THE HIDDEN SUBGROUP PROBLEM By ANALES DEBHAUMIK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 c 2010 Anales Debhaumik 2 I dedicate this to everyone who helped me in my research. 3 ACKNOWLEDGMENTS First and foremost I would like to express my sincerest gratitude to my advisor Dr Alexandre Turull. Without his guidance and persistent help this thesis would not have been possible. I would also like to thank sincerely my committee members for their help and constant encouragement. Finally I would like to thank my wife Manisha Brahmachary for her love and constant support. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1 INTRODUCTION ................................... 8 1.1 Preliminaries .................................. 8 1.2 Quantum Computing .............................. 11 1.3 Quantum Fourier Transform .......................... 12 1.4 Hidden Subgroup Problem .......................... 14 1.5 Distinguishability ................................ 18 2 HIDDEN SUBGROUPS FOR ALMOST ABELIAN GROUPS ........... 20 2.1 Algorithm to find the hidden subgroups .................... 20 2.2 An Almost Abelian Group of Order 3 2n ................... 22 · 3 KEMPE-SHALEV DISTINGUISHABLITY ..................... 33 4 ALGORITHMIC DISTINGUISHABILITY ...................... 43 4.1 An algorithm for distinguishability ....................... 43 4.2 Abelian Groups ................................. 45 4.3 Frobenius Groups ............................... 46 5 SOME CALCULATIONS ............................... 55 5.1 Kempe-Shalev distinguishability ........................ 55 5.2 Algorithmic Distinguishability ......................... 56 APPENDIX A TABLES FOR CHAPTER 4 ............................. 59 B TABLE FOR CHAPTER 5 .............................. 70 REFERENCES ....................................... 71 BIOGRAPHICAL SKETCH ................................ 73 5 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE HIDDEN SUBGROUP PROBLEM By Anales Debhaumik May 2010 Chair: Alexandre Turull Major: Mathematics The topic of my research is the Hidden Subgroup Problem. The problem can be stated as follows: Problem. (Hidden Subgroup Problem) Let G be a finite group and H a subgroup. Given a black-box function f : G S which is constant on (left)-cosets gH of H and takes → different values for different cosets, determine a set of generators for H. Efficient classical algorithms for the Hidden Subgroup Problem are not known. However, efficient quantum algorithms are known for this problem in some cases. One such algorithm implies Shor’s famous efficient method for breaking the RSA cryptosystem. An efficient solution of this problem in all cases would have wide implications in the field of theoretical computer science. For example it would most likely solve some classical problems which are neither NP complete nor are in P. A solution would also − imply a solution for Graph Isomorphism problem which is a long standing problem in computer science. In the present thesis, we study some quantum algorithms for the Hidden Sub- group Problem. We discuss Quantum Fourier Tranform and its applications to the Hidden Subgroup Problem. We discuss Almost Abelian groups and show that there is a quantum algorithm that solves the Hidden Subgroup Problem for them. We also study the decision version of the problem. We compare two different formalizations of 6 this concept. We show that these formalizations coincide in the case of abelian and Frobenius groups. We conclude with a family of groups where the formalizations may not coincide. 7 CHAPTER 1 INTRODUCTION 1.1 Preliminaries A quantum computer is an, at present theoretical, machine which performs computations on the basis of the laws of quantum mechanics. Such a device can prepare quantum states and perform measurements on them. As with classical computers, numerical data form the input and the output for the quantum computer. The processing of this data in a quantum computer may involve actions on them which are not classical. As a result, the output of a quantum computer is not fully determined by its input. Given a particular input, various outputs may occur with different probabilities. In this sense, quantum computers behave differently than classical ones. Though quantum computing as a field of research is still in its infancy significant amount of work has already been done in this field. In 1985 Deutsch gave a quantum algorithm for a very simple problem and showed that it worked better than a classical one. In his problem we are given a blackbox which computes a simple function. The box takes two bits as input and outputs two bits. It implements one of the four following functions from F2 F2: 2 → 2 f1(x, y)=(x, y) f2(x, y)=(x, y + 1) f3(x, y)=(x, x + y) f4(x, y)=(x, x + y + 1) The goal is to determine whether the function implemented by the box is in the set f , f or f , f . On a classical computer it will take 2 queries to find the answer to { 1 2} { 3 4} the question. But on a quantum computer it will take just 1 query to find the answer. In 1992 Deutsch and Jozsa (7) gave the first non-trivial quantum algorithm. Their problem is the generalization of the above problem. In this problem we are given a function n f : F F2. The goal is to find out how many queries to the box are needed in the 2 → 8 worst case to determine whether the function is a constant or balanced (0 on one half of the domain states and 1 on the other half). For the classical case the problem has exponential query complexity. But for the quantum case as was shown by Deutsch and Jozsa only a single query is needed. Motivated by this Berstein and Vazirani (3) gave a quantum algorithm which worked significantly faster (super-polynomial speedup) than the best classical algorithm. The Berstein Vazirani problem has a non-recursive and a n recursive version. In the non-recursive version we are given a function f : F F2. 2 → The function is fs (x) = x.s for some unknown s where x.s denotes the dot product. The goal is to find s. In the classical case the query complexity is at least n whereas for the quantum case only a single query is needed. The recursive version is a little more complex than the non-recursive one. Simon (24) then gave a quantum algorithm which performed exponentially faster than its classical counterpart. In Simon’s problem we are given an integer m 1 and a function f : Fm R, where R is a finite set. We also ≥ 2 → know that there exists a nonzero element s Fm such that for all x, y Fm f (x) = f (y) ∈ 2 ∈ 2 if and only if x=y or x = y + s. The goal is to find s. All these paved the way for Shor’s celebrated paper on factoring and discrete log (23). There is no known efficient classical algorithm for factoring a large number n. The security and robustness of RSA public-key cryptosystem is based on this fact. Shor showed in his paper that given a quantum computer factoring of a number n can be done in polynomial time. Simon’s and Shor’s algorithms laid the framework for a very important problem of quantum computation known as the Hidden Subgroup Problem. The problem can be stated as follows. Problem. (Hidden Subgroup Problem) Let G be a finite group and H a subgroup. Given a black-box function f : G S which is constant on (left)-cosets gH of H and takes → different values for different cosets, determine a set of generators for H. The Hidden Subgroup Problem is at the heart of Shor’s and Simon’s problems. In Shor’s problem given n let a be any integer such gcd(a, n)=1. Let r be the order of 9 a Z∗. The goal is to find r. In this problem G is Z and H is rZ. For practical purposes ∈ n l x we can work with Zm where m = 2 > n. The function f is given by f (x) = a + nZ. Simon’s problem reduces to a special case of the Hidden Subgroup Problem if we take G to be Fm and H = 0, s . 2 { } Efficient quantum algorithms for The Hidden Subgroup Problem are known for abelian groups thanks to the efforts of Shor, Simon, Deutsch etc (7). The non-abelian case still poses a challenge. Some positive results have been obtained for Dihedral group by M. Ettinger and P. Hoyer (8), G. Kuperberg (19), Regev (21), Dave Bacon, Andrew M. Childs and Wim van Dam (1). Efficient quantum algorithms have been obtained for semidirect product of the forms Zpr ⋊ Zp by Yoshifumi Inui and Francois r1 r2 rn Le Gall (15), for ZN ⋊ Zp where N = p p ...p and prime p does not divide pj 1 by 1 2 n − Dong Pyo Chi, Jeong San Kim and Soojoon Lee (5), for some metacyclic groups and r all groups of the form Zp ⋊ Zp for any prime p and a fixed r by Dave Bacon, Andrew M. n Childs and Wim van Dam (1), for Wreath product of the form Z Z2 by M. Rotteler and T. 2 ≀ Beth (4), for some solvable groups by K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen (10) and for groups with small commutators by G. Ivanyos, F. Magniez, and M. Santha (17). Unfortunately a unified solution has still not been found. An efficient solution of this problem would have wide implications in the field of theoretical computer science. For example it would most likely solve some classical problems which are neither NP complete nor are in P. A solution would also imply a − solution for Graph Isomorphism problem which is a long standing problem in computer science.