Sharp probability estimates for Shor's order-finding algorithm

Abstract: Let N be a (large) positive integer, let b be an integer satisfying 1 < b < N that is 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.