ua nt - p h / 06 07 14 8 v 2 4 A ug 2 00 6 Sharp probability estimates for Shor ’ s order - finding algorithm
نویسنده
چکیده
Abstract: Let N be a positive integer, let b < N be a positive integer relatively prime to N , and let r be the order of b modulo N . Finally, let QC be a quantum computer whose input register has the size specified in Shor’s original description of his order-finding algorithm. We prove that when Shor’s algorithm is implemented on QC, then the probability P of obtaining a (nontrivial) divisor of r exceeds .7 whenever N ≥ 211 and r ≥ 40, and we establish that .7736 is an asymptotic lower bound for P . When N is not a power of an odd prime, Gerjuoy has shown that P exceeds 90 percent for N and r sufficiently large. We give easily checked conditions on N and r for this 90 percent threshold to hold, and we establish an asymptotic lower bound for P of 2Si(4π)/π ≈ .9499 in this situation. More generally, for any nonnegative integer q, we show that when QC(q) is a quantum computer whose input register has q more qubits than does QC, and Shor’s algorithm is run on QC(q), then an asymptotic lower bound on P is 2Si(2q+2π)/π (if N is not a power of an odd prime). Our arguments are elementary and our lower bounds on P are carefully justified.
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