Chapter 2. Basic Principles of Quantum mechanics
|
|
- Jane Little
- 6 years ago
- Views:
Transcription
1 Chapter 2. Basic Principles of Quantum mechanics In this chapter we introduce basic principles of the quantum mechanics. Quantum computers are based on the principles of the quantum mechanics. In the classical computation we are familiar with bits and we use Boolean algebra. Please read the following analysis: In physics there are some models describe physical events in the macroscopic universe and microscopic universe. For instance, working principle of an engine based on laws of thermodynamics. If this sentence make sense then you realized the second law! Research and development in the physics, leads to the production to the high-technological devices, computers, scientific instruments, electronics technology, communication technology, as well as scientific papers and patents. I hope at the end of this course your point of view to technological developments will improve and you realize existence of a bridge between theory and practice. Quantum computation is based on SUPERPOSITION and COMPOSITION of the state. In the classical computation we cannot obtain superposition of two binary number (bit) 1 and 0. In quantum computation it is possible to obtain mixture of two quantum bit (qubits): i.e spin up and spin down electron. The representations of bit and qubit are: The representation is known as DIRAC notation which is called a KET vector. In the classical system we can not express a bit as decomposition of the bits. The ket vectors also represented by: The quantum theory is the "theory of everything". Note that one can complete all of the calculations of the quantum computation by using simple matrix and vector algebra! In this lecture we will discuss: Four postulates of QM States: Dirac notation (Bra-Ket notation), Pauli Spin Matrices Transformation of the states: Pauli Gates Measurement Composite systems & Entanglement States: Dirac notation (Bra-Ket notation), Pauli Spin Matrices A quantum mechanical system completely described by a state vector. A state vector include all information about the quantum system. In classical computation all computations based on a bit in the base 2 number system. In quantum computation theory the computation based on state (qubits). As you see that this open a new way to the computation theory! A stationary state corresponds to a quantum state with a single definite energy. A superposition state is mixture of stationary states. (For more details come to class) Consider two level quantum system (for example spin up and down state of the electron). Wave function of the system may be written as the superposition of the spins:
2 The wave should be normalized. Then BRA of the wave function can be written as Normalization yields: Note that are orthogonal vectors, then The expression reads as follows. The state is with probability and with a probability. Transformation of the states: Pauli Gates Time evolution of a quantum system in the Hilbert space is described by a unitary transformation, generally by using unitary matrices. Unitary matrix is defined as follows: If transpose conjugate of a matrix U is equal to the inverse of matrix then the matrix is known as unitary matrix. Note that transpose conjugate of U can be abbreviated as U + and satisfy the relation U + U=I (unit matrix). Examples: The matrices are unitary and first 3 matrices are known as Pauli Matrices. Transformation of the state can be realized as change of the physical situation of the quantum particle from one state to other. Pauli Matrices are mathematical representations of the transformation of spin state of the particles. Technically the spin of the particles can be changed by applying magnetic field. Consider a particle of zero (down) spin. By applying magnetic field we can change the direction of the spin to. This transformation mathematically represented as: Explicitly Generally, transformation of the state with a unitary matrix can be expressed as i.e: Means, the quantum particle has a spin up and down with probability a and b respectively. The state of the particle can be changed by applying an external field. New spin state of the particle is again up and down, but in this case the probabilities are b and a respectively. One of the important transformation can be done by the matrix:, is known as Hadamard Matrix. This transformation is:
3 Before going further, let us practically construct some simple quantum gates namely PAULI GATES. As it is mentioned before this gates physically realized by using material (ions, for instance) whose spins controlled by magnetic field. Similar devices also constructed by using semi transparent mirrors and polarization properties of the photos. The devices also constructed by using an appropriate crystal and again polarization properties of the photons. Pauli X-Gate X Pauli Y-Gate Y Pauli Z-Gate Z Exercise: Show that XY=iZ, YZ=iX, ZX=iY and X 2 =Y 2 =Z 2 =I. One can also construct a gate using the transformation matrix:. This gate is known as Walsh-Hadamard gate that is one ofthe important gate of the quantum computers. Walsh Hadamard H-Gate
4 H Measurement Measurement of a qubit is another postulates of a quantum mechanics and play an important role in the quantum computation. Consiter a state. We want to measure a or/and b in order to determine output of the calculation. Unfortunately quantum mechanics does not allow determination of a and b. However, quantum we can obtain an indirect information about the state. The measurement always disturb the system then only one of the output can be determined with a probalities: As an example for the given state the probabilities. The postulate can be stated as follows: Quantum measurements can be described by measuring transformation of the state by the operator, mathematically Consider operators.probability of measurement of the output is and the state vector Then the probability of measuring the state are. Composite systems & Entanglement Mathematical Supplement As in the digital computation system, using simple quantum logic gates we can design more complicated circuits. For multi uncoupled qubits (state), the state of the composite system is given by tensor product or direct product of the matrix representation. Consider tensor product of two qubit system: These two qubit states can be represented by matrices: Then tensor product of two matrices are given by:
5 The quantum computation requires this multiplication. Sometimes we might want to compute effect of a gate conditionally. The mathematical operation composing gates conditionally is the direct sum of the corresponding unitary matrices. Direct sum of two matrices can be described as: We will turn our attention to the composition of multi qubit systems in the next chapter. For a useful quantum computation device we have to construct multi-qubit quantum states. "The state space of a composite physical system is the tensor product of the state spaces of the component physical systems." Suppose that there are two states then tensor product produce multiple state (superposition of the states) such that For quantum systems tensor product captures the essence of superposition. Properties of tensor product will be introduced in the class!! Quantum entanglement Let us ask a question: how can we determine individual 1-qubit state of a 2-qubit quantum register: We recall the last postulate (composition) and define the states:, the tensor product If we ask the same question for the decomposition:, interestingly we can see that no individual 1-qubit state exist to form this decomposition. At first sight we can think that quantum mechanics does not allow this decomposition! If we decide to measure the first qubit then either will be measured with the probabilities respectively. In the computational basis if we measure qubit then without touching or measuring other qubit it is quantum state can be determined as. This shows that there is a mysterious connection between states!measurements of one qubit on the side effect another qubit. This shows that aforementioned entangled state can also be written as:. This effect seems to be in total contradiction with Einstain's theory of speed of light! Finally we define the entanglement state: If a multi qubit state can not be factored into individually one qubit state then the state is called entangled state. Entanglement is an important phenomenon in multi-qubit quantum memory registers and many quantum algorithms.
2.0 Basic Elements of a Quantum Information Processor. 2.1 Classical information processing The carrier of information
QSIT09.L03 Page 1 2.0 Basic Elements of a Quantum Information Processor 2.1 Classical information processing 2.1.1 The carrier of information - binary representation of information as bits (Binary digits).
More informationSome Introductory Notes on Quantum Computing
Some Introductory Notes on Quantum Computing Markus G. Kuhn http://www.cl.cam.ac.uk/~mgk25/ Computer Laboratory University of Cambridge 2000-04-07 1 Quantum Computing Notation Quantum Computing is best
More informationQuantum computing! quantum gates! Fisica dell Energia!
Quantum computing! quantum gates! Fisica dell Energia! What is Quantum Computing?! Calculation based on the laws of Quantum Mechanics.! Uses Quantum Mechanical Phenomena to perform operations on data.!
More informationShort Course in Quantum Information Lecture 2
Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture
More informationCS257 Discrete Quantum Computation
CS57 Discrete Quantum Computation John E Savage April 30, 007 Lect 11 Quantum Computing c John E Savage Classical Computation State is a vector of reals; e.g. Booleans, positions, velocities, or momenta.
More informationIntroduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871
Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871 Lecture 1 (2017) Jon Yard QNC 3126 jyard@uwaterloo.ca TAs Nitica Sakharwade nsakharwade@perimeterinstitute.ca
More informationQLang: Qubit Language
QLang: Qubit Language Christopher Campbell Clément Canonne Sankalpa Khadka Winnie Narang Jonathan Wong September 24, 24 Introduction In 965, Gordon Moore predicted that the number of transistors in integrated
More informationQuantum Entanglement and the Bell Matrix
Quantum Entanglement and the Bell Matrix Marco Pedicini (Roma Tre University) in collaboration with Anna Chiara Lai and Silvia Rognone (La Sapienza University of Rome) SIMAI2018 - MS27: Discrete Mathematics,
More informationPh 219b/CS 219b. Exercises Due: Wednesday 20 November 2013
1 h 219b/CS 219b Exercises Due: Wednesday 20 November 2013 3.1 Universal quantum gates I In this exercise and the two that follow, we will establish that several simple sets of gates are universal for
More informationFactoring on a Quantum Computer
Factoring on a Quantum Computer The Essence Shor s Algorithm Wolfgang Polak wp@pocs.com Thanks to: Eleanor Rieffel Fuji Xerox Palo Alto Laboratory Wolfgang Polak San Jose State University, 4-14-010 - p.
More information. Here we are using the standard inner-product over C k to define orthogonality. Recall that the inner-product of two vectors φ = i α i.
CS 94- Hilbert Spaces, Tensor Products, Quantum Gates, Bell States 1//07 Spring 007 Lecture 01 Hilbert Spaces Consider a discrete quantum system that has k distinguishable states (eg k distinct energy
More informationLecture 3: Hilbert spaces, tensor products
CS903: Quantum computation and Information theory (Special Topics In TCS) Lecture 3: Hilbert spaces, tensor products This lecture will formalize many of the notions introduced informally in the second
More informationQuantum Computing Lecture 3. Principles of Quantum Mechanics. Anuj Dawar
Quantum Computing Lecture 3 Principles of Quantum Mechanics Anuj Dawar What is Quantum Mechanics? Quantum Mechanics is a framework for the development of physical theories. It is not itself a physical
More informationHilbert Space, Entanglement, Quantum Gates, Bell States, Superdense Coding.
CS 94- Bell States Bell Inequalities 9//04 Fall 004 Lecture Hilbert Space Entanglement Quantum Gates Bell States Superdense Coding 1 One qubit: Recall that the state of a single qubit can be written as
More informationCSCI 2570 Introduction to Nanocomputing. Discrete Quantum Computation
CSCI 2570 Introduction to Nanocomputing Discrete Quantum Computation John E Savage November 27, 2007 Lect 22 Quantum Computing c John E Savage What is Quantum Computation It is very different kind of computation
More informationLecture 2: Introduction to Quantum Mechanics
CMSC 49: Introduction to Quantum Computation Fall 5, Virginia Commonwealth University Sevag Gharibian Lecture : Introduction to Quantum Mechanics...the paradox is only a conflict between reality and your
More informationChapter 10. Quantum algorithms
Chapter 10. Quantum algorithms Complex numbers: a quick review Definition: C = { a + b i : a, b R } where i = 1. Polar form of z = a + b i is z = re iθ, where r = z = a 2 + b 2 and θ = tan 1 y x Alternatively,
More information6. Quantum error correcting codes
6. Quantum error correcting codes Error correcting codes (A classical repetition code) Preserving the superposition Parity check Phase errors CSS 7-qubit code (Steane code) Too many error patterns? Syndrome
More informationQuantum Gates, Circuits & Teleportation
Chapter 3 Quantum Gates, Circuits & Teleportation Unitary Operators The third postulate of quantum physics states that the evolution of a quantum system is necessarily unitary. Geometrically, a unitary
More informationSingle qubit + CNOT gates
Lecture 6 Universal quantum gates Single qubit + CNOT gates Single qubit and CNOT gates together can be used to implement an arbitrary twolevel unitary operation on the state space of n qubits. Suppose
More informationComplex numbers: a quick review. Chapter 10. Quantum algorithms. Definition: where i = 1. Polar form of z = a + b i is z = re iθ, where
Chapter 0 Quantum algorithms Complex numbers: a quick review / 4 / 4 Definition: C = { a + b i : a, b R } where i = Polar form of z = a + b i is z = re iθ, where r = z = a + b and θ = tan y x Alternatively,
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationRichard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo
CS 497 Frontiers of Computer Science Introduction to Quantum Computing Lecture of http://www.cs.uwaterloo.ca/~cleve/cs497-f7 Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum
More informationChapter 11 Evolution by Permutation
Chapter 11 Evolution by Permutation In what follows a very brief account of reversible evolution and, in particular, reversible computation by permutation will be presented We shall follow Mermin s account
More information1 Measurements, Tensor Products, and Entanglement
Stanford University CS59Q: Quantum Computing Handout Luca Trevisan September 7, 0 Lecture In which we describe the quantum analogs of product distributions, independence, and conditional probability, and
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Part I Emma Strubell http://cs.umaine.edu/~ema/quantum_tutorial.pdf April 12, 2011 Overview Outline What is quantum computing? Background Caveats Fundamental differences
More informationReversible and Quantum computing. Fisica dell Energia - a.a. 2015/2016
Reversible and Quantum computing Fisica dell Energia - a.a. 2015/2016 Reversible computing A process is said to be logically reversible if the transition function that maps old computational states to
More informationLinear Algebra and Dirac Notation, Pt. 1
Linear Algebra and Dirac Notation, Pt. 1 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 1 / 13
More informationLecture 1: Introduction to Quantum Computing
Lecture : Introduction to Quantum Computing Rajat Mittal IIT Kanpur What is quantum computing? This course is about the theory of quantum computation, i.e., to do computation using quantum systems. These
More informationUnitary evolution: this axiom governs how the state of the quantum system evolves in time.
CS 94- Introduction Axioms Bell Inequalities /7/7 Spring 7 Lecture Why Quantum Computation? Quantum computers are the only model of computation that escape the limitations on computation imposed by the
More informationphys4.20 Page 1 - the ac Josephson effect relates the voltage V across a Junction to the temporal change of the phase difference
Josephson Effect - the Josephson effect describes tunneling of Cooper pairs through a barrier - a Josephson junction is a contact between two superconductors separated from each other by a thin (< 2 nm)
More informationQuantum Error Correction Codes-From Qubit to Qudit. Xiaoyi Tang, Paul McGuirk
Quantum Error Correction Codes-From Qubit to Qudit Xiaoyi Tang, Paul McGuirk Outline Introduction to quantum error correction codes (QECC) Qudits and Qudit Gates Generalizing QECC to Qudit computing Need
More informationQuantum Computing. 6. Quantum Computer Architecture 7. Quantum Computers and Complexity
Quantum Computing 1. Quantum States and Quantum Gates 2. Multiple Qubits and Entangled States 3. Quantum Gate Arrays 4. Quantum Parallelism 5. Examples of Quantum Algorithms 1. Grover s Unstructured Search
More informationPrinciples of Quantum Mechanics Pt. 2
Principles of Quantum Mechanics Pt. 2 PHYS 500 - Southern Illinois University February 9, 2017 PHYS 500 - Southern Illinois University Principles of Quantum Mechanics Pt. 2 February 9, 2017 1 / 13 The
More informationM.L. Dalla Chiara, R. Giuntini, R. Leporini, G. Sergioli. Qudit Spaces and a Many-valued Approach to Quantum Comp
Qudit Spaces and a Many-valued Approach to Quantum Computational Logics Quantum computational logics are special examples of quantum logic based on the following semantic idea: linguistic formulas are
More informationConcepts and Algorithms of Scientific and Visual Computing Advanced Computation Models. CS448J, Autumn 2015, Stanford University Dominik L.
Concepts and Algorithms of Scientific and Visual Computing Advanced Computation Models CS448J, Autumn 2015, Stanford University Dominik L. Michels Advanced Computation Models There is a variety of advanced
More informationQuantum Algorithms. Andreas Klappenecker Texas A&M University. Lecture notes of a course given in Spring Preliminary draft.
Quantum Algorithms Andreas Klappenecker Texas A&M University Lecture notes of a course given in Spring 003. Preliminary draft. c 003 by Andreas Klappenecker. All rights reserved. Preface Quantum computing
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More information)j > Riley Tipton Perry University of New South Wales, Australia. World Scientific CHENNAI
Riley Tipton Perry University of New South Wales, Australia )j > World Scientific NEW JERSEY LONDON. SINGAPORE BEIJING SHANSHAI HONG K0N6 TAIPEI» CHENNAI Contents Acknowledgments xi 1. Introduction 1 1.1
More informationQUANTUM COMPUTER SIMULATION
Chapter 2 QUANTUM COMPUTER SIMULATION Chapter 1 discussed quantum computing in non-technical terms and in reference to simple, idealized physical models. In this chapter we make the underlying mathematics
More information1 Readings. 2 Unitary Operators. C/CS/Phys C191 Unitaries and Quantum Gates 9/22/09 Fall 2009 Lecture 8
C/CS/Phys C191 Unitaries and Quantum Gates 9//09 Fall 009 Lecture 8 1 Readings Benenti, Casati, and Strini: Classical circuits and computation Ch.1.,.6 Quantum Gates Ch. 3.-3.4 Kaye et al: Ch. 1.1-1.5,
More informationDEVELOPING A QUANTUM CIRCUIT SIMULATOR API
ACTA UNIVERSITATIS CIBINIENSIS TECHNICAL SERIES Vol. LXVII 2015 DOI: 10.1515/aucts-2015-0084 DEVELOPING A QUANTUM CIRCUIT SIMULATOR API MIHAI DORIAN Stancu Ph.D. student, Faculty of Economic Sciences/Cybernetics
More informationQuantum computing and mathematical research. Chi-Kwong Li The College of William and Mary
and mathematical research The College of William and Mary Classical computing Classical computing Hardware - Beads and bars. Classical computing Hardware - Beads and bars. Input - Using finger skill to
More informationC/CS/Phys 191 Quantum Gates and Universality 9/22/05 Fall 2005 Lecture 8. a b b d. w. Therefore, U preserves norms and angles (up to sign).
C/CS/Phys 191 Quantum Gates and Universality 9//05 Fall 005 Lecture 8 1 Readings Benenti, Casati, and Strini: Classical circuits and computation Ch.1.,.6 Quantum Gates Ch. 3.-3.4 Universality Ch. 3.5-3.6
More informationLecture 1: Introduction to Quantum Computing
Lecture 1: Introduction to Quantum Computing Rajat Mittal IIT Kanpur Whenever the word Quantum Computing is uttered in public, there are many reactions. The first one is of surprise, mostly pleasant, and
More informationQuantum Information & Quantum Computation
CS9A, Spring 5: Quantum Information & Quantum Computation Wim van Dam Engineering, Room 59 vandam@cs http://www.cs.ucsb.edu/~vandam/teaching/cs9/ Administrivia Who has the book already? Office hours: Wednesday
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationSemiconductors: Applications in spintronics and quantum computation. Tatiana G. Rappoport Advanced Summer School Cinvestav 2005
Semiconductors: Applications in spintronics and quantum computation Advanced Summer School 1 I. Background II. Spintronics Spin generation (magnetic semiconductors) Spin detection III. Spintronics - electron
More informationPh 219b/CS 219b. Exercises Due: Wednesday 22 February 2006
1 Ph 219b/CS 219b Exercises Due: Wednesday 22 February 2006 6.1 Estimating the trace of a unitary matrix Recall that using an oracle that applies the conditional unitary Λ(U), Λ(U): 0 ψ 0 ψ, 1 ψ 1 U ψ
More informationLecture notes on Quantum Computing. Chapter 1 Mathematical Background
Lecture notes on Quantum Computing Chapter 1 Mathematical Background Vector states of a quantum system with n physical states are represented by unique vectors in C n, the set of n 1 column vectors 1 For
More informationIntroduction into Quantum Computations Alexei Ashikhmin Bell Labs
Introduction into Quantum Computations Alexei Ashikhmin Bell Labs Workshop on Quantum Computing and its Application March 16, 2017 Qubits Unitary transformations Quantum Circuits Quantum Measurements Quantum
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationQuantum Computing: Foundations to Frontier Fall Lecture 3
Quantum Computing: Foundations to Frontier Fall 018 Lecturer: Henry Yuen Lecture 3 Scribes: Seyed Sajjad Nezhadi, Angad Kalra Nora Hahn, David Wandler 1 Overview In Lecture 3, we started off talking about
More informationTeleportation of Quantum States (1993; Bennett, Brassard, Crepeau, Jozsa, Peres, Wootters)
Teleportation of Quantum States (1993; Bennett, Brassard, Crepeau, Jozsa, Peres, Wootters) Rahul Jain U. Waterloo and Institute for Quantum Computing, rjain@cs.uwaterloo.ca entry editor: Andris Ambainis
More informationQuantum Entanglement and Error Correction
Quantum Entanglement and Error Correction Fall 2016 Bei Zeng University of Guelph Course Information Instructor: Bei Zeng, email: beizeng@icloud.com TA: Dr. Cheng Guo, email: cheng323232@163.com Wechat
More informationLecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
More informationLecture 11 September 30, 2015
PHYS 7895: Quantum Information Theory Fall 015 Lecture 11 September 30, 015 Prof. Mark M. Wilde Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
More information1. Basic rules of quantum mechanics
1. Basic rules of quantum mechanics How to describe the states of an ideally controlled system? How to describe changes in an ideally controlled system? How to describe measurements on an ideally controlled
More informationBasic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi Module No. # 07 Bra-Ket Algebra and Linear Harmonic Oscillator - II Lecture No. # 01 Dirac s Bra and
More informationQuantum Computation. Alessandra Di Pierro Computational models (Circuits, QTM) Algorithms (QFT, Quantum search)
Quantum Computation Alessandra Di Pierro alessandra.dipierro@univr.it 21 Info + Programme Info: http://profs.sci.univr.it/~dipierro/infquant/ InfQuant1.html Preliminary Programme: Introduction and Background
More informationLecture 3: Constructing a Quantum Model
CS 880: Quantum Information Processing 9/9/010 Lecture 3: Constructing a Quantum Model Instructor: Dieter van Melkebeek Scribe: Brian Nixon This lecture focuses on quantum computation by contrasting it
More informationIntroduction to Quantum Computing for Folks
Introduction to Quantum Computing for Folks Joint Advanced Student School 2009 Ing. Javier Enciso encisomo@in.tum.de Technische Universität München April 2, 2009 Table of Contents 1 Introduction 2 Quantum
More informationQuantum Complexity Theory and Adiabatic Computation
Chapter 9 Quantum Complexity Theory and Adiabatic Computation 9.1 Defining Quantum Complexity We are familiar with complexity theory in classical computer science: how quickly can a computer (or Turing
More informationThe Postulates of Quantum Mechanics
p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits
More informationSUPERDENSE CODING AND QUANTUM TELEPORTATION
SUPERDENSE CODING AND QUANTUM TELEPORTATION YAQIAO LI This note tries to rephrase mathematically superdense coding and quantum teleportation explained in [] Section.3 and.3.7, respectively (as if I understood
More informationQuantum Computing. Vraj Parikh B.E.-G.H.Patel College of Engineering & Technology, Anand (Affiliated with GTU) Abstract HISTORY OF QUANTUM COMPUTING-
Quantum Computing Vraj Parikh B.E.-G.H.Patel College of Engineering & Technology, Anand (Affiliated with GTU) Abstract Formerly, Turing Machines were the exemplar by which computability and efficiency
More informationQuantum Mechanics II: Examples
Quantum Mechanics II: Examples Michael A. Nielsen University of Queensland Goals: 1. To apply the principles introduced in the last lecture to some illustrative examples: superdense coding, and quantum
More informationLecture 2: From Classical to Quantum Model of Computation
CS 880: Quantum Information Processing 9/7/10 Lecture : From Classical to Quantum Model of Computation Instructor: Dieter van Melkebeek Scribe: Tyson Williams Last class we introduced two models for deterministic
More informationState. Bell State Exercises
Bell State Exercises Frank Rioux he Bell states are maximally entangled superpositions of two-particle states. Consider two spin-/ particles created in the same event. here are four maximally entangled
More informationQubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable,
Qubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable, A qubit: a sphere of values, which is spanned in projective sense by two quantum
More informationarxiv: v4 [cs.dm] 19 Nov 2018
An Introduction to Quantum Computing, Without the Physics Giacomo Nannicini IBM T.J. Watson, Yorktown Heights, NY nannicini@us.ibm.com Last updated: November 0, 018. arxiv:1708.03684v4 [cs.dm] 19 Nov 018
More informationLecture 13B: Supplementary Notes on Advanced Topics. 1 Inner Products and Outer Products for Single Particle States
Lecture 13B: Supplementary Notes on Advanced Topics Outer Products, Operators, Density Matrices In order to explore the complexity of many particle systems a different way to represent multiparticle states
More informationC/CS/Phy191 Problem Set 6 Solutions 3/23/05
C/CS/Phy191 Problem Set 6 Solutions 3/3/05 1. Using the standard basis (i.e. 0 and 1, eigenstates of Ŝ z, calculate the eigenvalues and eigenvectors associated with measuring the component of spin along
More informationQuantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras
Quantum Mechanics- I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 4 Postulates of Quantum Mechanics I In today s lecture I will essentially be talking
More informationQuantum Computers. Todd A. Brun Communication Sciences Institute USC
Quantum Computers Todd A. Brun Communication Sciences Institute USC Quantum computers are in the news Quantum computers represent a new paradigm for computing devices: computers whose components are individual
More informationShor s Prime Factorization Algorithm
Shor s Prime Factorization Algorithm Bay Area Quantum Computing Meetup - 08/17/2017 Harley Patton Outline Why is factorization important? Shor s Algorithm Reduction to Order Finding Order Finding Algorithm
More informationQuantum Physics II (8.05) Fall 2002 Assignment 3
Quantum Physics II (8.05) Fall 00 Assignment Readings The readings below will take you through the material for Problem Sets and 4. Cohen-Tannoudji Ch. II, III. Shankar Ch. 1 continues to be helpful. Sakurai
More informationDECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS
DECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS SCOTT HOTTOVY Abstract. Quantum networks are used to transmit and process information by using the phenomena of quantum mechanics.
More informationQuantum Computation 650 Spring 2009 Lectures The World of Quantum Information. Quantum Information: fundamental principles
Quantum Computation 650 Spring 2009 Lectures 1-21 The World of Quantum Information Marianna Safronova Department of Physics and Astronomy February 10, 2009 Outline Quantum Information: fundamental principles
More informationIBM quantum experience: Experimental implementations, scope, and limitations
IBM quantum experience: Experimental implementations, scope, and limitations Plan of the talk IBM Quantum Experience Introduction IBM GUI Building blocks for IBM quantum computing Implementations of various
More informationQuantum Query Algorithm Constructions for Computing AND, OR and MAJORITY Boolean Functions
LATVIJAS UNIVERSITĀTES RAKSTI. 008, 733. sēj.: DATORZINĀTNE UN INFORMĀCIJAS TEHNOLOĢIJAS 15. 38. lpp. Quantum Query Algorithm Constructions for Computing AND, OR and MAJORITY Boolean Functions Alina Vasiljeva
More informationQuantum Information & Quantum Computing
Math 478, Phys 478, CS4803, February 9, 006 1 Georgia Tech Math, Physics & Computing Math 478, Phys 478, CS4803 Quantum Information & Quantum Computing Problems Set 1 Due February 9, 006 Part I : 1. Read
More information1 Measurement and expectation values
C/CS/Phys 191 Measurement and expectation values, Intro to Spin 2/15/05 Spring 2005 Lecture 9 1 Measurement and expectation values Last time we discussed how useful it is to work in the basis of energy
More informationDecomposition of Quantum Gates
Decomposition of Quantum Gates With Applications to Quantum Computing Dean Katsaros, Eric Berry, Diane C. Pelejo, Chi-Kwong Li College of William and Mary January 12, 215 Motivation Current Conclusions
More information6.896 Quantum Complexity Theory September 18, Lecture 5
6.896 Quantum Complexity Theory September 18, 008 Lecturer: Scott Aaronson Lecture 5 Last time we looked at what s known about quantum computation as it relates to classical complexity classes. Today we
More informationGates for Adiabatic Quantum Computing
Gates for Adiabatic Quantum Computing Richard H. Warren Abstract. The goal of this paper is to introduce building blocks for adiabatic quantum algorithms. Adiabatic quantum computing uses the principle
More informationUnitary Dynamics and Quantum Circuits
qitd323 Unitary Dynamics and Quantum Circuits Robert B. Griffiths Version of 20 January 2014 Contents 1 Unitary Dynamics 1 1.1 Time development operator T.................................... 1 1.2 Particular
More information6.896 Quantum Complexity Theory September 9, Lecture 2
6.96 Quantum Complexity Theory September 9, 00 Lecturer: Scott Aaronson Lecture Quick Recap The central object of study in our class is BQP, which stands for Bounded error, Quantum, Polynomial time. Informally
More informationD.5 Quantum error correction
D. QUANTUM ALGORITHMS 157 Figure III.34: E ects of decoherence on a qubit. On the left is a qubit yi that is mostly isoloated from its environment i. Ontheright,aweakinteraction between the qubit and the
More informationPh 219b/CS 219b. Exercises Due: Wednesday 21 November 2018 H = 1 ( ) 1 1. in quantum circuit notation, we denote the Hadamard gate as
h 29b/CS 29b Exercises Due: Wednesday 2 November 208 3. Universal quantum gates I In this exercise and the two that follow, we will establish that several simple sets of gates are universal for quantum
More informationA Quantum Associative Memory Based on Grover s Algorithm
A Quantum Associative Memory Based on Grover s Algorithm Dan Ventura and Tony Martinez Neural Networks and Machine Learning Laboratory (http://axon.cs.byu.edu) Department of Computer Science Brigham Young
More informationROM-BASED COMPUTATION: QUANTUM VERSUS CLASSICAL
arxiv:quant-ph/0109016v2 2 Jul 2002 ROM-BASED COMPUTATION: QUANTUM VERSUS CLASSICAL B. C. Travaglione, M. A. Nielsen Centre for Quantum Computer Technology, University of Queensland St Lucia, Queensland,
More informationTutorial on Quantum Computing. Vwani P. Roychowdhury. Lecture 1: Introduction
Tutorial on Quantum Computing Vwani P. Roychowdhury Lecture 1: Introduction 1 & ) &! # Fundamentals Qubits A single qubit is a two state system, such as a two level atom we denote two orthogonal states
More informationBasics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2016 Mika Hirvensalo Basics on quantum information 1 of 52 Brief
More informationarxiv: v3 [cs.dm] 13 Feb 2018
An Introduction to Quantum Computing, Without the Physics Giacomo Nannicini IBM T.J. Watson, Yorktown Heights, NY nannicini@us.ibm.com Last updated: February 14, 018. arxiv:1708.03684v3 [cs.dm] 13 Feb
More informationCompute the Fourier transform on the first register to get x {0,1} n x 0.
CS 94 Recursive Fourier Sampling, Simon s Algorithm /5/009 Spring 009 Lecture 3 1 Review Recall that we can write any classical circuit x f(x) as a reversible circuit R f. We can view R f as a unitary
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing The lecture notes were prepared according to Peter Shor s papers Quantum Computing and Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a
More informationIs Quantum Search Practical?
DARPA Is Quantum Search Practical? George F. Viamontes Igor L. Markov John P. Hayes University of Michigan, EECS Outline Motivation Background Quantum search Practical Requirements Quantum search versus
More informationAN INTRODUCTION TO QUANTUM COMPUTING
AN INTRODUCTION TO QUANTUM COMPUTING NOSON S YANOFSKY Abstract Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics It concerns a utilization of
More informationChapter 2. Mathematical formalism of quantum mechanics. 2.1 Linear algebra in Dirac s notation
Chapter Mathematical formalism of quantum mechanics Quantum mechanics is the best theory that we have to explain the physical phenomena, except for gravity. The elaboration of the theory has been guided
More information