1 Introduction 2 Classical Computer 2.1 Binary Representation 3 Quantum Computer 3.1 Qubit 4 Classical Gates and Circuits 5 Quantum Gates and Circuits 5.1 Hilbert space 5.2 Measurement 6 Deutsch Algorithm 7 Grover Algorithm 7.1 Grover algorithm: two-qubit 7.2 Grover algorithm: n-qubit 7.3 Grover diffusion and rotation gate G 7.4 Single Recursion: Two qubit 8 Deutsch-Josza Algorithm 9 Simon's Algorithm 9.1 Quantum Algorithm 9.2 An Illustrative Example 10 Quantum Fourier Transform (QFT) 51 10.1 Quantum circuit of QFT 11 Shor 11.1 Introduction 11.2 Understanding the classical algorithm 11.3 Quantum algorithm 12 Option Pricing 12.1 Quantum Algorithm for Option Pricing 12.2 Quadratic Improvement 12.3 Estimation of Phase 12.4 Call Option 13 Solving Linear Equations 13.1 Introduction 13.2 Harrow-Hassidim-Lloyd Algorithm 13.3 Specific Example 13.4 Other applications 14 Quantum-Classical Hybrid Algorithms 14.1 Why bother? 14.2 Overlap of Wavefunctions 14.3 Variational Quantum Eigensolvers 15 Quantum Error Correction 15.1 Introduction 15.2 Simple quantum errors 15.3 Kraus Operators 15.4 Nine-qubit Code 15.5 General properties of quantum error-correcting codes 15.6 Classical Linear Codes 15.7 CSS Codes 16 Efficiency of a Quantum Computer 16.1 So where does quantum computation take place? 16.2 Conclusions 16.3 Acknowledgements