Mathematics of Quantum Computation

Mathematics of Quantum Computation

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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.show more

Product details

  • Hardback | 448 pages
  • 156.5 x 246.4 x 29mm | 798.34g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 28 black & white illustrations
  • 1584882824
  • 9781584882824
  • 2,454,130

Table of contents

Preface PART I: QUANTUM ENTANGLEMENT ALGEBRAIC MEASURES OF ENTANGLEMENT, Jean-Luc Brylinski Introduction Rank of a Tensor Tensors in (C 2)A2 Tensors in (C 2)A3 Tensors in (C 2)A4 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 (R)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, Nanotechnologyshow more