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Unformatted text preview: Last revised 3/30/06 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481681, CS 483; Spring, 2005 c 2006, N. David Mermin IV. Searching with a Quantum Computer Suppose you know that exactly one nbit integer satisfies a certain condition, and suppose you have a blackboxed subroutine that acts on the N = 2 n different nbit integers, outputting 1 (“true”) if the integer is the special one and 0 (“false”) otherwise. In the absence of any other information, to find the special integer you can do no better with a classical computer than to apply repeatedly the subroutine to different random numbers until you hit on the special one. If you apply it to M different integers the probability of your finding the special number is M/N . You must test 1 2 N different integers to have a 50% chance of success. If, however, you have a quantum computer with a subroutine that performs such a test, then for large (but not astronomically large) N , you can find the special integer with a probability that is very close to 1, by a method that requires you to call the subroutine only ( π/ 4) √ N times. This very general capability of quantum computers was discovered by Lov Grover, and goes under the name of Grover’s search algorithm . Shor’s period finding algorithm, Grover’s search algorithm, along with their various modifications and extensions, constitute the two masterpieces of quantum computational software. One can think of the blackboxed subroutine in various ways. It could perform a mathematical calculation to determine whether the input integer is the special one. Here is an example: If an odd number p can be expressed as the sum of two squares, m 2 + n 2 , then since one of m or n must be even and the other odd, p must be of the form 4 k + 1. It is a fairly elementary theorem of number theory that if p is a prime number of the form 4 k + 1 then it can always be expressed as the sum of two squares in exactly one way. (Thus 5 = 4+1, 13 = 9+4, 17 = 16+1, 29 = 25 + 4, 37 = 36+1, 41 = 25+16, etc.) Given any such prime p , the simpleminded way to find the two squares is to take randomly selected integers x with 1 ≤ x ≤ N , with N the largest integer less than p p/ 2, until you find the one for which p p x 2 is an integer a . If p is of the order of a trillion, then following the simpleminded procedure you would have to calculate p p x 2 for nearly a million x to have a better than even chance of succeeding. But using Grover’s procedure with an appropriately programmed quantum computer you could succeed with a probability of success extremely close to 1 with by calling the quantum subroutine that evaluated p p x 2 fewer than a thousand times....
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This note was uploaded on 02/01/2008 for the course CS 483 taught by Professor Ginsparg during the Spring '08 term at Cornell.
 Spring '08
 Ginsparg

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