 2n 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 . P 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 be ±1. 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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