Mathematics of Quantum Computation and Quantum Technology by Louis Kauffman, Samuel J. Lomonaco

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 near to each side of technological know-how 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 examine of quantum technological know-how, arithmetic of Quantum Computation and Quantum know-how explores the speedy mathematical developments made during this box in recent times. Novel Viewpoints on various facets of Quantum Computing and expertise Edited via a well-respected workforce of specialists, this quantity compiles contributions from experts throughout a variety of disciplines. It includes 4 major elements, starting with themes in quantum computing that come with quantum algorithms and hidden subgroups, quantum seek, algorithmic complexity, and quantum simulation. the following part covers quantum know-how, similar to mathematical instruments, quantum wave services, superconducting quantum computing interference units (SQUIDs), and optical quantum computing. The part on quantum info bargains with blunders correction, cryptography, entanglement, and conversation. the ultimate half explores topological quantum computation, knot thought, type algebra, and common sense. The instruments you want to take on the subsequent iteration of Quantum expertise This booklet allows either the development of a standard quantum language and the improvement of interdisciplinary quantum recommendations, with a purpose to relief efforts within the pursuit of the last word goal-a "real" scalable quantum computing device.

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QHS ALGORITHMS 28 Intuitively, just as in Shor’s algorithm, the circle group R/Z is approximated with the finite cyclic group ZQ , where the group ZQ is identified with the additive group Q−1 0 1 , , . . , mod 1 , Q Q Q and where the hidden subgroup ZP is identified with the additive group P−1 0 1 , , . . , P P P mod 1 , with P = a2 . This is a physically implementable quantum algorithm. In a certain sense, it is actually faster than Shor’s algorithm because the last step of Shor’s algorithm uses the standard continued fraction algorithm to determine the unknown period.

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. The above quantum algorithm is then extended to a second quantum algorithm that finds the hidden period P of functions ϕ : R −→ R, where the unknown period P is a rational. Finally, the second algorithm is extended to a third algorithm which finds the hidden period P of functions ϕ : R −→ R, when P is an arbitrary real number.

Xn be a free group with free basis −1 x1 , x2 , . . , xn . Then a word is a ﬁnite string of the symbols x1 , x−1 1 , x2 , x2 , . . , xn , x−1 n . A reduced word is a word in which there is no substring of the form x j x−1 or x−1 j j x j . Two words are said to be equivalent if one can be transformed into the other by applying a ﬁnite number of −1 substring insertions or deletions of the form x j x−1 j or x j x j . We denote an arbitrary word w by w = a1 a2 · · · a , where each a j = x±1 k .

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