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Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 2009 Quantum Inductive Learning and Quantum Logic Synthesis Martin Lukac Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Lukac, Martin, "Quantum Inductive Learning and Quantum Logic Synthesis" (2009). Dissertations and Theses. Paper 2319. https://doi.org/10.15760/etd.2316 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. For more information, please contact [email protected]. QUANTUM INDUCTIVE LEARNING AND QUANTUM LOGIC SYNTHESIS by MARTIN LUKAC A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in ELECTRICAL AND COMPUTER ENGINEERING. Portland State University 2009 DISSERTATION APPROVAL The abstract and dissertation of Martin Lukac for the Doctor of Philosophy in Electrical and Computer Engineering were presented January 9, 2009, and accepted by the dissertation committee and the doctoral program. COMMITTEE APPROVALS: Irek Perkowski, Chair GarrisoH-Xireenwood -George ^Lendaris 5artM ?teven Bleiler Representative of the Office of Graduate Studies DOCTORAL PROGRAM APPROVAL: Malgorza /ska-Jeske7~Director Electrical Computer Engineering Ph.D. Program ABSTRACT An abstract of the dissertation of Martin Lukac for the Doctor of Philosophy in Electrical and Computer Engineering presented January 9, 2009. Title: Quantum Inductive Learning and Quantum Logic Synhesis Since Quantum Computer is almost realizable on large scale and Quantum Technology is one of the main solutions to the Moore Limit, Quantum Logic Synthesis (QLS) has become a required theory and tool for designing Quantum Logic Circuits. However, despite its growth, there is no any unified aproach to QLS as Quantum Computing is still being discovered and novel applications are being identified. The intent of this study is to experimentally explore principles of Quantum Logic Synthesis and its applications to Inductive Machine Learning. Based on algorithmic approach, I first design a Genetic Algorithm for Quantum Logic Synthesis that is used to prove and verify the methods proposed in this work. Based on results obtained from the evolutionary experimentation, I propose a fast, structure and cost based exhausitve search that is used for the design of a novel, least expensive universal familly of quantum gates. The results form both the evolutionary and heuristic search are used to for­ mulate an Inductive Learning Approach based on Quantum Logic Synthesis wih the intended application being the humanoid behavioral robotics. The presented approach illustrates a succesful algorithmic approach, where the search algorithm was able to invent/discover novel quantum circuits as well as novel principles in Quantum Logic Synthesis. To my parents and family. ACKNOWLEDGMENTS My sincere thanks to my advisor prof. Marek Perkowski and to each member of my dissertation committee for their guidance and help. CONTENTS Acknowledgments ii List of Tables x List of Figures xii 1 Introduction 1 1.1 Dissertation Statement 3 1.2 Contributions 3 2 Quantum Computing Basics and Concepts 8 2.1 Why quantum computing? 8 2.2 Mathematical Preliminaries to Quantum Computing 10 2.2.1 Bra-Ket notation 12 2.2.2 Matrix Trace 12 2.3 Quantum Mechanics 13 2.3.1 Bohr Particle Model 13 2.3.2 Quantum Model of Elementary Particle 15 CONTENTS iv 2.3.3 Schrodinger equation 20 2.3.4 Superposition of quantum states 21 2.4 Prom Quantum Mechanics to Quantum Logic 24 2.4.1 Simple Projective Measurement 29 2.4.2 Density Matrix and POVM 37 2.5 Quantum Logic, Quantum Gates and Quantum Logic Circuits 46 2.5.1 Single-qubit Quantum Gates 46 2.5.2 Multi-qubit and Controlled Quantum Gates 49 2.5.3 Quantum Circuits and Representation 52 2.5.4 Quantum Circuits and Sequential Logic 60 3 Quantum Logic Synthesis and Algorithmic Search 68 3.1 Introduction 68 3.2 Previous research on automated synthesis of quantum circuits 69 3.3 Function Classification for Logic Synthesis 71 3.4 Representation 72 3.5 Quantum-Based Synthesis: useful quantum circuits synthesis problems 75 3.5.1 Cost of quantum circuits 77 3.5.2 The Size of Quantum circuits 80 CONTENTS v 3.5.3 Quantum Logic Synthesis 83 3.5.4 NMR-based Quantum Logic Gates 86 3.6 Principles of Synthesis for NMR technology 88 3.6.1 V/T Gate - 'Level-generalization' 88 3.6.2 Insertion principle 91 3.6.3 The Divide and Conquer Principle 91 3.6.4 The Gate-collapsing principle 92 3.7 Examples of circuits obtained automatically for NMR technol­ ogy using methods from sections 3.5 and 3.6 94 3.8 Chapter Conclusion 100 4 Genetic Algorithm for Logic Synthesis 101 4.1 Introduction 101 4.2 Genetic algorithm 105 4.2.1 Encoding/Representation 105 4.2.2 Initialization steps of GA 107 4.3 Evaluation of Synthesis Errors Ill 4.3.1 Element Error Evaluation method (EE) 112 4.3.2 Measurement Evaluation method (ME) 113 4.3.3 Comparison of EE and ME 116 CONTENTS vi 4.4 Fitness functions of the GA 122 4.4.1 Simple Fitness Functions 123 4.4.2 Cost Based Fitness Functions 124 4.5 The Selection Process 129 4.6 Crossover and Mutation 133 4.6.1 Mutation 135 4.6.2 Additional GA tuning strategies 138 4.7 Chapter Conclusion 142 5 Evolutionary Search for Logic synthesis 144 5.1 Evolutionary Search 146 5.2 Experimental setup of GA 147 5.2.1 Input Sets of Quantum Primitives 147 5.3 Discussion of the Results of the Evolutionary Quantum Logic Synthesis using ME evaluation 150 5.3.1 Toffoli Gate 150 5.3.2 Synthesis of Fredkin Gate 159 5.3.3 Synthesis of Majority Gate 166 5.4 Discussion of the Results of the Evolutionary Quantum Logic Synthesis using EE evaluation 173 CONTENTS vii 5.4.1 Toffoli Gate 173 5.4.2 Predkin Gate 176 5.4.3 Entanglement Circuit 177 5.5 Discussion of the results of the Evolutionary Synthesis .... 180 5.5.1 Results Comparison 180 5.5.2 Encountered problems during the Evolutionary QLS . 181 5.6 Conclusion of the Evolutionary QLS 186 6 Structure Based Search for Universal Quantum Circuits 188 6.1 Introduction 188 6.2 The EX algorithm 191 6.3 Heuristic Search for Lowest Cost Function Class using the EX algorithm 195 6.3.1 New Quantum Logic Family 197 6.4 Discussion and Conclusion to the Structure Based Exhaustive search 201 7 Learning Quantum Behaviors 206 7.1 Quantum Behaviors 208 7.2 The concept of learning robotic behaviors from examples. 218 CONTENTS viu 7.2.1 Symbolic Quantum Synthesis of Single Output Quan­ tum Circuits 233 7.3 Experiments and Results for learning Quantum Behaviors . 239 7.3.1 Symbolic synthesis - Single output functions 239 7.4 Discussion on learning Quantum Benchmarks 242 7.5 Measurement Synthesis 246 7.5.1 Conclusions on the Measurement Dependent Quantum Logic Synthesis 265 8 Dissertation conclusion and contributions 268 References 271 Appendices 287 A Glossary 288 B Models of Quantum Computing 290 B.0.2 Quantum Cellular Automata 290 B.0.3 Quantum Finite Automata and Quantum State Machines301 B.0.4 Quantum Turing machine 316 B.0.5 Quantum Robots 317 CONTENTS ix C Cynthea, an interactive robotic system 321 C.l Definitions 321 C.2 Cynthea - the social interactive robotic framework 322 C.3 Learning and Adaptation 323 C.4 Sensors 327 C.4.1 Video-Sensor 328 C.4.2 Audio-Sensor 331 C.4.3 Voltage/Power-Sensor 333 C.5 Actuators 334 C.5.1 Servo-Control 336 C.5.2 Speech-Control 337 C.5.3 Network Control 338 C.5.4 Speech and Dialogue generation 339 C.6 Control and Scripting 341 C.6.1 CRL - The language 344 C.6.2 Scripting 355 C.7 Artificial Emotions 357 LIST OF TABLES 1 a). - K-map , b) - LUT 52 2 K-map of an incompletely specified 3x1 reversible quantum function before measurement 56 3 K-map of an incompletely specified 3x3 reversible quantum function before measurement with don't cares within single minterms 58 4 K-map of the Feynman gate 64 1 Cost models 78 1 Structure of a Genetic Algorithm 105 2 Example of the Majority gate encoding for the GA. Observe that each input minterm with qubit |g3) = |1) is a don't care. 115 1 Fixed parameters during search for Toffoli gate 150 2 Example of a 3 x 3 quantum probabilistic and a 3 x 1 determin­ istic functions generated by the circuit from Figure 5.4c. 159 3 Fixed parameters during search for Fredkin gate 159 4 Comparison of various fitness functions for all illustrated bench­ mark functions 181 1 Pseudo Code of EX Algorithm 192 LIST OF TABLES xi 6.2 The Peres gate Truth Table 196 6.3 The family of Peres gates 198 7.1 Braitenberg vehicle behaviors: (a) afraid of light, (b) attracted to light 209 7.2 A K-map of a completely specified reversible function 224 7.3 Incompletely specified reversible function f 227 7.4 Single output Boolean functions R to be synthesized (learned) 233 7.5 Example of analysis on circuit from Figure 7.11 236 7.6 Example of analysis on circuit from Figure 7.12 238 7.7 GA benchmark functions 239 LIST OF FIGURES 2.1 Bohr model of the atom (nucleus, orbiting electrons). Shown are light colors respective to the electron orbit transitions.
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