Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp probability estimates for Shor's order-finding algorithm   Let N be a (large positive integer, let b > 1 be an 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 0.7 whenever N exceeds 2^{11}-1 and r exceeds 39, and we
establish that 0.7736 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
(2/Pi) Si(4Pi), about .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 (2/Pi) Si(2^(q+2) Pi) (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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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat