chapter4 - Last revised 4/7/05 LECTURE NOTES ON QUANTUM...

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Last revised 4/7/05 LECTURE NOTES ON QUANTUM COMPUTATION Cornell University, Physics 481-681, CS 483; Spring, 2005 c ± 2005, N. David Mermin IV. Searching with a Quantum Computer Suppose you know that exactly one integer between 1 and N satisfes a certain con- dition, and suppose you have a black-boxed subroutine that takes integers between one and N as input, and outputs 1 (“true”) iF the integer is the special one and 0 (“False”) otherwise. In the absence oF any Further inFormation, to fnd the special integer you can do noth- ing better with a classical computer than repeatedly applying the subroutine to di±erent random numbers less than or equal to N until you fnd the one that outputs 1. You must test 1 2 N di±erent integers to achieve a 50% chance oF fnding the special one. IF, however, you have a quantum computer and have a quantum subroutine that per- Forms the test, then with probability very close to 1 you can fnd the special integer by calling the subroutine a number oF times that is only proportional to N . (More precisely, it is proportional to N log N . The source oF the logarithmic Factor is described in Sec- tion B below.) 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-fnding algo- rithm, Grover’s search algorithm, and their various modifcations and extensions constitute the two great triumphs oF quantum computational soFtware design. One can think oF the black-box in various ways. It could perForm a mathematical calculation to determine whether the input integer is the special one. ²or example iF an odd number p can be expressed as a 2 + b 2 then since one oF a or b must be even and the other odd, p must be oF the Form 4 n + 1. It is a Fairly elementary theorem oF number theory that iF p is a prime number oF the Form 4 n + 1 then it can indeed be expressed as such a sum oF two squares and in only one way. 1 Given any such prime p , the most simple-minded way 2 to fnd 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 fnd 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 1 Thus 5 = 4+1, 13 = 9+4, 17 = 16+1, 29 = 25 + 4, 37 = 36+1, 41 = 25+16, etc. 2 Mathematically well-inFormed Friends tell me that this example admits oF much more e³cient ways to proceed than random testing, but the quantum algorithm to be described enables even mathematical ignoramuses like me to do better than random testing by a Factor oF 1 / N , iF they have access to a quantum computer. And Grover will provide this speed-up even on problems that stump the mathematically well-inFormed.
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chapter4 - Last revised 4/7/05 LECTURE NOTES ON QUANTUM...

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