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Mathematics of Quantum Computation [Kõva köide]

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Edited by (Pennsylvania State University, Pennsylvania, USA), Edited by (Texas A&M University, College Station, USA)
  • Formaat: Hardback, 448 pages, kõrgus x laius: 234x156 mm, kaal: 990 g, 28 Illustrations, black and white
  • Ilmumisaeg: 14-Feb-2002
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584882824
  • ISBN-13: 9781584882824
Teised raamatud teemal:
Mathematics of Quantum ComputationSuurem pilt
  • Formaat: Hardback, 448 pages, kõrgus x laius: 234x156 mm, kaal: 990 g, 28 Illustrations, black and white
  • Ilmumisaeg: 14-Feb-2002
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1584882824
  • ISBN-13: 9781584882824
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781584882824.html
Märksõnad:
Among the most exciting developments in science today is the design and construction of the quantum computer. Its realization will be the result of multidisciplinary efforts, but ultimately, it is mathematics that lies at the heart of theoretical quantum computer science.

Mathematics of Quantum Computation brings together leading computer scientists, mathematicians, and physicists to provide the first interdisciplinary but mathematically focused exploration of the field's foundations and state of the art. Each section of the book addresses an area of major research, and does so with introductory material that brings newcomers quickly up to speed. Chapters that are more advanced include recent developments not yet published in the open literature.

Information technology will inevitably enter into the realm of quantum mechanics, and, more than all the atomic, molecular, optical, and nanotechnology advances, it is the device-independent mathematics that is the foundation of quantum computer and information science. Mathematics of Quantum Computation offers the first up-to-date coverage that has the technical depth and breadth needed by those interested in the challenges being confronted at the frontiers of research.
Preface v
Quantum Entanglement 1(2)
Algebraic measures of entanglement
3(22)
Jean-Luc Brylinski
Introduction
3(1)
Rank of a tensor
4(4)
Tensors in (C2)2
8(1)
Tensor in (C2)3
8(10)
Tensors in (C2)4
18(7)
Kinematics of qubit pairs
25(52)
Berthold-Georg Englert
Nasser Metwally
Introduction
25(3)
Preliminaries
28(6)
Basic classification of states
34(2)
Projectors and subspaces
36(7)
Rank 1
36(3)
Rank 2
39(2)
Rank 3
41(2)
Rank 4
43(1)
Positivity and separability
43(4)
Lewenstein-Sanpera decompositions
47(13)
Basic properties of optimal LSDs
51(3)
Optimal LSDs of truly positive states
54(6)
Examples
60(12)
Self-transposed states
60(3)
Generalized Werner states
63(3)
States of rank 2
66(6)
Acknowledgments
72(5)
Invariants for multiple qubits: the case of 3 qubits
77(22)
David A. Meyer
Noland Wallach
Introduction
77(2)
Invariants for compact Lie groups
79(3)
The simplest cases
82(3)
The case of 3 qubits
85(3)
A basic set of invariants for 3 qubits
88(7)
Some implications for other representations
95(4)
Universality of Quantum Gates 99(18)
Universal quantum gates
101(16)
Jean-Luc Brylinski
Ranee Brylinski
Statements of main results
101(3)
Examples and relations to works of other authors
104(2)
Proof of theorem 4.1 (outline)
106(1)
First step: from universality to exact universality
107(1)
Second step: reduction to n = 2
108(1)
Fourth Step: analyzing the Lie algebra g
109(1)
Fifth Step: the normalizer of H
110(2)
Proof of theorem 4.2
112(1)
A variant of theorem 4.1
113(4)
Quantum Search Algorithms 117(52)
From coupled pendulums to quantum search
119(16)
Lov K. Grover
Anirvan M. Sengupta
Introduction
120(1)
Classical analogy
120(1)
N Coupled pendulums
121(4)
The algorithm
125(2)
Rules of the game
125(1)
Algorithm
126(1)
Towards quantum searching
127(1)
The quantum search algorithm
128(2)
Why does it take O(√N) cycles?
130(1)
Applications and extensions
131(4)
Counting
131(1)
Mechanical applications
132(1)
Quantum mechanical applications
133(2)
Generalization of Grover's algorithm to multiobject search in quantum computing, Part I: continuous time and discrete time
135(26)
Goong Chen
Stephen A. Fulling
Jeesen Chen
Introduction
135(2)
Continuous time quantum computing algorithm for multiobject search
137(10)
Discrete time case: straightforward generalization of Grover's algorithm to multiobject search
147(14)
Generalization of Grover's algorithm to multiobject search in quantum computing, Part II: general unitary transformations
161(8)
Goong Chen
Shunhua Sun
Introduction
162(1)
Multiobject search algorithm using a general unitary transformation
163(6)
Quantum Computational Complexity 169(52)
Counting complexity and quantum computation
171(50)
Stephen A. Fenner
Introduction
171(2)
Preliminaries
173(22)
Qubits, quantum gates, and quantum circuits
175(3)
Classical complexity
178(14)
Classical computations on a quantum circuit
192(2)
Relativizing quantum computation
194(1)
Equivalence of FQP and GapP
195(5)
Strengths of the quantum model
200(7)
Oracle results
202(5)
Limitations of the quantum model
207(7)
Conclusions
214(7)
Quantum Error-Correcting Codes 221(54)
Algorithmic aspects of quantum error-correcting codes
223(30)
Markus Grassl
Introduction
223(1)
General quantum error-correcting codes
224(9)
General errors
224(5)
Local errors
229(4)
Binary quantum codes
233(8)
Construction
233(6)
Example: binary Hamming code
239(2)
Additive quantum codes
241(9)
Construction
241(5)
Example: quantum Hamming code
246(4)
Conclusions
250(3)
Clifford codes
253(22)
Andreas Klappenecker
Martin Rotteler
Introduction
253(2)
Motivation
255(1)
Quantum error control codes
256(3)
Nice error bases
259(3)
Stabilizer codes
262(2)
Clifford codes
264(1)
Clifford codes that are stabilizer codes
265(4)
A remarkable error group
269(1)
A weird error group
269(1)
Conclusions
270(5)
Quantum Computing Algebraic and Geometric Structures 275(46)
Invariant Polynomial functions on k qudits
277(10)
Jean-Luc Brylinski
Ranee Brylinski
Introduction
277(2)
Polynomial invariants of tensor states
279(2)
The generalized determinant function
281(1)
Asymptotics as k → ∞
282(1)
Quartic invariants of k qudits
283(4)
Z2-systolic freedom and quantum codes
287(34)
Michael H. Freedman
David A. Meyer
Feng Luo
Preliminaries and statement of results
287(7)
Mapping torus constructions
294(7)
Verification of freedom and curvature estimates
301(7)
Quantum codes from Riemannian manifolds
308(13)
Quantum Teleportation 321(36)
Quantum teleportation
323(34)
Kishore T. Kapale
M. Suhail Zubairy
Introduction
323(3)
Teleportation of a two-state system
326(11)
The formal scheme
327(3)
Cavity QED implementation
330(7)
Discrete N-state quantum systems
337(3)
Entangled state teleportation
340(6)
Two-qubit entangled state
341(3)
N-qubit entangled state
344(2)
Continuous quantum variable states
346(5)
Nonlocal measurements
346(2)
Wigner functions
348(3)
Concluding remarks
351(6)
Quantum Secure Communication and Quantum Cryptography 357(46)
Communicating with qubit pairs
359(44)
Almut Beige
Berthold-Georg Englert
Christian Kurtsiefer
Harald Weinfurter
Introduction
360(1)
The mean king's problem
361(10)
The Vaidman-Aharonov-Albert puzzle
361(1)
The stranded physicist's solution
362(5)
The mean king's second challenge
367(2)
A different perspective
369(2)
BB84: cryptography with single qubits
371(6)
Description of the scheme
372(1)
Eavesdropping: minimizing the error probability
373(2)
Eavesdropping: maximizing the raw information
375(2)
Cryptography with qubit pairs
377(8)
Description of the scheme
377(3)
Eavesdropping: minimizing the error probability
380(4)
Eavesdropping: maximizing the raw information
384(1)
Idealized single-photon schemes
385(7)
BB84 scheme with two state pairs
385(3)
Qubit-pair scheme with four state pairs
388(4)
Direct communication with qubit pairs
392(6)
Description of the scheme
392(3)
Minimal error probability
395(3)
Acknowledgments
398(5)
Commentary on Quantum Computing 403(18)
Transgressing the boundaries of quantum computation: a contribution to the hermeneutics of the NMR paradigm
405(16)
Stephen A. Fulling
Review of NMR quantum computing
406(1)
Review of modular arithmetic
407(2)
A proposed ``quantum'' implementation
409(3)
Aftermath
412(9)
Index 421


Goong Chen, Ranee K. Brylinski