Mathematics of Quantum Computation: 1st Edition (Hardback) book cover

Mathematics of Quantum Computation

1st Edition

Edited by Ranee K. Brylinski, Goong Chen

Chapman and Hall/CRC

448 pages | 28 B/W Illus.

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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

About the Authors/Editor

Goong Chen, Ranee K. Brylinski

Subject Categories

BISAC Subject Codes/Headings:
COM051300
COMPUTERS / Programming / Algorithms
MAT000000
MATHEMATICS / General
MAT003000
MATHEMATICS / Applied