Basic Concepts in Quantum Computation
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On Quaslfree States of CAR and Bogoliubov Automorphisms
Publ. RIMS Kyoto Univ. Vol. 6 (1970/71), 385-442 On Quaslfree States of CAR and Bogoliubov Automorphisms By Huzihiro ARAKI Abstract A necessary and sufficient condition for two quasifree states of CAR to be quasiequivalent is obtained. Quasif ree states is characterized as the unique KMS state of a Bogoliubov automorphism of CAR. The structure of the group of all inner Bogoliubov automorphisms of CAR is clarified. §1. Introduction A classification of gauge invariant quasifree states of the canonical anticommutation relations (CAR) up to quasi and unitary equivalence is recently obtained by Powers and St0rmer [12]. We shall generalize their result to arbitrary quasifree states. We use the formalism developped earlier [2] and study quasifree state <ps of a self dual CAR algebra. It is then shown that <pSi and <pS2 are quasiequivalent if and only if Sllz — Sllz is in the Hiibert Schmidt class. For a gauge invariant quasifree state <pA in the paper of Powers and St0rmer, S = A@(1 — A) and hence our result is a direct generali- zation of Powers and St0rmer. The quasifree primary state <ps for which 5 does not have eigen- value 1 is shown to be the unique KMS state for the one parameter group r(£/GO) of Bogoiiubov * automorphisms of CAR, where r(t/GO) corresponds to a unitary transformation U(£)=ex.piAH of the direct sum of testing function spaces of creation and annihilation operators and H is related to 5 by S= (l + e"*)"1. This is used to simplify some of arguments. A quasifree state q>s is primary unless 1/2 is an isolated point spectrum of 5, has an odd multiplicity and S(l — S1) is in the Received May 6, 1970 and in revised form November 17, 1970. -
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. -
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. -
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. -
Fault-Tolerant Interface Between Quantum Memories and Quantum Processors
ARTICLE DOI: 10.1038/s41467-017-01418-2 OPEN Fault-tolerant interface between quantum memories and quantum processors Hendrik Poulsen Nautrup 1, Nicolai Friis 1,2 & Hans J. Briegel1 Topological error correction codes are promising candidates to protect quantum computa- tions from the deteriorating effects of noise. While some codes provide high noise thresholds suitable for robust quantum memories, others allow straightforward gate implementation 1234567890 needed for data processing. To exploit the particular advantages of different topological codes for fault-tolerant quantum computation, it is necessary to be able to switch between them. Here we propose a practical solution, subsystem lattice surgery, which requires only two-body nearest-neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary topological subsystem codes in two dimensions and beyond. In particular, it can be employed to create a simple interface, a quantum bus, between noise resilient surface code memories and flexible color code processors. 1 Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, 6020 Innsbruck, Austria. 2 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Correspondence and requests for materials should be addressed to H.P.N. (email: [email protected]) NATURE COMMUNICATIONS | 8: 1321 | DOI: 10.1038/s41467-017-01418-2 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01418-2 oise and decoherence can be considered as the major encoding k = n − s qubits. We denote the normalizer of S by Nobstacles for large-scale quantum information processing. -
Encryption of Binary and Non-Binary Data Using Chained Hadamard Transforms
Encryption of Binary and Non-Binary Data Using Chained Hadamard Transforms Rohith Singi Reddy Computer Science Department Oklahoma State University, Stillwater, OK 74078 ABSTRACT This paper presents a new chaining technique for the use of Hadamard transforms for encryption of both binary and non-binary data. The lengths of the input and output sequence need not be identical. The method may be used also for hashing. INTRODUCTION The Hadamard transform is an example of generalized class of Fourier transforms [4], and it may be defined either recursively, or by using binary representation. Its symmetric form lends itself to applications ranging across many technical fields such as data encryption [1], signal processing [2], data compression algorithms [3], randomness measures [6], and so on. Here we consider an application of the Hadamard matrix to non-binary numbers. To do so, Hadamard transform is generalized such that all the values in the matrix are non negative. Each negative number is replaced with corresponding modulo number; for example while performing modulo 7 operations -1 is replaced with 6 to make the matrix non-binary. Only prime modulo operations are performed because non prime numbers can be divisible with numbers other than 1 and itself. The use of modulo operation along with generalized Hadamard matrix limits the output range and decreases computation complexity. In the method proposed in this paper, the input sequence is split into certain group of bits such that each group bit count is a prime number. As we are dealing with non-binary numbers, for each group of binary bits their corresponding decimal value is calculated. -
Realization of the Quantum Toffoli Gate
Realization of the quantum Toffoli gate Realization of the quantum Toffoli gate Based on: Monz, T; Kim, K; Haensel, W; et al Realization of the quantum Toffoli gate with trapped ions Phys. Rev. Lett. 102, 040501 (2009) Lanyon, BP; Barbieri, M; Almeida, MP; et al. Simplifying quantum logic using higher dimensional Hilbert spaces Nat. Phys. 5, 134 (2009) Realization of the quantum Toffoli gate Outline 1. Motivation 2. Principles of the quantum Toffoli gate 3. Implementation with trapped ions 4. Implementation with photons 5. Comparison and conclusion 6. Summary 6. Dezember 2010 Jakob Buhmann Jeffrey Gehrig 1 Realization of the quantum Toffoli gate 1. Motivation • Universal quantum logic gate sets are needed to implement algorithms • Implementation of algorithms is difficult due to the finite fidelity and large amount of gates • Use of other degrees of freedom to store information • Reduction of complexity and runtime 6. Dezember 2010 Jakob Buhmann Jeffrey Gehrig 2 Realization of the quantum Toffoli gate 2. Principles of the Toffoli gate • Three-qubit gate (C1, C2, T) • Logic flip of T depending on (C1 AND C2) Truth table Matrix form Input Output C1 C2 T C1 C2 T 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 6. Dezember 2010 Jakob Buhmann Jeffrey Gehrig 3 Realization of the quantum Toffoli gate 2. Principles of the Toffoli gate • Qubit Implementation • Qutrit Implementation Qutrit states: and 6. -
Dominant Strategies of Quantum Games on Quantum Periodic Automata
Computation 2015, 3, 586-599; doi:10.3390/computation3040586 OPEN ACCESS computation ISSN 2079-3197 www.mdpi.com/journal/computation Article Dominant Strategies of Quantum Games on Quantum Periodic Automata Konstantinos Giannakis y;*, Christos Papalitsas y, Kalliopi Kastampolidou y, Alexandros Singh y and Theodore Andronikos y Department of Informatics, Ionian University, 7 Tsirigoti Square, Corfu 49100, Greece; E-Mails: [email protected] (C.P.); [email protected] (K.K.); [email protected] (A.S.); [email protected] (T.A.) y These authors contributed equally to this work. * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +30-266-108-7712. Academic Editor: Demos T. Tsahalis Received: 29 August 2015 / Accepted: 16 November 2015 / Published: 20 November 2015 Abstract: Game theory and its quantum extension apply in numerous fields that affect people’s social, political, and economical life. Physical limits imposed by the current technology used in computing architectures (e.g., circuit size) give rise to the need for novel mechanisms, such as quantum inspired computation. Elements from quantum computation and mechanics combined with game-theoretic aspects of computing could open new pathways towards the future technological era. This paper associates dominant strategies of repeated quantum games with quantum automata that recognize infinite periodic inputs. As a reference, we used the PQ-PENNY quantum game where the quantum strategy outplays the choice of pure or mixed strategy with probability 1 and therefore the associated quantum automaton accepts with probability 1. We also propose a novel game played on the evolution of an automaton, where players’ actions and strategies are also associated with periodic quantum automata. -
Machine Learning for Designing Fast Quantum Gates
University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2016-01-26 Machine Learning for Designing Fast Quantum Gates Zahedinejad, Ehsan Zahedinejad, E. (2016). Machine Learning for Designing Fast Quantum Gates (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26805 http://hdl.handle.net/11023/2780 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Machine Learning for Designing Fast Quantum Gates by Ehsan Zahedinejad A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY CALGARY, ALBERTA January, 2016 c Ehsan Zahedinejad 2016 Abstract Fault-tolerant quantum computing requires encoding the quantum information into logical qubits and performing the quantum information processing in a code-space. Quantum error correction codes, then, can be employed to diagnose and remove the possible errors in the quantum information, thereby avoiding the loss of information. Although a series of single- and two-qubit gates can be employed to construct a quan- tum error correcting circuit, however this decomposition approach is not practically desirable because it leads to circuits with long operation times. An alternative ap- proach to designing a fast quantum circuit is to design quantum gates that act on a multi-qubit gate.