Quantum Computers: Theory and Algorithms

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