By Arthur O. Pittenger (auth.)

In 1994 Peter Shor [65] released a factoring set of rules for a quantum machine that reveals the major elements of a composite integer N extra successfully than is feasible with the recognized algorithms for a classical com puter. because the hassle of the factoring challenge is essential for the se curity of a public key encryption method, curiosity (and investment) in quan tum computing and quantum computation all at once blossomed. Quan tum computing had arrived. The examine of the function of quantum mechanics within the thought of computa tion turns out to have all started within the early Eighties with the guides of Paul Benioff [6]' [7] who thought of a quantum mechanical version of pcs and the computation strategy. A similar query used to be mentioned presently thereafter by means of Richard Feynman [35] who begun from a special perspec tive through asking what sort of computing device will be used to simulate physics. His research led him to the idea that with an appropriate category of "quantum machines" you may imitate any quantum system.

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**Extra resources for An Introduction to Quantum Computing Algorithms**

**Example text**

Hence Λf is spanned over Z by the 2g periods γj , f = aj x + bj yi. Let Λ be the Z-span in Z2 of the 2g pairs (aj , bj ), and let (λ1 , µ1 ), (λ2 , µ2 ) be a 34 II. MODULAR SYMBOL ALGORITHMS Z-basis for Λ. 2) ωj = λj x + µj yi (j = 1, 2). Thus ω1 and ω2 form a Z-basis for Λf . We may compute (λ1 , µ1 ) and (λ2 , µ2 ) from v + and v − using the Euclidean algorithm in Z. In fact it is easy to see that there are only two possibilities, since v ± are determined within the subspace they generate by being the +1 and −1 eigenvectors for an involution.

5) y= √ L(f ⊗ χ, 1) P (l, f ) = . l − m (l, f ) im− (l, f ) Assuming that N is not a perfect square, we find the smallest primes l + ≡ 1 (mod 4) and l− ≡ 3 (mod 4) (not dividing N ) such that m+ = m+ (l+ , f ) and m− = m− (l− , f ) are nonzero. A necessary (but not sufficient) condition for this to be true is that for the associated quadratic characters, χ1 (−N ) = χ2 (−N ) = −εN ; for if χ(−N ) = εN then the sign of the functional equation for L(f ⊗ χ, s) is −1, and hence L(f ⊗ χ, 1) = 0. Suitable primes always exist, provided that N is not a perfect square, by a theorem of Murty and Murty (see [44]).

2) as before to obtain the periods ω 1 and ω2 . If N is a square, however, then χ(−N ) = χ(−1) for all primes l not dividing 2N ; hence we will only be able to find the real period this way if εN = −1, and only the imaginary period if εN = +1. Rather than seek a way round this difficulty we always use the “direct” method to compute the periods when N is square. 3) we clearly want to choose l as small as possible. 3) for L(f ⊗ χ, 1) at a certain point n = nmax . In practice we may use this to estimate the number of eigenvalues ap needed to obtain the desired accuracy.