Quantum Computing: Lecture Notes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Theory and Application of Fermi Pseudo-Potential in One Dimension
Theory and application of Fermi pseudo-potential in one dimension The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Wu, Tai Tsun, and Ming Lun Yu. 2002. “Theory and Application of Fermi Pseudo-Potential in One Dimension.” Journal of Mathematical Physics 43 (12): 5949–76. https:// doi.org/10.1063/1.1519940. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41555821 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA CERN-TH/2002-097 Theory and application of Fermi pseudo-potential in one dimension Tai Tsun Wu Gordon McKay Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A., and Theoretical Physics Division, CERN, Geneva, Switzerland and Ming Lun Yu 41019 Pajaro Drive, Fremont, California, U.S.A.∗ Abstract The theory of interaction at one point is developed for the one-dimensional Schr¨odinger equation. In analog with the three-dimensional case, the resulting interaction is referred to as the Fermi pseudo-potential. The dominant feature of this one-dimensional problem comes from the fact that the real line becomes disconnected when one point is removed. The general interaction at one point is found to be the sum of three terms, the well-known delta-function potential arXiv:math-ph/0208030v1 21 Aug 2002 and two Fermi pseudo-potentials, one odd under space reflection and the other even. -
Security of Quantum Key Distribution with Entangled Qutrits
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Security of Quantum Key Distribution with Entangled Qutrits. Thomas Durt,1 Nicolas J. Cerf,2 Nicolas Gisin,3 and Marek Zukowski˙ 4 1 Toegepaste Natuurkunde en Fotonica, Theoeretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium 2 Ecole Polytechnique, CP 165, Universit´e Libre de Bruxelles, 1050 Brussels, Belgium 3 Group of Applied Physics - Optique, Universit´e de Gen`eve, 20 rue de l’Ecole de M´edecine, Gen`eve 4, Switzerland, 4Instytut Fizyki Teoretycznej i Astrofizyki Uniwersytet, Gda´nski, PL-80-952 Gda´nsk, Poland July 2002 Abstract The study of quantum cryptography and quantum non-locality have traditionnally been based on two-level quantum systems (qubits). In this paper we consider a general- isation of Ekert's cryptographic protocol[20] where qubits are replaced by qutrits. The security of this protocol is related to non-locality, in analogy with Ekert's protocol. In order to study its robustness against the optimal individual attacks, we derive the infor- mation gained by a potential eavesdropper applying a cloning-based attack. Introduction Quantum cryptography aims at transmitting a random key in such a way that the presence of an eavesdropper that monitors the communication is revealed by disturbances in the transmission of the key (for a review, see e.g. [21]). Practically, it is enough in order to realize a crypto- graphic protocol that the key signal is encoded into quantum states that belong to incompatible bases, as in the original protocol of Bennett and Brassard[4]. -
An Introduction Quantum Computing
An Introduction to Quantum Computing Michal Charemza University of Warwick March 2005 Acknowledgments Special thanks are given to Steve Flammia and Bryan Eastin, authors of the LATEX package, Qcircuit, used to draw all the quantum circuits in this document. This package is available online at http://info.phys.unm.edu/Qcircuit/. Also thanks are given to Mika Hirvensalo, author of [11], who took the time to respond to some questions. 1 Contents 1 Introduction 4 2 Quantum States 6 2.1 CompoundStates........................... 8 2.2 Observation.............................. 9 2.3 Entanglement............................. 11 2.4 RepresentingGroups . 11 3 Operations on Quantum States 13 3.1 TimeEvolution............................ 13 3.2 UnaryQuantumGates. 14 3.3 BinaryQuantumGates . 15 3.4 QuantumCircuits .......................... 17 4 Boolean Circuits 19 4.1 BooleanCircuits ........................... 19 5 Superdense Coding 24 6 Quantum Teleportation 26 7 Quantum Fourier Transform 29 7.1 DiscreteFourierTransform . 29 7.1.1 CharactersofFiniteAbelianGroups . 29 7.1.2 DiscreteFourierTransform . 30 7.2 QuantumFourierTransform. 32 m 7.2.1 Quantum Fourier Transform in F2 ............. 33 7.2.2 Quantum Fourier Transform in Zn ............. 35 8 Random Numbers 38 9 Factoring 39 9.1 OneWayFunctions.......................... 39 9.2 FactorsfromOrder.......................... 40 9.3 Shor’sAlgorithm ........................... 42 2 CONTENTS 3 10 Searching 46 10.1 BlackboxFunctions. 46 10.2 HowNotToSearch.......................... 47 10.3 Single Query Without (Grover’s) Amplification . ... 48 10.4 AmplitudeAmplification. 50 10.4.1 SingleQuery,SingleSolution . 52 10.4.2 KnownNumberofSolutions. 53 10.4.3 UnknownNumberofSolutions . 55 11 Conclusion 57 Bibliography 59 Chapter 1 Introduction The classical computer was originally a largely theoretical concept usually at- tributed to Alan Turing, called the Turing Machine. -
Completeness of the ZX-Calculus
Completeness of the ZX-calculus Quanlong Wang Wolfson College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 2018 Acknowledgements Firstly, I would like to express my sincere gratitude to my supervisor Bob Coecke for all his huge help, encouragement, discussions and comments. I can not imagine what my life would have been like without his great assistance. Great thanks to my colleague, co-auhor and friend Kang Feng Ng, for the valuable cooperation in research and his helpful suggestions in my daily life. My sincere thanks also goes to Amar Hadzihasanovic, who has kindly shared his idea and agreed to cooperate on writing a paper based one his results. Many thanks to Simon Perdrix, from whom I have learned a lot and received much help when I worked with him in Nancy, while still benefitting from this experience in Oxford. I would also like to thank Miriam Backens for loads of useful discussions, advertising for my talk in QPL and helping me on latex problems. Special thanks to Dan Marsden for his patience and generousness in answering my questions and giving suggestions. I would like to thank Xiaoning Bian for always being ready to help me solve problems in using latex and other softwares. I also wish to thank all the people who attended the weekly ZX meeting for many interesting discussions. I am also grateful to my college advisor Jonathan Barrett and department ad- visor Jamie Vicary, thank you for chatting with me about my research and my life. I particularly want to thank my examiners, Ross Duncan and Sam Staton, for their very detailed and helpful comments and corrections by which this thesis has been significantly improved. -
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. -
Arxiv:1803.04114V2 [Quant-Ph] 16 Nov 2018
Learning the quantum algorithm for state overlap Lukasz Cincio,1 Yiğit Subaşı,1 Andrew T. Sornborger,2 and Patrick J. Coles1 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. Short-depth algorithms are crucial for reducing computational error on near-term quantum com- puters, for which decoherence and gate infidelity remain important issues. Here we present a machine-learning approach for discovering such algorithms. We apply our method to a ubiqui- tous primitive: computing the overlap Tr(ρσ) between two quantum states ρ and σ. The standard algorithm for this task, known as the Swap Test, is used in many applications such as quantum support vector machines, and, when specialized to ρ = σ, quantifies the Renyi entanglement. Here, we find algorithms that have shorter depths than the Swap Test, including one that has a constant depth (independent of problem size). Furthermore, we apply our approach to the hardware-specific connectivity and gate sets used by Rigetti’s and IBM’s quantum computers and demonstrate that the shorter algorithms that we derive significantly reduce the error - compared to the Swap Test - on these computers. I. INTRODUCTION hyperparameters, we optimize the algorithm in a task- oriented manner, i.e., by minimizing a cost function that quantifies the discrepancy between the algorithm’s out- Quantum supremacy [1] may be coming soon [2]. put and the desired output. The task is defined by a While it is an exciting time for quantum computing, de- training data set that exemplifies the desired computa- coherence and gate fidelity continue to be important is- tion. -
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. -
On the Tightness of the Buhrman-Cleve-Wigderson Simulation
On the tightness of the Buhrman-Cleve-Wigderson simulation Shengyu Zhang Department of Computer Science and Engineering, The Chinese University of Hong Kong. [email protected] Abstract. Buhrman, Cleve and Wigderson gave a general communica- 1 1 n n tion protocol for block-composed functions f(g1(x ; y ); ··· ; gn(x ; y )) by simulating a decision tree computation for f [3]. It is also well-known that this simulation can be very inefficient for some functions f and (g1; ··· ; gn). In this paper we show that the simulation is actually poly- nomially tight up to the choice of (g1; ··· ; gn). This also implies that the classical and quantum communication complexities of certain block- composed functions are polynomially related. 1 Introduction Decision tree complexity [4] and communication complexity [7] are two concrete models for studying computational complexity. In [3], Buhrman, Cleve and Wigderson gave a general method to design communication 1 1 n n protocol for block-composed functions f(g1(x ; y ); ··· ; gn(x ; y )). The basic idea is to simulate the decision tree computation for f, with queries i i of the i-th variable answered by a communication protocol for gi(x ; y ). In the language of complexity measures, this result gives CC(f(g1; ··· ; gn)) = O~(DT(f) · max CC(gi)) (1) i where CC and DT are the communication complexity and the decision tree complexity, respectively. This simulation holds for all the models: deterministic, randomized and quantum1. It is also well known that the communication protocol by the above simulation can be very inefficient. -
Thesis Is Owed to Him, Either Directly Or Indirectly
UvA-DARE (Digital Academic Repository) Quantum Computing and Communication Complexity de Wolf, R.M. Publication date 2001 Document Version Final published version Link to publication Citation for published version (APA): de Wolf, R. M. (2001). Quantum Computing and Communication Complexity. Institute for Logic, Language and Computation. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 Quantum Computing and Communication Complexity Ronald de Wolf Quantum Computing and Communication Complexity ILLC Dissertation Series 2001-06 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Quantum Computing and Communication Complexity Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magni¯cus prof.dr. -
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/. -
Introductory Topics in Quantum Paradigm
Introductory Topics in Quantum Paradigm Subhamoy Maitra Indian Statistical Institute [[email protected]] 21 May, 2016 Subhamoy Maitra Quantum Information Outline of the talk Preliminaries of Quantum Paradigm What is a Qubit? Entanglement Quantum Gates No Cloning Indistinguishability of quantum states Details of BB84 Quantum Key Distribution Algorithm Description Eavesdropping State of the art: News and Industries Walsh Transform in Deutsch-Jozsa (DJ) Algorithm Deutsch-Jozsa Algorithm Walsh Transform Relating the above two Some implications Subhamoy Maitra Quantum Information Qubit and Measurement A qubit: αj0i + βj1i; 2 2 α; β 2 C, jαj + jβj = 1. Measurement in fj0i; j1ig basis: we will get j0i with probability jαj2, j1i with probability jβj2. The original state gets destroyed. Example: 1 + i 1 j0i + p j1i: 2 2 After measurement: we will get 1 j0i with probability 2 , 1 j1i with probability 2 . Subhamoy Maitra Quantum Information Basic Algebra Basic algebra: (α1j0i + β1j1i) ⊗ (α2j0i + β2j1i) = α1α2j00i + α1β2j01i + β1α2j10i + β1β2j11i, can be seen as tensor product. Any 2-qubit state may not be decomposed as above. Consider the state γ1j00i + γ2j11i with γ1 6= 0; γ2 6= 0. This cannot be written as (α1j0i + β1j1i) ⊗ (α2j0i + β2j1i). This is called entanglement. Known as Bell states or EPR pairs. An example of maximally entangled state is j00i + j11i p : 2 Subhamoy Maitra Quantum Information Quantum Gates n inputs, n outputs. Can be seen as 2n × 2n unitary matrices where the elements are complex numbers. Single input single output quantum gates. Quantum input Quantum gate Quantum Output αj0i + βj1i X βj0i + αj1i αj0i + βj1i Z αj0i − βj1i j0i+j1i j0i−|1i αj0i + βj1i H α p + β p 2 2 Subhamoy Maitra Quantum Information Quantum Gates (contd.) 1 input, 1 output. -
Progress in Satellite Quantum Key Distribution
www.nature.com/npjqi REVIEW ARTICLE OPEN Progress in satellite quantum key distribution Robert Bedington 1, Juan Miguel Arrazola1 and Alexander Ling1,2 Quantum key distribution (QKD) is a family of protocols for growing a private encryption key between two parties. Despite much progress, all ground-based QKD approaches have a distance limit due to atmospheric losses or in-fibre attenuation. These limitations make purely ground-based systems impractical for a global distribution network. However, the range of communication may be extended by employing satellites equipped with high-quality optical links. This manuscript summarizes research and development which is beginning to enable QKD with satellites. It includes a discussion of protocols, infrastructure, and the technical challenges involved with implementing such systems, as well as a top level summary of on-going satellite QKD initiatives around the world. npj Quantum Information (2017) 3:30 ; doi:10.1038/s41534-017-0031-5 INTRODUCTION losses that increase exponentially with distance, greatly limiting Quantum key distribution (QKD) is a relatively new cryptographic the secure key rates that can be achieved over long ranges. primitive for establishing a private encryption key between two For any pure-loss channel with transmittance η, it has been parties. The concept has rapidly matured into a commercial shown that the secure key rate per mode of any QKD protocol 5, 6, 7 technology since the first proposal emerged in 1984,1 largely scales linearly with η for small η. This places a fundamental because of a very attractive proposition: the security of QKD is not limit to the maximum distance attainable by QKD protocols based on the computational hardness of solving mathematical relying on direct transmission.