Gates, States, and Circuits
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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. -
Quantum Information
Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding . -
Lecture Notes: Qubit Representations and Rotations
Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1. -
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. -
Introduction to Quantum Information and Computation
Introduction to Quantum Information and Computation Steven M. Girvin ⃝c 2019, 2020 [Compiled: May 2, 2020] Contents 1 Introduction 1 1.1 Two-Slit Experiment, Interference and Measurements . 2 1.2 Bits and Qubits . 3 1.3 Stern-Gerlach experiment: the first qubit . 8 2 Introduction to Hilbert Space 16 2.1 Linear Operators on Hilbert Space . 18 2.2 Dirac Notation for Operators . 24 2.3 Orthonormal bases for electron spin states . 25 2.4 Rotations in Hilbert Space . 29 2.5 Hilbert Space and Operators for Multiple Spins . 38 3 Two-Qubit Gates and Entanglement 43 3.1 Introduction . 43 3.2 The CNOT Gate . 44 3.3 Bell Inequalities . 51 3.4 Quantum Dense Coding . 55 3.5 No-Cloning Theorem Revisited . 59 3.6 Quantum Teleportation . 60 3.7 YET TO DO: . 61 4 Quantum Error Correction 63 4.1 An advanced topic for the experts . 68 5 Yet To Do 71 i Chapter 1 Introduction By 1895 there seemed to be nothing left to do in physics except fill out a few details. Maxwell had unified electriciy, magnetism and optics with his theory of electromagnetic waves. Thermodynamics was hugely successful well before there was any deep understanding about the properties of atoms (or even certainty about their existence!) could make accurate predictions about the efficiency of the steam engines powering the industrial revolution. By 1895, statistical mechanics was well on its way to providing a microscopic basis in terms of the random motions of atoms to explain the macroscopic predictions of thermodynamics. However over the next decade, a few careful observers (e.g. -
Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes
Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes Daniel Litinski and Felix von Oppen Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany We present a planar surface-code-based bilizers requires more potentially faulty controlled-not scheme for fault-tolerant quantum computation (CNOT) gates. which eliminates the time overhead of single- The main drawback of surface codes in comparison qubit Clifford gates, and implements long-range to color codes is the absence of transversal single-qubit multi-target CNOT gates with a time overhead Clifford gates, i.e., the gates that are products of the that scales only logarithmically with the control- Hadamard gate H and the phase gate S. While the target separation. This is done by replacing transversal Clifford gates of color codes provide them hardware operations for single-qubit Clifford with fast logical H and S gates, defect-based propos- gates with a classical tracking protocol. Inter- als for surface codes [14] implement the H gate via a qubit communication is added via a modified multi-step measurement protocol, and the S gate via a lattice surgery protocol that employs twist de- distilled ancilla qubit. In order to lower the overhead fects of the surface code. The long-range multi- of single-qubit Clifford gates, surface code qubits can target CNOT gates facilitate magic state distil- be encoded using twist defects [15], which are essen- lation, which renders our scheme fault-tolerant tially Majoranas that can be braided via code defor- and universal. mation [16]. -
Studies in the Geometry of Quantum Measurements
Irina Dumitru Studies in the Geometry of Quantum Measurements Studies in the Geometry of Quantum Measurements Studies in Irina Dumitru Irina Dumitru is a PhD student at the department of Physics at Stockholm University. She has carried out research in the field of quantum information, focusing on the geometry of Hilbert spaces. ISBN 978-91-7911-218-9 Department of Physics Doctoral Thesis in Theoretical Physics at Stockholm University, Sweden 2020 Studies in the Geometry of Quantum Measurements Irina Dumitru Academic dissertation for the Degree of Doctor of Philosophy in Theoretical Physics at Stockholm University to be publicly defended on Thursday 10 September 2020 at 13.00 in sal C5:1007, AlbaNova universitetscentrum, Roslagstullsbacken 21, and digitally via video conference (Zoom). Public link will be made available at www.fysik.su.se in connection with the nailing of the thesis. Abstract Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally- Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs. This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements. -
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. -
Elementary Gates for Quantum Computation
Elementary gates for quantum computation Adriano Barenco Charles H. Bennett Richard Cleve y z Oxford University IBM Research University of Calgary David P. DiVincenzo Norman Margolus Peter Shor x { y IBM Research MIT AT&T Bell Labs Tycho Sleator John Smolin Harald Weinfurter yy k UCLA Univ. of Innsbruck New York Univ. submitted to Physical Review A, March 22, 1995 (AC5710) Abstract We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Bo olean values (x; y )to(x; x y )) is universal in the sense that all unitary op erations on arbitrarily many bits n n (U(2 )) can b e expressed as comp ositions of these gates. Weinvestigate the numb er of the ab ove gates required to implement other gates, such as gener- alized Deutsch-To oli gates, that apply a sp eci c U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satis ed. These gates play a central role in many prop osed constructions of quantum com- putational networks. We derive upp er and lower b ounds on the exact number Clarendon Lab oratory, Oxford OX1 3PU, UK; [email protected]. y Yorktown Heights, New York, NY 10598, USA; b ennetc/[email protected]. z Department of Computer Science, Calgary, Alb erta, Canada T2N 1N4; [email protected]. x Lab oratory for Computer Science, Cambridge MA 02139 USA; [email protected]. -
A Gentle Introduction to Quantum Computing Algorithms With
A Gentle Introduction to Quantum Computing Algorithms with Applications to Universal Prediction Elliot Catt1 Marcus Hutter1,2 1Australian National University, 2Deepmind {elliot.carpentercatt,marcus.hutter}@anu.edu.au May 8, 2020 Abstract In this technical report we give an elementary introduction to Quantum Computing for non- physicists. In this introduction we describe in detail some of the foundational Quantum Algorithms including: the Deutsch-Jozsa Algorithm, Shor’s Algorithm, Grocer Search, and Quantum Count- ing Algorithm and briefly the Harrow-Lloyd Algorithm. Additionally we give an introduction to Solomonoff Induction, a theoretically optimal method for prediction. We then attempt to use Quan- tum computing to find better algorithms for the approximation of Solomonoff Induction. This is done by using techniques from other Quantum computing algorithms to achieve a speedup in com- puting the speed prior, which is an approximation of Solomonoff’s prior, a key part of Solomonoff Induction. The major limiting factors are that the probabilities being computed are often so small that without a sufficient (often large) amount of trials, the error may be larger than the result. If a substantial speedup in the computation of an approximation of Solomonoff Induction can be achieved through quantum computing, then this can be applied to the field of intelligent agents as a key part of an approximation of the agent AIXI. arXiv:2005.03137v1 [quant-ph] 29 Apr 2020 1 Contents 1 Introduction 3 2 Preliminaries 4 2.1 Probability ......................................... 4 2.2 Computability ....................................... 5 2.3 ComplexityTheory................................... .. 7 3 Quantum Computing 8 3.1 Introduction...................................... ... 8 3.2 QuantumTuringMachines .............................. .. 9 3.2.1 Church-TuringThesis .............................. -
Quantum Inductive Learning and Quantum Logic Synthesis
Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 2009 Quantum Inductive Learning and Quantum Logic Synthesis Martin Lukac Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Lukac, Martin, "Quantum Inductive Learning and Quantum Logic Synthesis" (2009). Dissertations and Theses. Paper 2319. https://doi.org/10.15760/etd.2316 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. For more information, please contact [email protected]. QUANTUM INDUCTIVE LEARNING AND QUANTUM LOGIC SYNTHESIS by MARTIN LUKAC A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in ELECTRICAL AND COMPUTER ENGINEERING. Portland State University 2009 DISSERTATION APPROVAL The abstract and dissertation of Martin Lukac for the Doctor of Philosophy in Electrical and Computer Engineering were presented January 9, 2009, and accepted by the dissertation committee and the doctoral program. COMMITTEE APPROVALS: Irek Perkowski, Chair GarrisoH-Xireenwood -George ^Lendaris 5artM ?teven Bleiler Representative of the Office of Graduate Studies DOCTORAL PROGRAM APPROVAL: Malgorza /ska-Jeske7~Director Electrical Computer Engineering Ph.D. Program ABSTRACT An abstract of the dissertation of Martin Lukac for the Doctor of Philosophy in Electrical and Computer Engineering presented January 9, 2009. Title: Quantum Inductive Learning and Quantum Logic Synhesis Since Quantum Computer is almost realizable on large scale and Quantum Technology is one of the main solutions to the Moore Limit, Quantum Logic Synthesis (QLS) has become a required theory and tool for designing Quantum Logic Circuits. -
Arxiv:Quant-Ph/0002039V2 1 Aug 2000
Methodology for quantum logic gate construction 1,2 3,2 2 Xinlan Zhou ∗, Debbie W. Leung †, and Isaac L. Chuang ‡ 1 Department of Applied Physics, Stanford University, Stanford, California 94305-4090 2 IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120 3 Quantum Entanglement Project, ICORP, JST Edward Ginzton Laboratory, Stanford University, Stanford, California 94305-4085 (October 29, 2018) We present a general method to construct fault-tolerant quantum logic gates with a simple primitive, which is an analog of quantum teleportation. The technique extends previous results based on traditional quantum teleportation (Gottesman and Chuang, Nature 402, 390, 1999) and leads to straightforward and systematic construction of many fault-tolerant encoded operations, including the π/8 and Toffoli gates. The technique can also be applied to the construction of remote quantum operations that cannot be directly performed. I. INTRODUCTION Practical realization of quantum information processing requires specific types of quantum operations that may be difficult to construct. In particular, to perform quantum computation robustly in the presence of noise, one needs fault-tolerant implementation of quantum gates acting on states that are block-encoded using quantum error correcting codes [1–4]. Fault-tolerant quantum gates must prevent propagation of single qubit errors to multiple qubits within any code block so that small correctable errors will not grow to exceed the correction capability of the code. This requirement greatly restricts the types of unitary operations that can be performed on the encoded qubits. Certain fault-tolerant operations can be implemented easily by performing direct transversal operations on the encoded qubits, in which each qubit in a block interacts only with one corresponding qubit, either in another block or in a specialized ancilla.