Mathematics of Quantum Computation (Hardcover)

Preface PART I: QUANTUM ENTANGLEMENTALGEBRAIC MEASURES OF ENTANGLEMENT, Jean-Luc BrylinskiIntroductionRank of a Tensor Tensors in (C 2)A2Tensors in (C 2)A3Tensors in (C 2)A4KINEMATICS OF QUBIT PAIRS, Berthold-Geor Englert and Nasser MetwallyIntroductionBasic 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 WallachIntroductionInvariants for Compact Lie Groups The Simplest Cases The Case of Three QubitsA Basic Set of Invariants for Three QubitsSome Implications for Other RepresentationsPART II: UNIVERSALITY OF QUANTUM GATESUNIVERSAL QUANTUM GATES, Jean-Luc Brylinski and Ranee BrylinskiStatements of Main ResultsExamples and Relations to Works of Other AuthorsFrom Universality to Exact Universality Analyzing the Lie Algebra g Normalizer of HPART III: QUANTUM SEARCH ALGORITHMSFROM COUPLED PENDULUMS TO QUANTUM SEARCH Lov K. Grover and Anirvan M. SenguptaIntroductionClassical Analogy N Coupled PendulumsThe AlgorithmTowards Quantum Searching The Quantum Search Algorithm Why Does it Take O(vN) cycles?Applications and ExtensionsGENERALIZATION OF GROVER'S ALGORITHM TO MULTIOBJECT SEARCH IN QUANTUM COMPUTING, Part I: Continuous Time and Discrete Time, Goon Chen, Stephen A,. Fulling, and Jeesen ChenIntroductionAnalog Multiobject Quantum Search AlgorithmDiscrete Time or "Digital" Case GENERALIZATION OF GROVER'S ALGORITHM TO MULTIOBJECT SEARCH IN QUANTUM COMPUTING, Part II: General Unitary Transformations, Goon Chen and Shunhua SunIntroductionMultiobject Search Algorithm PART III: QUANTUM COMPUTATIONAL COMPLEXITYCOUNTING COMPLEXITY AND QUANTUM COMPUTATION, Stephen A. FennerIntroductionEquivalence of FQP and GapP Strengths of the Quantum ModelLimitations of the Quantum Model PART IV: QUANTUM ERROR-CORRECTING CODESALGORITHMIC ASPECTS OF QUANTUM ERROR-CORRECTING CODES, Markus GrasslIntroductionGeneral Quantum Error-Correcting CodesBinary Quantum CodesAdditive Quantum Codes ConclusionsCLIFFORD CODES, Andreas Klappenecker and Martin RottelerMotivationQuantum Error Control CodesNice Error Bases Stabilizer Codes Clifford Codes Clifford Codes that are Stabilizer Codes A Remarkable Error Group A Weird Error GroupConclusionsPART V: QUANTUM COMPUTING ALGEBRAIC AND GEOMETRIC STRUCTURESINVARIANT POLYNOMIAL FUNCTIONS ON K QUDITS, Jean-Luc Brylinski and Ranee BrylinskiIntroduction Polynomial Invariants of Tensor StatesThe Generalized Determinant FunctionAsymptotics as k ®8 Quartic Invariants of k QubitsZs-SYSTOLIC FREEDOM AND QUANTUM CODES, Michael H. Freedman, David A. Meyer, and Feng LuoPreliminaries and Statement of ResultsMapping Torus ConstructionsVerification of Freedom and Curvature Estimates Quantum Codes from Riemannian ManifoldsPART VI: QUANTUM TELEPORTATION, Kishore T. Kapale and M. Suhail ZubairyIntroductionTeleportation of a 2-State System Discrete N-State Quantum SystemsQuantum Teleportation of Entangled State Continuous Quantum Variable States Concluding Remarks PART VII: QUANTUM SECURE COMMUNICATION AND QUANTUM CRYPTOGRAPHYCOMMUNICATING WITH QUBIT PAIRS, Almut Beige, Berthold-Georg Engler, Christian Kurtsiefer, and Harald WeinfurterIntroductionThe Mean King's ProblemCryptography with Single QubitsCryptography with Qubit PairsIdealized Single-Photon SchemesDirect Communication with Qubit PairsPART VIII: COMMENTARY ON QUANTUM COMPUTINGTRANSGRESSING THE BOUNDARIES OF QUANTUM COMPUTATION: A CONTRIBUTION TO THE HERMENEUTICS OF THE NMR PARADIGM, Stephen A. FullingReview of NMR Quantum ComputingReview of Modular Arithmetic A Proposed "Quantum" Implementation AftermathKeywords: Nanoscience, Nanotechnology