CALIFORNIA STATE UNIVERSITY, NORTHRIDGE ANALYSIS ON QUANTUM COMPUTATION WITH A SPECIAL EMPHASIS ON EMPIRICAL SIMULATION OF QUANTUM ALGORITHMS A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Computer Science By ARINDAM MISHRA MAY 2020 ©Copyright by ARINDAM MISHRA, 2020 All Rights Reserved ii The thesis of Arindam Mishra is approved: ___________________________________________ ________________ Katya Mkrtchyan, Ph.D. Date ___________________________________________ ________________ Robert D McIlhenny, Ph.D. Date ___________________________________________ ________________ John Noga, Ph.D., Chair Date California State University, Northridge iii DEDICATION I dedicate this paper to my grandmother Shivani Mishra, my parents Milan and Deepali Mishra, my uncle and aunt Kalyan and Chitra Mishra, and my two sweet sisters Ishani and Aishwarya Mishra, with my gratitude for their support and motivations throughout my life. iv Table of Contents COPYRIGHT ii DEDICATION ....................................................................................................................................... iv LIST OF TABLES ................................................................................................................................ vii LIST OF FIGURES ............................................................................................................................. viii ABSTRACT ........................................................................................................................................... ix Chapter One ............................................................................................................................................ 1 1.1 INTRODUCTION .................................................................................................................................. 1 1.1.1 BACKGROUND ............................................................................................................................ 1 1.1.2 THESIS COVERAGE AND APPROACH .......................................................................................... 2 1.1.3 MOTIVATION .............................................................................................................................. 4 1.2 TOOLS USED ...................................................................................................................................... 5 1.2.1 IBM Q EXPERIENCE ................................................................................................................... 5 1.2.2 QISKIT ....................................................................................................................................... 6 Chapter Two............................................................................................................................................ 7 2.1 QUBIT – QUANTUM ANALOGOUS TO BIT ........................................................................................ 7 2.1.1 DEFINITION ................................................................................................................................. 7 2.1.2 QUBIT REPRESENTATION ............................................................................................................ 7 2.2 QUANTUM GATES .............................................................................................................................. 8 2.2.1 QUANTUM GATE TYPES .............................................................................................................. 8 2.2.2 CLASSICAL AND QUANTUM GATES ............................................................................................ 9 2.3 QUBIT ROBUSTNESS ........................................................................................................................ 10 2.3.1 DIVINCENZO CRITERIA ............................................................................................................ 10 2.3.2 QUANTUM COHERENCE AND GATE TIME ................................................................................. 11 2.4 QUBIT MODALITY ........................................................................................................................... 12 2.4.1 QUBIT TECHNOLOGIES ............................................................................................................. 12 2.4.2 ELECTRON AND NUCLEAR SPIN ................................................................................................ 12 2.4.3 NEUTRAL ATOMS AND TRAPPED IONS ...................................................................................... 14 2.4.4 SUPERCONDUCTING QUBITS .................................................................................................... 15 Chapter Three........................................................................................................................................ 17 3.1 QUANTUM ALGORITHMS ................................................................................................................. 17 3.1.1 SIMON ALGORITHM .................................................................................................................. 17 IMPLEMENTATION ............................................................................................................................. 18 3.1.2 QUANTUM FOURIER TRANSFORMATION ALGORITHM ............................................................. 21 IMPLEMENTATION ............................................................................................................................. 22 v 3.1.3 SHOR’S ALGORITHM................................................................................................................. 22 3.2. QUANTUM SUPREMACY ................................................................................................................. 23 Chapter Four ......................................................................................................................................... 24 4.1 DEUTSCH–JOZSA ALGORITHM ................................................................................................ 24 4.1.1 THE ALGORITHM ...................................................................................................................... 24 4.1.2 IMPLEMENTATION .................................................................................................................... 24 4.1.3 ANALYSIS AND INTERPRETATION OF RESULTS ........................................................................ 31 4.1.4 OUTCOME AND ASSOCIATED FUTURISTIC WORKS .................................................................... 32 Chapter Five .......................................................................................................................................... 33 5.1 PRACTICAL QUANTUM COMPUTERS AND CURRENT STANDING IN THE QUANTUM WORLD ........... 33 5.1.1 NISQ ERA ................................................................................................................................. 33 5.1.1 FEW IMPORTANT EXPERIMENTS WORTH MENTION ................................................................... 34 5.2 FUTURE SCOPES AND POSSIBILITIES IN THE QUANTUM COMPUTATION.......................................... 35 5.2.1 QUANTUM COMPUTATIONS ADVANTAGE ................................................................................ 35 5.2.2 FUTURE SCOPE .......................................................................................................................... 35 Chapter Six............................................................................................................................................ 37 6.1 FUTURE WORKS .............................................................................................................................. 37 Chapter Seven ....................................................................................................................................... 38 7.1 CONCLUSION ................................................................................................................................... 38 REFERENCES ..................................................................................................................................... 39 APPENDIX ........................................................................................................................................... 40 A.1 LIST OF DIRAC NOTATIONS ........................................................................................................ 40 A.2 LINEAR ALGEBRA ....................................................................................................................... 40 A.3 PYTHON CODE FOR QASM ......................................................................................................... 41 vi LIST OF TABLES Table 2. 1 Quantum Gates with Matrix representation ........................................................................................... 9 Table 2. 2 Truth Table for Toffoli gate ................................................................................................................. 10 Table 2. 3 DiVincenzo Criteria ............................................................................................................................. 11 Table 3. 1Simon's Algorithm input ....................................................................................................................... 18 Table