THE HIDDEN SUBGROUP PROBLEM FOR GENERALIZED QUATERNIONS by JULIA TUMASOVA UPTON A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2009 Copyright Julia Tumasova Upton 2009 ALL RIGHTS RESERVED ABSTRACT The hidden subgroup problem is a pivotal problem in quantum computation since it reflects the structure of tasks for which quantum algorithms significantly outperform classical algorithms. In this dissertation, a quantum algorithm that solves the hidden subgroup problem over the general- ized quaternion group is developed. The algorithm employs the abelian quantum Fourier transform and Kuperberg sieve to reveal the hidden subgroup. ii DEDICATION I dedicate this dissertation to my grandmother, Tamara Semenovna Babakhanova, an elec- trical engineer extraordinaire, who showered me with love and inspired my interest in mathematics and science. I love you. I miss you. I also dedicate this work to my husband Mark, who supported me through every step of this journey. Without you, my love, this would not have been possible. iii ACKNOWLEDGMENTS I would like to thank, first and foremost, my adviser, Dr. Jon Corson, who enthusiastically ventured with me into the exciting new world of quantum computation. Our many discussions of mathematics, physics, technology, and the human condition inspired and guided me throughout this undertaking. I also thank my committee members, Dr. Joseph Neggers, Dr. Martin Evans, Dr. Martyn Dixon, and Dr. Allen Stern for their support. I am indebted to my family for the constant encouragement, and the sacrifices made to help me succeed in this endeavor. I thank my husband Mark for the inspiration I drew from our numerous discussions and for being there when I needed him most. I am grateful to my daughter Sophia for bringing such joy into my life and making it all worthwhile. iv CONTENTS ABSTRACT . ii DEDICATION . iii ACKNOWLEDGMENTS . iv LIST OF TABLES . vii LIST OF FIGURES . viii 1 INTRODUCTION . 1 1.1 The Power of A Quantum Computer . 1 1.2 A Brief History of Quantum Computing . 3 1.3 The Hidden Subgroup Problem in Quantum Computing . 4 1.3.1 General Formulation and Special Cases . 4 1.3.2 The Dihedral Hidden Subgroup Problem . 5 2 A QUANTUM MODEL OF COMPUTATION . 6 2.1 The Quantum Mechanical Framework . 6 2.2 The No-Cloning Theorem . 11 2.3 Quantum Gates . 11 2.3.1 Single-qubit gates . 12 2.3.2 Multi-qubit gates . 16 2.3.3 Universal Quantum Gates . 20 2.4 Probabilistic Nature of Quantum Algorithms . 21 3 SHOR’S FACTORIZATION ALGORITHM . 24 v 3.1 Reduction of Factorization to Order Finding . 24 3.2 The Quantum Part: Order Finding . 25 3.2.1 The Main Ingredient: Quantum Fourier Transform . 25 3.2.2 Quantum Algorithm for Order Finding . 29 3.3 Shor’s Algorithm Applied: An Illustration . 31 3.4 Shor’s Algorithm as a Special Case of the Abelian HSP . 35 4 HIDDEN SUBGROUP PROBLEM FOR QUATERNIONS . 37 4.1 Tablets From On High: The Story of Quaternions . 37 4.1.1 The Origins . 37 4.1.2 The Struggle . 42 4.1.3 The Revival . 43 4.2 The Group Q4 ..................................... 44 4.3 The Group Q8 ..................................... 46 4.3.1 Group Presentation . 46 4.3.2 Subgroup Classification . 48 4.3.3 Group Construction . 49 4.4 The Generalized Quaternions . 52 4.4.1 Group Presentation and Construction . 52 4.4.2 Subgroup Structure . 55 4.5 Hidden Subgroup Algorithm for Generalized Quaternions . 56 4.5.1 Hidden Subgroup Problem for Generalized Quaternions . 56 4.5.2 Coset Sample Generation Procedure . 58 4.5.3 The Kuperberg Sieve . 60 REFERENCES . 64 vi LIST OF TABLES 1.1 Special Cases of The Hidden Subgroup Problem . 4 2.1 The AND Operation . 19 1 2.2 The Chernoff bounds for d = 4 ............................ 23 3.1 Registers’ States . 32 4.1 Cayley Table for Q8 .................................. 47 4.2 Multiplication Table for Q8 .............................. 51 4.3 “Twisted” Multiplication . 52 vii LIST OF FIGURES 2.1 A Simple Quantum Circuit . 14 2.2 Hadamard Gates in Parallel . 15 2.3 CNOT Gate . 16 2.4 The Toffoli Gate . 19 3.1 Quantum Circuit for QFT † .............................. 29 3.2 Quantum Circuit for Period Finding . 29 3.3 Probability Plot . 33 3.4 Shor’s Algorithm as a Case of HSP . 35 3.5 Implementable Shor’s Algorithm as a Case of HSP . 36 4.1 Quaternion Plaque on Broom Bridge in Dublin . 39 4.2 Quantum Circuit for Period Finding over Q4 ..................... 44 4.3 Reduction of QHSP . 57 viii CHAPTER 1 INTRODUCTION Reality is that which, when you stop believing in it, does not go away. Philip K. Dick 1.1 The Power of A Quantum Computer A quantum computer is a computing device whose operation is based on the uniquely quantum-mechanical effects of interference and entanglement. The quantum bits (qubits), named after a cubit, an ancient unit of length, are prepared in a superposition of classical computation states which are combined to interfere producing the computation output. Qubits are manipulated by quantum gates, most of which represent unitary transformations. The general state of a single qubit is a unit vector in a two-dimensional Hilbert space and with a standard choice of the computational basis fj0i;j1ig can be written as 0 1 a B C jyi = a j0i + b j1i = @ A; (1.1) b where amplitudes a and b are complex numbers such that jaj2 + jbj2 = 1 (1.2) The state jyi is in a superposition of the states j0i and j1i, i.e., it exists in both states simultane- ously. 1 The quantum state for a system of n qubits is a unit vector in a 2n-dimensional Hilbert space and can be expressed in the computational basis as 1 1 (n) jy i = ∑ ::: ∑ ai0;:::;in−1 ji0 :::in−1i (1.3) i0=0 in−1=0 where jy(n)i is normalized. A system of n qubits represents a superposition of 2n states, allowing a quantum computer to operate on states simultaneously. This phenomenon of quantum parallelism is the source of the quantum computer’s overwhelming power that allows for the execution of algorithms which no classical computer can efficiently handle. However, the only way to extract information from qubits is to subject them to measurement, which reduces the system to a single state in accordance with the Born rule. The rule states that for a qubit in a superposition (1.1) the measurement result is 0 with probability jaj2 or 1 with probability jbj2. Thus, the post-measurement state no longer contains any information pertaining to the amplitudes of the pre-measurement superposition state, other than indicating that the particular amplitude corresponding to the actual output state was not 0, and likely not exceedingly small. Moreover, a measurement gate is not unitary, i.e., it is irreversible. For an input x covering the range 0 ≤ x < 2n and a computed function f , the measurement of the input register produces a single randomly chosen x0, while the measurement of the output register gives a single corresponding value f (x0). To take advantage of superposition and thus achieve an exponential speedup over the clas- sical computer, quantum algorithms involve judicial arrangements of unitary gates, in some cases supplemented by intermediate measurement gates acting on subsets of a system’s qubits. In most cases, this approach leads to an extraction of information about global properties of f , which man- ifest themselves in the relations between the values of f for many distinct values of x, information that a classical computer can only produce by making many independent evaluations. 2 1.2 A Brief History of Quantum Computing The Turing machine is the standard mathematical model of a classical computer. A math- ematical model of a quantum computer, just as independent of a physical realization, was first discussed by Manin [1] in 1980. A few years later, Feynman [2, 3] pointed out that simulation of certain quantum systems on a classical computer requires time that scales exponentially with the number of particles in a simulated system. Feynman proposed that a quantum system might be efficiently simulated by a computer that exploits quantum effects. In 1985, a concrete mathematical model, the quantum Turing machine, was introduced by Deutsch [4], who followed it up in 1989 [5] by an equivalent but more convenient model, the quantum circuit. These ideas led to the search for further applications of a quantum computer, initially resulting in solutions to somewhat contrived problems, e.g., the Deutsch-Jozsa algorithm [6], the Simon algorithm [7], and the Bernstein-Varzani algorithm [8]. Nonetheless, these early results demonstrated the existence of specific problems that can be handled dramatically faster by a quantum computer. A breakthrough that elevated quantum computation from a primarily academic exercise to a matter of great practical significance came in 1994 when Shor [9] introduced polynomial- time quantum algorithms for factoring numbers and computing the discrete logarithm. The widely used public key cryptography systems RSA and Diffie-Hellman rely on the fact that there are no polynomial-time classical algorithms for these problems. When implemented on a quantum computer, Shor’s algorithms will break these encryption schemes in a matter of seconds. Another masterpiece of quantum-computational software, Grover’s algorithm [10], was introduced in 1996, offering a solution for unstructured search problems with quadratic speedup over classical search algorithms. Recently Hallgren [11] developed a quantum algorithm for Pell’s equation with exponential speedup over any existing classical algorithm. Quantum algorithms that outperform classical algorithms can be separated into three cate- gories [12]: 1.