)j > Riley Tipton Perry University of New South Wales, Australia. World Scientific CHENNAI

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1 Riley Tipton Perry University of New South Wales, Australia )j > World Scientific NEW JERSEY LONDON. SINGAPORE BEIJING SHANSHAI HONG K0N6 TAIPEI» CHENNAI

2 Contents Acknowledgments xi 1. Introduction What is Quantum Computing? Why Another Quantum Computing Tutorial? Quantum Computation and Quantum Information 2 2. Computer Science Introduction History Turing Machines Binary Numbers and Formal Languages Turing Machines in Action The Universal Turing Machine The Halting Problem Circuits Common Gates Combinations of Gates Relevant Properties Universality Computational Resources and Efficiency Quantifying Computational Resources Standard Complexity Classes The Strong Church-Turing Thesis Quantum Turing Machines Energy and Computation 24 V

3 vi Quantum Computing from the Ground Up Reversibility Irreversibility Landauer's Principle Maxwell's Demon Reversible Computation Reversible Gates Reversible Circuits Mathematics for Quantum Computing Introduction Polynomials Logical Symbols Trigonometry Review Right Angled Triangles Converting Between Degrees and Radians Inverses Angles in Other Quadrants Visualisations and Identities Logs Complex Numbers Polar Coordinates and Complex Conjugates Rationalising and Dividing Exponential Form Matrices Matrix Operations Vectors and Vector Spaces Introduction Column Notation The Zero Vector Properties of Vectors in C" The Dual Vector Linear Combinations Linear Independence Spanning Set Basis Probability Theory Probability Amplitudes The Inner Product Orthogonality 63

4 Contents vii The Unit Vector Bases for C" The Gram Schmidt Method Linear Operators Outer Products and Projectors The Adjoint Eigenvalues and Eigenvectors Trace Normal Operators Unitary Operators Hermitian and Positive Operators Diagonalisable Matrix The Commutator and Anti-Commutator Polar Decomposition Spectral Decomposition Tensor Products Fourier Transforms The Fourier Series The Discrete Fourier Transform Quantum Mechanics History Classical Physics Important Concepts Statistical Mechanics Important Experiments The Photoelectric Effect Bright Line Spectra Proto Quantum Mechanics The New Theory of Quantum Mechanics Important Principles for Quantum Computing Linear Algebra Superposition Dirac Notation Ill Representing Information Uncertainty Entanglement 113

5 viii Quantum Computing from the Ground Up Quantum Computing Elements of Quantum Computing Introduction History Bits and Qubits Entangled States Quantum Circuits Important Properties of Quantum Circuits Common Circuits The Reality of Building Circuits Building a. Programmable Quantum Computer 5.4 The Four Postulates of Quantum Mechanics Postulate One Postulate Two Postulate Three Postulate Four Information Theory Introduction History Shannon's Communication Model Channel Capacity Classical Information Sources Independent Information Sources Classical Redundancy and Compression Shannon's Noiseless Coding Theorem Quantum Information Sources Pure and Mixed States Schumacher's Quantum Noiseless Coding Theorem Noise and Error Correction Quantum Noise Quantum Error Correction Bell States Same Measurement Direction Different Measurement Directions Bell's Inequality Cryptology Classical Cryptography 195

6 Contents ix Quantum Cryptography Alternative Models of Computation Quantum Algorithms Introduction Deutsch's Algorithm The Problem Denned The Classical Solution The Quantum Solution Physical Implementations The Deutsch-Josza Algorithm The Problem Defined The Quantum Solution Shor's Algorithm The Quantum Fourier Transform Fast Factorisation Order Finding Grover's Algorithm The Travelling Salesman Problem Quantum Searching Using Quantum Mechanical Devices and Recent Developments Introduction Physical Realisation Implementation Technologies Quantum Computer Languages Encryption Devices Recent Developments Hardware and Architecture Cryptography Algorithms 231 Bibliography 233 Index 239