Southern Methodist University SMU Scholar Electrical Engineering Theses and Dissertations Electrical Engineering Fall 12-21-2019 Technology-dependent Quantum Logic Synthesis and Compilation Kaitlin Smith Southern Methodist University, [email protected] Follow this and additional works at: https://scholar.smu.edu/engineering_electrical_etds Part of the Other Electrical and Computer Engineering Commons Recommended Citation Smith, Kaitlin, "Technology-dependent Quantum Logic Synthesis and Compilation" (2019). Electrical Engineering Theses and Dissertations. 30. https://scholar.smu.edu/engineering_electrical_etds/30 This Dissertation is brought to you for free and open access by the Electrical Engineering at SMU Scholar. It has been accepted for inclusion in Electrical Engineering Theses and Dissertations by an authorized administrator of SMU Scholar. For more information, please visit http://digitalrepository.smu.edu. TECHNOLOGY-DEPENDENT QUANTUM LOGIC SYNTHESIS AND COMPILATION Approved by: Dr. Mitchell Thornton - Committee Chairman Dr. Jennifer Dworak Dr. Gary Evans Dr. Duncan MacFarlane Dr. Theodore Manikas Dr. Ronald Rohrer TECHNOLOGY-DEPENDENT QUANTUM LOGIC SYNTHESIS AND COMPILATION A Dissertation Presented to the Graduate Faculty of the Lyle School of Engineering Southern Methodist University in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy with a Major in Electrical Engineering by Kaitlin N. Smith (B.S., EE, Southern Methodist University, 2014) (B.S., Mathematics, Southern Methodist University, 2014) (M.S., EE, Southern Methodist University, 2015) December 21, 2019 ACKNOWLEDGMENTS I am grateful for the many people in my life who made the completion of this dissertation possible. First, I would like to thank Dr. Mitch Thornton for introducing me to the field of quantum computation and for directing me during my graduate studies. I would also like to thank my committee for supporting my research and for all of the suggestions and guidance that helped me to develop my skills as a scientist. To my family and friends: thank you for being there. You have no idea how much your constant encouragement, advice, and love have meant to me over the years as I completed this degree. To my Mom and Dad: thank you for always being my biggest fans and for always believing in me. You have both taught me so much, and have given me the courage to chase my dreams. I love you. iii Smith , Kaitlin N. B.S., EE, Southern Methodist University, 2014 B.S., Mathematics, Southern Methodist University, 2014 M.S., EE, Southern Methodist University, 2015 Technology-dependent Quantum Logic Synthesis and Compilation Advisor: Dr. Mitchell Thornton - Committee Chairman Doctor of Philosophy degree conferred December 21, 2019 Dissertation completed September XX, 2019 The models and rules of quantum computation and quantum information processing (QIP) differ greatly from those that govern classical computation, and these differences have caused the implementation of quantum processing devices with a variety of new technologies. Many platforms have been developed in parallel, but at the time of writing, one method of quantum computing has not shown to be superior to the rest. Because of the variation that exists between quantum platforms, even between those of the same technology, there must be a way to automatically synthesize technology-independent quantum designs into forms that are capable of physical realization on a quantum computer (QC) with specific operating parameters. Additionally, results of synthesis must be formally verified to con- firm that output technology-dependent specifications are logically identical to their original, technology-independent forms. The first contribution of this work to the field of quantum computing is the creation of such a methodology. Quantum technology mapping and op- timization for machines with fixed coupling maps and libraries of gates can be performed with an automatic quantum compiler, and the development and test of this compiler will be explored in this dissertation. Furthermore, this compiler can be considered in a more general context to be a synthesis tool for QIP circuits in a specific realization technology, many of which are capable of implementing systems where the radix of computation, r, is greater than two. As a result of this ability, the second contribution of this work is the presentation of architectures for higher-dimensional quantum entanglement. iv TABLE OF CONTENTS ACKNOWLEDGMENTS........................................................... iii LIST OF FIGURES................................................................. viii LIST OF TABLES..................................................................x LIST OF ABBREVIATIONS........................................................ xii CHAPTER 1. Introduction..................................................................1 1.1. Classical Computation and Limitations...................................2 1.2. Contribution.............................................................3 2. Quantum Information.........................................................4 2.1. The Qubit...............................................................4 2.2. Physical Quantum Implementations......................................6 2.2.1. Transmons........................................................7 2.2.2. Photonics.........................................................8 2.3. The Superposition Principle.............................................. 10 2.4. The Wavefunction and Quantum Computing.............................. 11 2.5. Quantum Operations..................................................... 14 2.6. Requirements for Quantum Computation................................. 17 2.7. Entanglement............................................................ 18 3. Quantum Logic Synthesis Considerations...................................... 22 3.1. No-Cloning Theorem..................................................... 22 3.2. Reversible Logic.......................................................... 24 3.3. Gate Libraries and Coupling Constraints.................................. 26 3.4. Current Physical Quantum Technology................................... 27 v 3.4.1. IBM Q............................................................ 27 3.4.2. Rigetti............................................................ 29 3.4.3. Quantum with Photonic Devices................................... 30 3.5. Quantum Cost........................................................... 33 3.6. Quantum Multiple-valued Decision Diagrams............................. 35 3.7. Zero-supressed Decision Diagrams........................................ 36 4. Technology Mapping Algorithms............................................... 39 4.1. Connectivity Tree Reroute................................................ 39 4.2. Zero-suppressed Decision Diagram Technology Mapping................... 42 4.2.1. Problem Formulation with ZDD Mapping.......................... 42 4.2.2. Finding Maximal Partitions....................................... 43 4.2.3. ZDD mapping in the Quantum Compilation Flow.................. 47 4.2.4. Experimental Results............................................. 48 5. Formally-verified Synthesis Methods and Experiments......................... 52 5.1. IBM..................................................................... 53 5.1.1. Methodology...................................................... 53 5.1.2. Experimental Results............................................. 55 5.2. Rigetti................................................................... 62 5.2.1. Methodology...................................................... 62 5.2.2. Experimental Results............................................. 65 6. Higher Dimensioned Quantum Logic Synthesis................................. 68 6.1. Qudit Information........................................................ 71 6.2. Qudit Superposition...................................................... 73 6.2.1. The Hadamard Gate.............................................. 74 6.2.2. The Chrestenson Gate............................................ 74 vi 6.3. Single Qudit Basis Permutation........................................... 78 6.4. Controlled Qudit Operators.............................................. 79 7. Higher Dimensioned Entanglement Generators................................. 84 7.1. Partial Entanglement of Qudit Pairs...................................... 85 7.2. Maximal Entanglement Generators for Qudit Pairs........................ 87 7.3. Maximal Entanglement of Qudit Groups.................................. 97 7.3.1. Synthesis of Qudit Entanglement States........................... 100 8. Conclusion.................................................................... 106 8.1. Summary................................................................ 106 8.2. Future Work............................................................. 107 APPENDIX A. The Radix-4 Chrestenson Gate................................................ 109 A.0.1. Quantum Optics.................................................. 110 A.1. The Four-port Coupler................................................... 111 A.2. Physical Realizations of the Four-port Coupler............................ 115 A.2.1. Fabrication....................................................... 117 A.2.2. Characterization.................................................. 117 A.3. Implementing Qudit Quantum Operations with the Coupler..............