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We call the set X a free basis of the group F, and frequently denote the group F by F x1 , x2 , . . , xn , . 9 is unique. ) In other words, a group is free provided it has a set of generators such that the only relations among those generators are those required for F to be a group. For example, • xi x−1 i = 1 is an allowed relation. • xi x j = x j xi is not an allowed relation for i = j. • x3i = 1 is not an allowed relation. g. group with a ﬁnite set of n generators {g1 , g2 , . , gn }, and let F = F x1 , x2 , .

20 1. a. a vintage Shor algorithm. This algorithm finds the entire hidden quotient group Hϕ . 7. WANDERING SHOR ALGORITHMS 21 plexity13 O n2 (lg N)3 (lg lg N)n+1 , where n is the rank of the free abelian group A. This can be readily deduced from the abbreviated flowchart given in Fig. 6. 6: Abbreviated flowchart for the first wandering Shor algorithm. , log2 . 1. 8 Continuous (variable) Shor algorithms In [27] and in [29], the algorithmic primitives found in above sections of this paper were used to create a class of algorithms called continuous Shor algorithms.

Perhaps some reader of this paper will be able to answer this question. Acknowledgement This work is partially supported by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-01-2-0522. S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. This work is REFERENCES 43 also partially supported by the Institute for Scientific Interchange (ISI), Torino, the National Institute of Standards and Technology (NIST), the Mathematical Sciences Research Institute (MSRI), the Isaac Newton Institute for Mathematical Sciences, and the L-O-O-P fund.