DOCSLIB.ORG

Signature Redacted a U Th O R

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Signature Redacted a U Th O R Three Complexity Classification Questions at the ITUTE Quantum/Classical Boundary OFTENO by OCT 0 3 2019 Daniel Grier LIBRARIES ARCHIVES Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2019 @ Massachusetts Institute of Technology 2019. All rights reserved. Signature redacted A u th o r ................................................................ Department of Electrical Engineering and Computer Science August 30, 2019 redacted Certified by.. Signature Scott Aaronson Professor of Computer Science, The University of Texas at Austin Thesis Supervisor Signature redacted A ccepted by ............... ............... Ac b /Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Three Complexity Classification Questions at the Quantum/Classical Boundary by Daniel Grier Submitted to the Department of Electrical Engineering and Computer Science on August 30, 2019, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract A central promise of quantum computers is their ability to solve some problems dramatically more efficiently than their classical counterparts. Thus, to understand feasible computation in our physical world, we must turn to quantum rather than classical complexity theory. That said, classical complexity theory has a long and successful history of developing tools and techniques to analyze the power of various computing models. Can we use classical complexity theory to aid our understanding of the quantum world? As it turns out, the answer is yes. There is actually a very fruitful connection between quantum and classical complexity theory, each field informing the other. We will add to this perspective through the lens of classification-attempts to categorize all variations of the object of study as thoroughly and completely as possible. First, we will show that every regular language has quantum query complexity E(1), 5(Vi), or O(n). Combining quantum query complexity with these fundamental classical languages not only reveals new structure in these languages, but also leads to a generalization of Grover's famous quantum search algorithm. Second, we will discuss the complexity of computing the permanent over various matrix groups. In particular, this will show that computing the permanent of a unitary matrix is #P-hard. The theorem statement is classical, and yet, the proof is almost entirely the result of exploiting well-known theorems in quantum linear optics. Finally, we give a complete classification of Clifford operations over qubits. Although the Clifford operations are classically simulable, they also exhibit distinct quantum behavior, making them a particularly interesting gate set at the quantum/classical boundary. Thesis Supervisor: Scott Aaronson Title: Professor of Computer Science, The University of Texas at Austin 3 4 Acknowledgments I should probably start my acknowledgments by stating how inadequate they will be in expressing my gratitude for all the people who have contributed to my graduate school experience. My advisor Scott Aaronson might refer to that sentiment as "forehead- bangingly obvious," but sometimes you just have to set the record straight. Of course, Scott is on the top of my list when it comes to people to thank. For all my years, Scott has been an academic heavyweight role model. He bleeds interesting research ideas, maintains a highly-successful blog, has two rambunctious children, all while being one of the most supportive advisors I've seen in the business. Even when forces drew him towards UT Austin during my third year, he fought to keep me financially supported at MIT as long as possible. I will always be grateful to have had Scott as an advisor, and I hope to never lose his infectious passion for science. On the topic of advisors, I would also like to thank my undergraduate advisor Stephen Fenner at the University of South Carolina for introducing me to computa- tional complexity theory, and more generally to the world of research in theoretical computer science. It seems rather unlikely that I'd be sitting here typing these ac- knowledgments in this thesis were it not for him giving me some problem about the complexity of a strange combinatorial game many years ago. I will always appreciate his patience and encouragement during these early research years, and he remains a friend and collaborator to this day. There are many faculty members at MIT who helped me somewhere along the way. First, I would like to thank both Aram Harrow and Ryan Williams for serving on my thesis committee. I would also like to thank those professors who helped to fund my studies through a TA position after Scott moved. Amongst them, I would especially like to mention Ronitt Rubinfeld for her support during this time. Finally, I am incredibly grateful for the kindness of Srini Devadas, who single-handedly made it possible for me to do research full-time for my final semester. I hope the work I did during that time would make him proud. Compared to many of my peers, I have had relatively few distinct collaborators 5 during my time at MIT, which, I suppose, means that I am all the more grateful for the time I've worked with them: Scott Aaronson, Adam Bouland, Matt Coudron, Stephen Fenner, Ramis Movassagh, Luke Schaeffer, Aarthi Sundaram, and John Watrous. By far, my closest collaborator was the inimitable Luke Schaeffer. Almost no day went by during my time at MIT that didn't have some sort of research check-in with Luke, often lasting two, three, four hours. He is on all but a few of my published papers, including all three I will discuss in this thesis. I can't really imagine what grad school would have been like without Luke, but I'm guessing it would have been less fun and less successful. I'd also like to thank the entire theory group at MIT. It was an intensely social group of people that knew how to play a mean game of soccer, sing their hearts out during karaoke, and were even pretty good at math if you gave them the chance. I have no doubt that many of them will go on (or have already gone on) to be world class researchers, and I feel privileged to have been part of the group. Purely out of fear of leaving anybody out, I will not try to enumerate all of my theory friends, but needless to say... I will miss you guys. I'd also like to thank the various organizations I was part of during my time at MIT that made it that much more enjoyable. First, to the cycling club team at MIT, consider this my formal apology for not training harder. I didn't deserve to be part of such a fun and successful team, but I sure did enjoy being along for the ride. I'd also like to thank my hockey team, who took a chance on me when I messaged them out of the blue asking to join. Sports have always been a large part of my life, and I'm really lucky to have have been part of two great teams. Finally, I would like to thank the EECS Communications Lab for hiring me on as a writing advisor. I would especially like to thank my manager Alison Takemura, who not only helped me with my own communication tasks, but also helped me think about helping others as well. Finally, I would like to thank my entire family-Jon, Marion, and Ben Grier- and my girlfriend Amy for the incredible support and good times over the years. To say that they've helped me become a better and happier person would just be forehead-bangingly obvious. 6 Previously Published Material The three main results of this thesis are based on previous papers. Chapter 3 is from a conference paper in joint work with Scott Aaronson and Luke Schaeffer [7]. Chapter 4 and Chapter 5 are based on a conference and workshop paper, respectively, with Luke Schaeffer [49, 50]. 7 8 Contents 1 Introduction 13 1.1 R esults .................................. 15 1.1.1 Quantum query complexity of regular languages ....... 15 1.1.2 Hardness results for matrix permanents ............ 16 1.1.3 Classification of Clifford operations .............. 17 1.2 Related Work .............................. 17 2 Background 21 2.1 Quantum Computers .......................... 21 2.1.1 States .............................. 22 2.1.2 Operations. ........................... 23 2.1.3 Measurement .......................... 24 2.2 Query Complexity ........................... 24 2.2.1 Relationships ................ .......... 26 2.3 Complexity Classes ........................... 27 2.3.1 Decision classes ......................... 27 2.3.2 Function classes ............... .......... 28 3 A Quantum Query Complexity 'richotomy for Regular Languages 33 3.1 Results Overview ............................ 36 3.1.1 Proof Techniques ..... ................... 38 3.1.2 Related Work .......................... 38 3.2 Background ..... 40 9 3.2.1 Regular languages ... ............ .......... 40 3.2.2 Query complexity with non-binary alphabets ... ....... 43 3.3 Applications of Star-free Algorithm .... ...... ...... ... 44 3.3.1 Dynamic AND-OR .... ........ ........ .... 44 3.3.2 Bounded Dyck language .. ........... ........ 45 3.3.3 Addition ...... ......... ........ ....... 45 3.3.4 Length-2 Word Break ...... ............ ..... 46 3.3.5 Grid problems .............. ............. 48 3.4 Formal Statement of Trichotomy Theorem ............... 49 3.4.1 Flattening .................... ......... 50 3.4.2 Trichotomy Theorem ................... .... 53 3.4.3 Equivalence of algebraic and regular expression definitions .. 54 3.4.4 Monotonic query complexity .. ............ ..... 56 3.4.5 Structure of the proof ....... ............ ... 57 3.5 Upper Bounds . ......................... ..... 57 3.5.1 Proof techniques .. ....................... 58 3.5.2 O(Vi) algorithm for star-free languages ............ 64 3.6 Dichotomy Theorems ............ ............... 67 3.7 Lower Bounds ........ ............ ........... 71 3.8 Context-Free Languages ..... .................... 74 3.8.1 Context-free languages do not obey the trichotomy .....
Load more

Recommended publications
  • Varying Constants, Gravitation and Cosmology
    Varying constants, Gravitation and Cosmology Jean-Philippe Uzan Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Universit´ePierre et Marie Curie, 98 bis bd Arago, 75014 Paris (France) and Department of Mathematics and Applied Mathematics, Cape Town University, Rondebosch 7701 (South Africa) and National Institute for Theoretical Physics (NITheP), Stellenbosch 7600 (South Africa). email: [email protected] http//www2.iap.fr/users/uzan/ September 29, 2010 Abstract Fundamental constants are a cornerstone of our physical laws. Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to mat- ter. This will induce a violation of the universality of free fall. It is thus of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy. We thus detail the relations between the constants, the tests of the local posi- tion invariance and of the universality of free fall. We then review the main experimental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, Solar system observations, meteorites dating, quasar absorption spectra, stellar physics, pul- sar timing, the cosmic microwave background and big bang nucleosynthesis. At each step we arXiv:1009.5514v1 [astro-ph.CO] 28 Sep 2010 describe the basics of each system, its dependence with respect to the constants, the known systematic effects and the most recent constraints that have been obtained. We then describe the main theoretical frameworks in which the low-energy constants may actually be varying and we focus on the unification mechanisms and the relations between the variation of differ- ent constants.
    [Show full text]
  • Limits on Efficient Computation in the Physical World
    Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Bachelor of Science (Cornell University) 2000 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Umesh Vazirani, Chair Professor Luca Trevisan Professor K. Birgitta Whaley Fall 2004 The dissertation of Scott Joel Aaronson is approved: Chair Date Date Date University of California, Berkeley Fall 2004 Limits on Efficient Computation in the Physical World Copyright 2004 by Scott Joel Aaronson 1 Abstract Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Doctor of Philosophy in Computer Science University of California, Berkeley Professor Umesh Vazirani, Chair More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational complexity theory to help determine which are which. In the first part of the thesis, I attack the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known. In partic- ular, any quantum algorithm that solves the collision problem—that of deciding whether a sequence of n integers is one-to-one or two-to-one—must query the sequence Ω n1/5 times.
    [Show full text]
  • The Complexity Zoo
    The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/.
    [Show full text]
  • Ab Initio Investigations of the Intrinsic Optical Properties of Germanium and Silicon Nanocrystals
    Ab initio Investigations of the Intrinsic Optical Properties of Germanium and Silicon Nanocrystals D i s s e r t a t i o n zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von Dipl.-Phys. Hans-Christian Weißker geboren am 12. Juli 1971 in Greiz Gutachter: 1. Prof. Dr. Friedhelm Bechstedt, Jena. 2. Dr. Lucia Reining, Palaiseau, Frankreich. 3. Prof. Dr. Victor Borisenko, Minsk, Weißrußland. Tag der letzten Rigorosumsprüfung: 12. August 2004. Tag der öffentlichen Verteidigung: 19. Oktober 2004. Why is warrant important to knowledge? In part because true opinion might be reached by arbitrary, unreliable means. Peter Railton1 1Explanation and Metaphysical Controversy, in P. Kitcher and W.C. Salmon (eds.), Scientific Explanation, Vol. 13, Minnesota Studies in the Philosophy of Science, Minnesota, 1989. Contents 1 Introduction 7 2 Theoretical Foundations 13 2.1 Density-Functional Theory . 13 2.1.1 The Hohenberg-Kohn Theorem . 13 2.1.2 The Kohn-Sham scheme . 15 2.1.3 Transition to system without spin-polarization . 17 2.1.4 Physical interpretation by comparison to Hartree-Fock . 17 2.1.5 LDA and LSDA . 19 2.1.6 Forces in DFT . 19 2.2 Excitation Energies . 20 2.2.1 Quasiparticles . 20 2.2.2 Self-energy corrections . 22 2.2.3 Excitation energies from total energies . 25 QP 2.2.3.1 Conventional ∆SCF method: Eg = I − A . 25 QP 2.2.3.2 Discussion of the ∆SCF method Eg = I − A . 25 2.2.3.3 ∆SCF with occupation constraint .
    [Show full text]
  • Quantum Computing : a Gentle Introduction / Eleanor Rieffel and Wolfgang Polak
    QUANTUM COMPUTING A Gentle Introduction Eleanor Rieffel and Wolfgang Polak The MIT Press Cambridge, Massachusetts London, England ©2011 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email [email protected] This book was set in Syntax and Times Roman by Westchester Book Group. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Rieffel, Eleanor, 1965– Quantum computing : a gentle introduction / Eleanor Rieffel and Wolfgang Polak. p. cm.—(Scientific and engineering computation) Includes bibliographical references and index. ISBN 978-0-262-01506-6 (hardcover : alk. paper) 1. Quantum computers. 2. Quantum theory. I. Polak, Wolfgang, 1950– II. Title. QA76.889.R54 2011 004.1—dc22 2010022682 10987654321 Contents Preface xi 1 Introduction 1 I QUANTUM BUILDING BLOCKS 7 2 Single-Qubit Quantum Systems 9 2.1 The Quantum Mechanics of Photon Polarization 9 2.1.1 A Simple Experiment 10 2.1.2 A Quantum Explanation 11 2.2 Single Quantum Bits 13 2.3 Single-Qubit Measurement 16 2.4 A Quantum Key Distribution Protocol 18 2.5 The State Space of a Single-Qubit System 21 2.5.1 Relative Phases versus Global Phases 21 2.5.2 Geometric Views of the State Space of a Single Qubit 23 2.5.3 Comments on General Quantum State Spaces
    [Show full text]
  • Plasma-Surface Interactions and Impact on Electron Energy Distribution Function
    Plasma-Surface Interactions and Impact on Electron Energy Distribution Function N. Fox-Lyon(a), N. Ning(b), D.B. Graves(b), V. Godyak(c) and G.S. Oehrlein (a) (a) University of Maryland, College Park (b) University of California, Berkeley (c) RF Plasma Consulting, Brookline, MA Motivation Control of plasma distribution functions (DF): Clarify issues when plasma-surface interactions at plasma boundaries change strongly, e.g. non -reactive vs. reactive discharges and/or surfaces 2 Laboratory for Plasma Processing of Materials Motivation OES MS MS Ion MS Ion Wall Ar/HH22 Plasma Probe Erosion Transport C, CHn Redeposition Si, SiHn’ CHm or SiHm’ Different Surface Carbon or Silicon Surface Modified Surface Real-time Layer ellipsometry (intensity, extent) • Ar/H 2 plasma interacting w/ a-C:H • Characterize impact of surface-generated species on f(v,r,t) • Probe, plasma sampling and OES measurements of f(v,r,t) for modified situations • Characterize surface processes using ellipsometry, and compare results with MD simulations of surface processes • Interpret data by developing overall model 3 Laboratory for Plasma Processing of Materials Wrinkling-induced Surface Roughness Formation Wrinkle wavelength and amplitude calculated using measured using damaged properties vs. AFM derived properties R. L. Bruce, et al., J. Appl. Phys. 107, 084310 (2010) 4 Helium Plasma Pre-Treatment Possible approach: Plasma pretreatment (PPT) UV plasma radiation reduces plane strain modulus E s (chain-scissioning) and densifies without stress before actual PE Helium
    [Show full text]
  • Quantum Lower Bound for Recursive Fourier Sampling
    Quantum Lower Bound for Recursive Fourier Sampling Scott Aaronson Institute for Advanced Study, Princeton [email protected] Abstract One of the earliest quantum algorithms was discovered by Bernstein and Vazirani, for a problem called Recursive Fourier Sampling. This paper shows that the Bernstein-Vazirani algorithm is not far from optimal. The moral is that the need to “uncompute” garbage can impose a fundamental limit on efficient quantum computation. The proof introduces a new parameter of Boolean functions called the “nonparity coefficient,” which might be of independent interest. Like a classical algorithm, a quantum algorithm can solve problems recursively by calling itself as a sub- routine. When this is done, though, the algorithm typically needs to call itself twice for each subproblem to be solved. The second call’s purpose is to uncompute ‘garbage’ left over by the first call, and thereby enable interference between different branches of the computation. Of course, a factor of 2 increase in running time hardly seems like a big deal, when set against the speedups promised by quantum computing. The problem is that these factors of 2 multiply, with each level of recursion producing an additional factor. Thus, one might wonder whether the uncomputing step is really necessary, or whether a cleverly designed algorithm might avoid it. This paper gives the first nontrivial example in which recursive uncomputation is provably necessary. The example concerns a long-neglected problem called Recursive Fourier Sampling (henceforth RFS), which was introduced by Bernstein and Vazirani [5] in 1993 to prove the first oracle separation between BPP and BQP. Many surveys on quantum computing pass directly from the Deutsch-Jozsa algorithm [8] to the dramatic results of Simon [14] and Shor [13], without even mentioning RFS.
    [Show full text]
  • Limits on Efficient Computation in the Physical World by Scott Joel
    Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Bachelor of Science (Cornell University) 2000 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY Committee in charge: Professor Umesh Vazirani, Chair Professor Luca Trevisan Professor K. Birgitta Whaley Fall 2004 The dissertation of Scott Joel Aaronson is approved: Chair Date Date Date University of California, Berkeley Fall 2004 Limits on Efficient Computation in the Physical World Copyright 2004 by Scott Joel Aaronson 1 Abstract Limits on Efficient Computation in the Physical World by Scott Joel Aaronson Doctor of Philosophy in Computer Science University of California, Berkeley Professor Umesh Vazirani, Chair More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational complexity theory to help determine which are which. In the first part of the thesis, I attack the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known. In partic- ular, any quantum algorithm that solves the collision problem—that of deciding whether a sequence of n integers is one-to-one or two-to-one—must query the sequence Ω n1/5 times.
    [Show full text]
  • Algebraic Modelling of Quantum Mechanical Equations in the Finite- and Infinite-Dimensional Hilbert Spaces
    Algebraic modelling of quantum mechanical equations in the finite- and infinite-dimensional Hilbert spaces Megill, Norman Dwight Doctoral thesis / Disertacija 2011 Degree Grantor / Ustanova koja je dodijelila akademski / stručni stupanj: University of Zagreb, Faculty of Science / Sveučilište u Zagrebu, Prirodoslovno-matematički fakultet Permanent link / Trajna poveznica: https://urn.nsk.hr/urn:nbn:hr:217:788633 Rights / Prava: In copyright Download date / Datum preuzimanja: 2021-09-26 Repository / Repozitorij: Repository of Faculty of Science - University of Zagreb PHYSICS DEPARTMENT OF THE FACULTY OF SCIENCE Norman Dwight Megill ALGEBRAIC MODELLING OF QUANTUM MECHANICAL EQUATIONS IN THE FINITE- AND INFINITE-DIMENSIONAL HILBERT SPACES DOCTORAL THESIS Zagreb, 2011 FIZICKIˇ ODSJEK PRIRODOSLOVNO-MATEMATICKOGˇ FAKULTETA Norman Dwight Megill ALGEBARSKO MODELIRANJE KVANTNO-MEHANICKIHˇ JEDNADŽBI U KONACNOˇ I BESKONACNOˇ DIMENZIONALNIM HILBERTOVIM PROSTORIMA DOKTORSKI RAD Zagreb, 2011 PHYSICS DEPARTMENT OF THE FACULTY OF SCIENCE Norman Dwight Megill ALGEBRAIC MODELLING OF QUANTUM MECHANICAL EQUATIONS IN THE FINITE- AND INFINITE-DIMENSIONAL HILBERT SPACES DOCTORAL THESIS Supervisor: Prof. Mladen Paviciˇ c´ Zagreb, 2011 FIZICKIˇ ODSJEK PRIRODOSLOVNO-MATEMATICKOGˇ FAKULTETA Norman Dwight Megill ALGEBARSKO MODELIRANJE KVANTNO-MEHANICKIHˇ JEDNADŽBI U KONACNOˇ I BESKONACNOˇ DIMENZIONALNIM HILBERTOVIM PROSTORIMA DOKTORSKI RAD Mentor: Prof. Mladen Paviciˇ c´ Zagreb, 2011 Committees This thesis, entitled Algebraic Modelling of Quantum Mechanical Equations in the Finite- and Infinite-Dimensional Hilbert Spaces and written by Norman Dwight Megill, has been approved for the Department of Physics of the Faculty of Science of the University of Zagreb by the following committee: Prof. Hrvoje Šikic´ (Department of Mathematics, Faculty of Science, University of Zagreb), Prof. Mladen Paviciˇ c´ (Chair of Physics, Faculty of Civil Engineering, University of Zagreb), Prof.
    [Show full text]
  • Quantum Computing: Lecture Notes
    Quantum Computing: Lecture Notes Ronald de Wolf Preface These lecture notes were formed in small chunks during my “Quantum computing” course at the University of Amsterdam, Feb-May 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 45 minutes, with an additional 45-minute lecture for exercises and × homework. The first half of the course (Chapters 1–7) covers quantum algorithms, the second half covers quantum complexity (Chapters 8–9), stuff involving Alice and Bob (Chapters 10–13), and error-correction (Chapter 14). A 15th lecture about physical implementations and general outlook was more sketchy, and I didn’t write lecture notes for it. These chapters may also be read as a general introduction to the area of quantum computation and information from the perspective of a theoretical computer scientist. While I made an effort to make the text self-contained and consistent, it may still be somewhat rough around the edges; I hope to continue polishing and adding to it. Comments & constructive criticism are very welcome, and can be sent to [email protected] Attribution and acknowledgements Most of the material in Chapters 1–6 comes from the first chapter of my PhD thesis [71], with a number of additions: the lower bound for Simon, the Fourier transform, the geometric explanation of Grover. Chapter 7 is newly written for these notes, inspired by Santha’s survey [62]. Chapters 8 and 9 are largely new as well. Section 3 of Chapter 8, and most of Chapter 10 are taken (with many changes) from my “quantum proofs” survey paper with Andy Drucker [28].
    [Show full text]
  • Equiangular Vectors Approach to Mutually Unbiased Bases
    Entropy 2013, 15, 1726-1737; doi:10.3390/e15051726 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Equiangular Vectors Approach to Mutually Unbiased Bases Maurice R. Kibler 1;2;3 1 Faculte´ des Sciences et Technologies, Universite´ de Lyon, 37 rue du repos, 69361 Lyon, France 2 Departement´ de Physique, Universite´ Claude Bernard Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France 3 Groupe Theorie,´ Institut de Physique Nucleaire,´ CNRS/IN2P3, 4 rue Enrico Fermi, 69622 Villeurbanne, France; E-Mail: [email protected]; Tel: 33(0)472448235, FAX: 33(0)472448004 Received: 26 April 2013 / Accepted: 6 May 2013 / Published: 8 May 2013 Abstract: Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the 1 other equals d . The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d + 1) vectors in the d2-dimensional 2 2 space Cd (without a modulus for the inner product). The transformation from Cd to Cd corresponds to the passage from equiangular lines to equiangular vectors.
    [Show full text]
  • Quantum Complexity, Relativized Worlds, Oracle Separations
    Quantum Complexity, Relativized Worlds, and Oracle Separations by Dimitrios I. Myrisiotis A thesis presented to the Department of Mathematics, of the School of Science, of the National and Kapodistrian University of Athens, in fulfillment of the thesis requirement for the degree of Master of Science in Logic, and the Theory of Algorithms and Computation. Athens, Greece, 2016 c Dimitrios I. Myrisiotis 2016 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii H Παρούσα Diπλωματική ErgasÐa ekpoνήθηκε sto plaÐsio twn spoud¸n gia thn απόκτηση tou Metαπτυχιακού Dipl¸matoc EidÐkeushc sth Loγική kai JewrÐa AlgorÐjmwn kai Upologiσμού pou aponèmei to Τμήμα Majhmatik¸n tou Εθνικού kai Kapodisτριακού PanepisthmÐou Ajhn¸n EgkrÐjhke thn από thn Exetasτική Epiτροπή apoτελούμενη από touc: Onomatep¸numo BajmÐda Upoγραφή 1. 2. 3. 4. iii iv Notational Conventions True Denotes the notion of the logical truth. False Denotes the notion of the logical falsity. Yes It is used, instead of True, to denote the acceptance, of some element, by some algorithm that produces the answer to some decision problem. No It is used, instead of False, to denote the rejection, of some element, by some algorithm that produces the answer to some decision problem. :P Expresses the negation of the logical proposition P, that is, the assertion “it is not the case that P.” P ^ Q Expresses the conjuction of the logical proposition P with the logical proposition Q.
    [Show full text]