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Unformatted text preview: March 2, 2006 Physics 681-481; CS 483: Assignment #4 (please hand in after the lecture, Thursday, March 16th) I. Probabilities for solving Simons problem. As described on pages 16-18 of Chapter 2, to estimate how many times a quantum computer has to invoke the subroutine U f to solve Simons problem, we must answer a purely mathematical question. We have an n-dimensional space of vectors whose com- ponents are either 0 or 1, on which vector addition and inner products are both carried out with modulo 2 arithmetic. We are interested in the ( n- 1)-dimensional subspace of vectors orthogonal to a given vector a . We have a quantum computer program that gives us a random vector y in that subspace. If we run the program n + x times, what is the probability q that n- 1 of the vectors y will be linearly independent? I argue in Chapter 2 that q = 1- 1 2 2+ x 1- 1 2 3+ x 1- 1 2 n + x . (1) Consider the case n = 3, x = 1, and a = 111. There are 4 different y s (including y = 0) that satisfy...
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