lect notes on quantum computation(lect4)

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Last revised 3/30/06 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2005 c 2006, N. David Mermin IV. Searching with a Quantum Computer Suppose you know that exactly one n -bit integer satisfies a certain condition, and suppose you have a black-boxed subroutine that acts on the N = 2 n different n -bit 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 black-boxed 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 simple-minded 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 simple-minded 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. Mathematically well-informed friends tell me that this particular example admits of much more efficient ways to proceed than random testing, but the quantum algorithm to be 1
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described below enables even mathematical ignoramuses like me, equipped with a quantum computer, to do better than random testing by a factor of 1 / N . And Grover’s algorithm
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