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N − 1 . Grover’s oracle is essentially given by the unitary transformation I j : H −→ |0 |j H −→ (−1) f ( j) | j , where I j = I − 2 j0 j0 is inversion in the hyperplane orthogonal to | j . |0 Let W denote the Hadamard transformation on the Hilbert space H. Then Grover’s algorithm is as follows: Grover’s Algorithm Step 0 (Initialization) |ψ 1 N−1 ←− W |0 = √ ∑ | j N j=0 k ←− 0 . √ Step 1 Loop until k ≈ π N/4 |ψ k ←− Q |ψ = −W I|0 W I j |ψ |0 ←− k + 1 . 1. QHS ALGORITHMS 38 Step 2 Measure |ψ with respect to the standard basis |0 , |1 , |2 , .

Before we can describe the above results, we need to review a number of definitions. 7 [Universal Deﬁnition] A group F is said to be free of ﬁnite rank n if there exists a ﬁnite set of n generators X = x1 , x2 , . . , xn such that, for every group G and for every map f : X −→ G of the set X into the group G, the map f extends to a morphism f : F −→ G. 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.

Actually, the papers [27], [29] give in succession three such continuous Shor algorithms, each successively more general than the previous. For the first algorithm, we assume that the unknown hidden period P is an integer. 4 of this paper. In Step 5 of QR AND(ϕ ), the observable ∞ A= dy −∞ Qy |y y| Q is measured, where Q is an integer chosen so that Q ≥ 2P2 . It then follows that the output of this algorithm is a rational m/Q which is a convergent of the continued fraction expansion of a rational of the form n/P.