Introduction to Quantum Complexity Theory
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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. -
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. -
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/. -
On the Power of Quantum Fourier Sampling
On the Power of Quantum Fourier Sampling Bill Fefferman∗ Chris Umansy Abstract A line of work initiated by Terhal and DiVincenzo [TD02] and Bremner, Jozsa, and Shepherd [BJS10], shows that restricted classes of quantum computation can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov [AA13] take this further by considering a distribution that can be sampled ef- ficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled classically within bounded total variation distance, unless the PH collapses. In this work we use Quantum Fourier Sampling to construct a class of distributions that can be sampled exactly by a quantum computer. We then argue that these distributions cannot be approximately sampled classically, unless the PH collapses, under variants of the Aaronson-Arkhipov conjectures. In particular, we show a general class of quantumly sampleable distributions each of which is based on an “Efficiently Specifiable” polynomial, for which a classical approximate sampler implies an average-case approximation. This class of polynomials contains the Permanent but also includes, for example, the Hamiltonian Cycle polynomial, as well as many other familiar #P-hard polynomials. Since our distribution likely requires the full power of universal quantum computation, while the Aaronson-Arkhipov distribution uses only linear optical quantum computation with noninteracting bosons, why is this result interesting? We can think of at least three reasons: 1. Since the conjectures required in [AA13] have not yet been proven, it seems worthwhile to weaken them as much as possible. -
Entangled Many-Body States As Resources of Quantum Information Processing
ENTANGLED MANY-BODY STATES AS RESOURCES OF QUANTUM INFORMATION PROCESSING LI YING A thesis submitted for the Degree of Doctor of Philosophy CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. LI YING 23 July 2013 Acknowledgments I am very grateful to have spent about four years at CQT working with Leong Chuan Kwek. He always brings me new ideas in science and has helped me to establish good collaborative relationships with other scien- tists. Kwek helped me a lot in my life. I am also very grateful to Simon C. Benjamin. He showd me how to do high quality researches in physics. Simon also helped me to improve my writing and presentation. I hope to have fruitful collaborations in the near future with Kwek and Simon. For my project about the ground-code MBQC (Chapter2), I am thank- ful to Tzu-Chieh Wei, Daniel E. Browne and Robert Raussendorf. In one afternoon, Tzu-Chieh showed me the idea of his recent paper in this topic in the quantum cafe, which encouraged me to think about the ground- code MBQC. Dan and Robert have a high level of comprehension on the subject of the MBQC. And we had some very interesting discussions and communications. I am grateful to Sean D. -
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 . -
QIP 2010 Tutorial and Scientific Programmes
QIP 2010 15th – 22nd January, Zürich, Switzerland Tutorial and Scientific Programmes asymptotically large number of channel uses. Such “regularized” formulas tell us Friday, 15th January very little. The purpose of this talk is to give an overview of what we know about 10:00 – 17:10 Jiannis Pachos (Univ. Leeds) this need for regularization, when and why it happens, and what it means. I will Why should anyone care about computing with anyons? focus on the quantum capacity of a quantum channel, which is the case we understand best. This is a short course in topological quantum computation. The topics to be covered include: 1. Introduction to anyons and topological models. 15:00 – 16:55 Daniel Nagaj (Slovak Academy of Sciences) 2. Quantum Double Models. These are stabilizer codes, that can be described Local Hamiltonians in quantum computation very much like quantum error correcting codes. They include the toric code This talk is about two Hamiltonian Complexity questions. First, how hard is it to and various extensions. compute the ground state properties of quantum systems with local Hamiltonians? 3. The Jones polynomials, their relation to anyons and their approximation by Second, which spin systems with time-independent (and perhaps, translationally- quantum algorithms. invariant) local interactions could be used for universal computation? 4. Overview of current state of topological quantum computation and open Aiming at a participant without previous understanding of complexity theory, we will discuss two locally-constrained quantum problems: k-local Hamiltonian and questions. quantum k-SAT. Learning the techniques of Kitaev and others along the way, our first goal is the understanding of QMA-completeness of these problems. -
Research Statement Bill Fefferman, University of Maryland/NIST
Research Statement Bill Fefferman, University of Maryland/NIST Since the discovery of Shor's algorithm in the mid 1990's, it has been known that quan- tum computers can efficiently solve integer factorization, a problem of great practical relevance with no known efficient classical algorithm [1]. The importance of this result is impossible to overstate: the conjectured intractability of the factoring problem provides the basis for the se- curity of the modern internet. However, it may still be a few decades before we build universal quantum computers capable of running Shor's algorithm to factor integers of cryptographically relevant size. In addition, we have little complexity theoretic evidence that factoring is com- putationally hard. Consequently, Shor's algorithm can only be seen as the first step toward understanding the power of quantum computation, which has become one of the primary goals of theoretical computer science. My research focuses not only on understanding the power of quantum computers of the indefinite future, but also on the desire to develop the foundations of computational complexity to rigorously analyze the capabilities and limitations of present-day and near-term quantum devices which are not yet fully scalable quantum computers. Furthermore, I am interested in using these capabilities and limitations to better understand the potential for cryptography in a fundamentally quantum mechanical world. 1 Comparing quantum and classical nondeterministic computation Starting with the foundational paper of Bernstein and Vazirani it has been conjectured that quantum computers are capable of solving problems whose solutions cannot be found, or even verified efficiently on a classical computer [2]. -
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 -
Manipulating Entanglement Sudden Death in Two Coupled Two-Level
Manipulating entanglement sudden death in two coupled two-level atoms interacting off-resonance with a radiation field: an exact treatment Gehad Sadieka,1, 2 Wiam Al-Drees,3 and M. Sebaweh Abdallahb4 1Department of Applied Physics and Astronomy, University of Sharjah, Sharjah 27272, UAE 2Department of Physics, Ain Shams University, Cairo 11566, Egypt 3Department of Physics and Astronomy, King Saud University, Riyadh 11451, Saudi Arabia 4Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia arXiv:1911.08208v1 [quant-ph] 19 Nov 2019 a Corresponding author: [email protected] b Deceased. 1 Abstract We study a model of two coupled two-level atoms (qubits) interacting off-resonance (at non-zero detuning) with a single mode radiation field. This system is of special interest in the field of quan- tum information processing (QIP) and can be realized in electron spin states in quantum dots or Rydberg atoms in optical cavities and superconducting qubits in linear resonators. We present an exact analytical solution for the time evolution of the system starting from any initial state. Uti- lizing this solution, we show how the entanglement sudden death (ESD), which represents a major threat to QIP, can be efficiently controlled by tuning atom-atom coupling and non-zero detuning. We demonstrate that while one of these two system parameters may not separately affect the ESD, combining the two can be very effective, as in the case of an initial correlated Bell state. However in other cases, such as a W-like initial state, they may have a competing impacts on ESD. Moreover, their combined effect can be used to create ESD in the system as in the case of an anti-correlated initial Bell state. -
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 -
Quantum Information Processing II: Implementations
Lecture Quantum Information Processing II: Implementations spring term (FS) 2017 Lectures & Exercises: Andreas Wallraff, Christian Kraglund Andersen, Christopher Eichler, Sebastian Krinner Please take a seat offices:in HPFthe frontD 8/9 (AW),center D part14 (CKA), of the D lecture2 (SK), D hall 17 (SK) ifETH you Hoenggerberg do not mind. email: [email protected] web: www.qudev.ethz.ch moodle: https://moodle-app2.let.ethz.ch/course/view.php?id=3151 What is this lecture about? Introduction to experimental realizations of systems for quantum information processing (QIP) Questions: • How can one use quantum physics to process and communicate information more efficiently than using classical physics only? ► QIP I: Concepts • How does one design, build and operate physical systems for this purpose? ► QIP II: Implementations Is it really interesting? Goals of the Lecture • generally understand how quantum physics is used for – quantum information processing – quantum communication – quantum simulation – quantum sensing • know basic implementations of important quantum algorithms – Deutsch-Josza Algorithm – searching in a database (Grover algorithm) Concept Question You are searching for a specific element in an unsorted database with N entries (e.g. the phone numbers in a regular phone book). How many entries do you have to look at (on average) until you have found the item you were searching for? 1. N^2 2. N Log(N) 3. N 4. N/2 5. Sqrt(N) 6. 1 Goals of the Lecture • generally understand how quantum physics is used for – quantum information