Odd orders in Shor’s factoring algorithm

Odd orders in Shor’s factoring algorithm Shor’s factoring algorithm (SFA) finds the prime factors of a number, $$N=p_1 p_2$$ N = p 1 p 2 , exponentially faster than the best known classical algorithm. Responsible for the speedup is a subroutine called the quantum order finding algorithm (QOFA) which calculates the order—the smallest integer, $$r$$ r , satisfying $$a^r \mod N =1$$ a r mod N = 1 , where $$a$$ a is a randomly chosen integer coprime to $$N$$ N (meaning their greatest common divisor is one, $$\gcd (a, N) =1$$ gcd ( a , N ) = 1 ). Given $$r$$ r , and with probability not $$<$$ < 1/2, the factors are given by $$p_1 = \gcd (a^{\frac{r}{2}} - 1, N)$$ p 1 = gcd ( a r 2 - 1 , N ) and $$p_2 = \gcd (a^{\frac{r}{2}} + 1, N)$$ p 2 = gcd ( a r 2 + 1 , N ) . For odd $$r$$ r , it is assumed that the factors cannot be found (since $$a^{\frac{r}{2}}$$ a r 2 is not generally integer) and the QOFA is relaunched with a different value of $$a$$ a . But a recent paper (Martin-Lopez et al. Nat. Photonics 6:773, 2012) noted that the factors can sometimes be found from odd orders if the coprime is square. This raises the question of improving SFA’s success probability by considering odd orders. We show that an improvement is possible, though it is small. We present two techniques for retrieving the order from apparently useless runs of the QOFA: not discarding odd orders; and looking out for new order finding relations in the case of failure. In terms of efficiency, using our techniques is equivalent to avoiding square coprimes and disregarding odd orders, which is simpler in practice. Even still, our techniques may be useful in the near future, while demonstrations are restricted to factoring small numbers. The most convincing demonstrations of the QOFA are those that return a non-power-of-two order, making odd orders that lead to the factors attractive to experimentalists. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Odd orders in Shor’s factoring algorithm

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-014-0910-z
Publisher site
See Article on Publisher Site

Abstract

Shor’s factoring algorithm (SFA) finds the prime factors of a number, $$N=p_1 p_2$$ N = p 1 p 2 , exponentially faster than the best known classical algorithm. Responsible for the speedup is a subroutine called the quantum order finding algorithm (QOFA) which calculates the order—the smallest integer, $$r$$ r , satisfying $$a^r \mod N =1$$ a r mod N = 1 , where $$a$$ a is a randomly chosen integer coprime to $$N$$ N (meaning their greatest common divisor is one, $$\gcd (a, N) =1$$ gcd ( a , N ) = 1 ). Given $$r$$ r , and with probability not $$<$$ < 1/2, the factors are given by $$p_1 = \gcd (a^{\frac{r}{2}} - 1, N)$$ p 1 = gcd ( a r 2 - 1 , N ) and $$p_2 = \gcd (a^{\frac{r}{2}} + 1, N)$$ p 2 = gcd ( a r 2 + 1 , N ) . For odd $$r$$ r , it is assumed that the factors cannot be found (since $$a^{\frac{r}{2}}$$ a r 2 is not generally integer) and the QOFA is relaunched with a different value of $$a$$ a . But a recent paper (Martin-Lopez et al. Nat. Photonics 6:773, 2012) noted that the factors can sometimes be found from odd orders if the coprime is square. This raises the question of improving SFA’s success probability by considering odd orders. We show that an improvement is possible, though it is small. We present two techniques for retrieving the order from apparently useless runs of the QOFA: not discarding odd orders; and looking out for new order finding relations in the case of failure. In terms of efficiency, using our techniques is equivalent to avoiding square coprimes and disregarding odd orders, which is simpler in practice. Even still, our techniques may be useful in the near future, while demonstrations are restricted to factoring small numbers. The most convincing demonstrations of the QOFA are those that return a non-power-of-two order, making odd orders that lead to the factors attractive to experimentalists.

Journal

Quantum Information ProcessingSpringer Journals

Published: Jan 10, 2015

References

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