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Mathematics of Quantum Computation





ISBN 9781584882824
Published February 14, 2002 by Chapman and Hall/CRC
448 Pages 28 B/W Illustrations

 
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Book Description

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.

Table of Contents

Preface
PART I: QUANTUM ENTANGLEMENT
ALGEBRAIC MEASURES OF ENTANGLEMENT, Jean-Luc Brylinski
Introduction
Rank of a Tensor
Tensors in (C 2)Ä2
Tensors in (C 2)Ä3
Tensors in (C 2)Ä4
KINEMATICS OF QUBIT PAIRS, Berthold-Geor Englert and Nasser Metwally
Introduction
Basic Classification of States
Projectors and Subspaces
Positivity and Separability
Lewenstein-Sanpera Decompositions
Examples
INVARIANTS FOR MULTIPLE QUBITS: The Case of 3 Qubits, David A. Meyer and Noland Wallach
Introduction
Invariants for Compact Lie Groups
The Simplest Cases
The Case of Three Qubits
A Basic Set of Invariants for Three Qubits
Some Implications for Other Representations
PART II: UNIVERSALITY OF QUANTUM GATES
UNIVERSAL QUANTUM GATES, Jean-Luc Brylinski and Ranee Brylinski
Statements of Main Results
Examples and Relations to Works of Other Authors
From Universality to Exact Universality
Analyzing the Lie Algebra g
Normalizer of H
PART III: QUANTUM SEARCH ALGORITHMS
FROM COUPLED PENDULUMS TO QUANTUM SEARCH Lov K. Grover and Anirvan M. Sengupta
Introduction
Classical Analogy
N Coupled Pendulums
The Algorithm
Towards Quantum Searching
The Quantum Search Algorithm
Why Does it Take O(vN) cycles?
Applications and Extensions
GENERALIZATION OF GROVER'S ALGORITHM TO MULTIOBJECT SEARCH IN QUANTUM COMPUTING, Part I: Continuous Time and Discrete Time, Goon Chen, Stephen A,. Fulling, and Jeesen Chen
Introduction
Analog Multiobject Quantum Search Algorithm
Discrete Time or "Digital" Case
GENERALIZATION OF GROVER'S ALGORITHM TO MULTIOBJECT SEARCH IN QUANTUM COMPUTING, Part II: General Unitary Transformations, Goon Chen and Shunhua Sun
Introduction
Multiobject Search Algorithm
PART III: QUANTUM COMPUTATIONAL COMPLEXITY
COUNTING COMPLEXITY AND QUANTUM COMPUTATION, Stephen A. Fenner
Introduction
Equivalence of FQP and GapP
Strengths of the Quantum Model
Limitations of the Quantum Model
PART IV: QUANTUM ERROR-CORRECTING CODES
ALGORITHMIC ASPECTS OF QUANTUM ERROR-CORRECTING CODES, Markus Grassl
Introduction
General Quantum Error-Correcting Codes
Binary Quantum Codes
Additive Quantum Codes
Conclusions
CLIFFORD CODES, Andreas Klappenecker and Martin Rotteler
Motivation
Quantum Error Control Codes
Nice Error Bases
Stabilizer Codes
Clifford Codes
Clifford Codes that are Stabilizer Codes
A Remarkable Error Group
A Weird Error Group
Conclusions
PART V: QUANTUM COMPUTING ALGEBRAIC AND GEOMETRIC STRUCTURES
INVARIANT POLYNOMIAL FUNCTIONS ON K QUDITS, Jean-Luc Brylinski and Ranee Brylinski
Introduction
Polynomial Invariants of Tensor States
The Generalized Determinant Function
Asymptotics as k ®8
Quartic Invariants of k Qubits
Zs-SYSTOLIC FREEDOM AND QUANTUM CODES, Michael H. Freedman, David A. Meyer, and Feng Luo
Preliminaries and Statement of Results
Mapping Torus Constructions
Verification of Freedom and Curvature Estimates
Quantum Codes from Riemannian Manifolds
PART VI: QUANTUM TELEPORTATION, Kishore T. Kapale and M. Suhail Zubairy
Introduction
Teleportation of a 2-State System
Discrete N-State Quantum Systems
Quantum Teleportation of Entangled State
Continuous Quantum Variable States
Concluding Remarks
PART VII: QUANTUM SECURE COMMUNICATION AND QUANTUM CRYPTOGRAPHY
COMMUNICATING WITH QUBIT PAIRS, Almut Beige, Berthold-Georg Engler, Christian Kurtsiefer, and Harald Weinfurter
Introduction
The Mean King's Problem
Cryptography with Single Qubits
Cryptography with Qubit Pairs
Idealized Single-Photon Schemes
Direct Communication with Qubit Pairs
PART VIII: COMMENTARY ON QUANTUM COMPUTING
TRANSGRESSING THE BOUNDARIES OF QUANTUM COMPUTATION: A CONTRIBUTION TO THE HERMENEUTICS OF THE NMR PARADIGM, Stephen A. Fulling
Review of NMR Quantum Computing
Review of Modular Arithmetic
A Proposed "Quantum" Implementation
Aftermath


Keywords: Nanoscience, Nanotechnology

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Editor(s)

Biography

Goong Chen, Ranee K. Brylinski