By Louis Kauffman, Samuel J. Lomonaco

Книга arithmetic of Quantum Computation and Quantum know-how arithmetic of Quantum Computation and Quantum TechnologyКниги Физика Автор: Goong Chen, Louis Kauffman, Samuel J. Lomonaco Год издания: 2007 Формат: pdf Издат.:Chapman & Hall/CRC Страниц: 624 Размер: 3,6 ISBN: 1584888997 Язык: Английский0 (голосов: zero) Оценка:Research and improvement within the pioneering box of quantum computing contain with reference to each aspect of technology and engineering, together with the numerous components of arithmetic and physics. according to the enterprise knowing that arithmetic and physics are equivalent companions within the carrying on with research of quantum technological know-how, arithmetic of Quantum Computation and Quantum know-how explores the quick mathematical developments made during this box lately. Novel Viewpoints on quite a few facets of Quantum Computing and know-how Edited through a well-respected group of specialists, this quantity compiles contributions from experts throughout quite a few disciplines. It includes 4 major components, starting with subject matters in quantum computing that come with quantum algorithms and hidden subgroups, quantum seek, algorithmic complexity, and quantum simulation. the subsequent part covers quantum expertise, akin to mathematical instruments, quantum wave capabilities, superconducting quantum computing interference units (SQUIDs), and optical quantum computing. The part on quantum info offers with errors correction, cryptography, entanglement, and verbal exchange. the ultimate half explores topological quantum computation, knot thought, classification algebra, and common sense. The instruments you must take on the subsequent iteration of Quantum know-how This e-book allows either the development of a standard quantum language and the improvement of interdisciplinary quantum suggestions, for you to reduction efforts within the pursuit of the last word goal-a "real" scalable quantum computing device.

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For more details, we refer the reader to [31]. 13 Is Grover’s algorithm a QHS algorithm? In this section, our objective is to factor Grover’s algorithm into the QHS primitives developed in the previous sections of this paper. As a result, we will show that Grover’s algorithm is more closely related to Shor’s algorithm than one might at first expect. 13. IS GROVER’S ALGORITHM A QHS ALGORITHM? 37 will refer to as the Grover HSP. However, we will then show that the standard QHS algorithm for this HSP cannot possibly find a solution.

THE QUANTUM CIRCLE AND SHOR ALGORITHMS 27 Hence, ψ3 = ∑ |n dx e−2π inx |Φ (x) n∈Z 1/a = ∑ |n n∈Z = dx e−2π inx |Φ (x) δn=0 mod a 0 ⎛ 1/a ∑| a dx e−2π inx |Φ (x) ⎠ = ⎝ ∈Z ⎞ ∑| a 0 ∈Z |Ω ( a) . Step 4 Measurement of ψ3 = ∑ | a |Ω ( a) ∈HZ ⊗HC ∈Z with respect to the observable ∑ n |n n| n∈Z to produce a random eigenvalue a. 8 The above quantum circle algorithm can be extended to a quantum algorithm which ﬁnds the hidden period α of a function Φ : R/Z −→ C, when α is an arbitrary real number mod 1.

We assume that the hidden subgroup K is normal and of finite index in F. ) • Choose a finite group probe G with presentation (x1 , x2 , . . , xn : r1 , r2 , . . , rm )ν , where the subscript ν denotes the epimorphism ν : F −→ G induced by the map x j −→ x jCons r2 , . . , rm . 1. QHS ALGORITHMS 36 • Choose a minimal 2-sided Schreier transversal τ : G −→ F of the epimorphism ν : F −→ G. • Finally, construct the push ϕ = Push (ϕ ) = ϕ ◦ τ : G −→ S. Our generalized Shor algorithm for the free group F consists of the following steps: Shor’s Algorithm Generalized to Free Groups Step 1 Call QR AND(ϕ ) to produce a word s j in F close to a word s j lying in ϕ −1 ϕ (1).