QUANTUM COMPUTATION USING GEOMETRIC ALGEBRA APPROVED BY SUPERVISORY COMMITTEE Copyright 2002 Douglas J. Matzke All Rights Reserved To all the great thinkers who aspire to understand the mysteries of quantum mechanics QUANTUM COMPUTATION USING GEOMETRIC ALGEBRA by Douglas J. Matzke, BSEE, MSEE DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Dallas in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN ELECTRICAL ENGINEERING THE UNIVERSITY OF TEXAS AT DALLAS May 2002 ACKNOWLEDGEMENTS My first thank you goes to my committee chair Dr. Cyrus Cantrell, who gave me a chance three years ago when I approached him with a very vague idea about mapping Boolean logic into linear mathematics. Choosing this kind of subject (ultimately, quantum computing) is traditionally extremely mathematical and my approaching it with a very pragmatic engineering perspective must have caused him some amusement while requiring the patience of a saint. My most heartfelt thanks goes to my like-minded friend and committee co-chair Michael Manthey, whom I first met at the PhysComp94 conference, and who lives in the mountains of Crestone, Colorado. I visited Mike in Colorado for three days in the summer of 2000, where he taught me the fundamentals of geometric algebra by hand-cranking the equations. Based on that tedious manual experience, I returned to Dallas and a month later created the prototype ga.pl tool described in this dissertation. Once this and subsequent GA tools were available, the quantum representation practically revealed itself in another month. I am eternally indebted to Mike for enabling one of the most creative and fulfilling periods of my life. Another very big thank you also goes to Nick Lawrence of Lawrence Technologies who hired me in the summer of 2001, due to his interests in high-dimensional software technology, which enabled me to write the bulk of this dissertation in only two months. My other committee members William Pervin, Wolfgang Rindler, and Samuel Villareal were fabulous and very supportive. A very special thanks goes to my spouse Patricia, who kept our home running smoothly while I was enjoying this time-consuming, yet creative, endeavor. v QUANTUM COMPUTATION USING GEOMETRIC ALGEBRA Publication No. __________ Douglas J. Matzke, Ph. D. The University of Texas at Dallas, 2002 Supervising Professor: Cyrus Cantrell This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication. Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anti-commutative outer products into a real-valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools. vi TABLE OF CONTENTS ACKNOWLEDEMENTS ........................................................................................................v ABSTRACT..............................................................................................................................vi LIST OF TABLES .....................................................................................................................x LIST OF FIGURES ............................................................................................................... xiii CHAPTER 1 INTRODUCTION TO QUANTUM COMPUTING ......................................... 1 1.1 Problem Definition ......................................................................................................... 1 1.1.1 Why Quantum Computing is Important ................................................................. 2 1.1.2 Understanding Quantum Computing is Difficult .................................................... 3 1.1.3 Expanding the Quantum Computing Industry ........................................................ 4 1.2 Intended Audience ......................................................................................................... 4 1.3 Improving Understanding and Building Tools .............................................................. 5 1.4 Summary of Results ....................................................................................................... 6 CHAPTER 2 THE UNIVERSE IS A QUANTUM COMPUTER .......................................... 7 2.1 Information is Primitive and Physical ............................................................................ 7 2.2 High Dimensional Spaces .............................................................................................. 9 2.2.1 High Dimensional Metrics versus Intuition .......................................................... 11 2.2.2 Space and Time are Linked ................................................................................... 12 2.2.3 Hilbert Spaces and Tensor Products ..................................................................... 14 2.2.4 The Geometric Algebra Alternative ...................................................................... 15 2.3 Classical Versus Quantum Computing ........................................................................ 16 2.3.1 States and Vectors for Bits versus Qubits ............................................................. 17 2.3.2 Superposition and Entanglement .......................................................................... 18 2.3.3 Classical versus Quantum Algorithms .................................................................. 20 2.3.4 Quantum Computers ............................................................................................. 21 2.4 Steps in Quantum Computing ....................................................................................... 21 CHAPTER 3 BOOLEAN LOGIC IN QUANTUM COMPUTING ...................................... 23 3.1 Unitary Transforms and Rotations ............................................................................... 23 3.2 Reversible Computing ................................................................................................. 24 3.3 Universal Logic and Quantum Gates ........................................................................... 25 3.4 Toffoli and Fredkin Reversible Gates .......................................................................... 26 vii 3.5 Boolean Logic and Operators Using Linear Mathematics ........................................... 28 3.5.1 Boolean Logic in Galois Fields ............................................................................. 28 3.5.2 XOR Dominated Logic ......................................................................................... 30 3.5.3 Toffoli and Fredkin Gates in GF(2) ...................................................................... 31 3.5.4 Boolean Logic is formally non-linear ................................................................... 33 CHAPTER 4 GEOMETRIC ALGEBRA FOUNDATIONS ................................................. 35 4.1 Matrix versus Algebraic Notation ................................................................................ 35 4.2 Co-Occurrence and Co-Exclusion ............................................................................... 36 4.3 Geometric Algebra Principles ...................................................................................... 40 4.3.1 Scalars and Vectors ............................................................................................... 41 4.3.2 Bivectors and Anticommutative Multiplication .................................................... 42 4.3.3 N-vectors and Mixed Rank Expressions ............................................................... 45 4.3.4 Number of Elements in a Geometric Algebra ....................................................... 47 4.3.5 Quaternions in Geometric Algebra ....................................................................... 48 4.3.6 Inner Product Definition ....................................................................................... 50 4.3.7 Outer Product and Inner Product Examples........................................................... 52 4.4 Geometric Algebra Tools for Boolean Logic .............................................................. 54 4.4.1 Boolean Logic in GA ............................................................................................ 54 4.4.2 GA Evaluator ........................................................................................................ 57 4.4.3 Universal Logic Decode in GA ............................................................................. 60 4.4.4 GA Generator ........................................................................................................ 68 4.5 GA Expression Solver Tool ........................................................................................