1st Edition

Mathematics of Quantum Computation

Edited By Ranee K. Brylinski, Goong Chen Copyright 2002
    448 Pages 28 B/W Illustrations
    by Chapman & Hall

    448 Pages 28 B/W Illustrations
    by Chapman & Hall

    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
    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

    Biography

    Goong Chen, Ranee K. Brylinski