Errata list, Nielsen & Chuang. rrata/errata.html
|
|
- Erin Harper
- 6 years ago
- Views:
Transcription
1 Errata list, Nielsen & Chuang rrata/errata.html
2 Part II, Nielsen & Chuang Quantum circuits (Ch 4) SK Quantum algorithms (Ch 5 & 6) Göran Johansson Physical realisation of quantum computers (Ch 7) Andreas Walther
3 Quantum computation and quantum information Chapter 4 Quantum circuits
4 Chapter 4 Quantum circuits Quantum circuits provide us with a language for describing quantum algorithms We can quantify the resources needed for a quantum algorithm in terms of gates, operations etc. It also provides a toolbox for algorithm design. Simulation of quantum systems
5 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
6 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
7 Why are there few quantum algorithms Algorithm design is difficult Quantum algorithms need to be better than classical algorithms to be interesting Our intuition works better for classical algorithms, making obtaining ideas about quantum algorithms still harder
8 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
9 Chapter 4, Quantum circuits 4.2 Single qubit operations We will analyse arbitrary single qubit and controlled single qubit operations
10 In quantum information data is represented by quantum bits (qubits) A qubit is a quantum mechanical systems with two states > and > that can be in any arbitrary superposition of those states +
11 Qubit representation e i cos e i sin 2 2
12 The Bloch sphere i i e cos e sin 2 2
13 Single qubit gates Single qubit gates, U, are unitary
14 .3. Single qubit gates
15 Fig 4.2, page 77
16 Single qubit gates on Bloch sphere X-gate 8 rotation around x-axis Y-gate 8 rotation around y-axis Z-gate 8 rotation around z-axis
17 The Bloch sphere i i e cos e sin 2 2
18 Fig 4.2, page 77
19 Single qubit gates on Bloch S-gate sphere 9 rotation around z-axis T-gate 45 rotation around z-axis H-gate 9 rotation around the y-axis followed by 8 rotation around the x-axis
20 Rotation an arbitrary angle around axes x, y and z
21 Rotation an arbitrary angle around axes x, y and z H-gate 9 rotation around y-axis followed by 8 rotation around x-axis Show this from Eqs 4.4 and 4.5 (does not commute)
22 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
23 CNOT is a MODULO-2 addition of the control bit to the target bit c t c t c
24 Fig 4.4, control-u c t c U c t
25 Controlled arbitrary rotation c t c U c t Lets start with inputs c> = > and t> = >, what is the output? Now we choose c> = > and t> = >, what is the output?
26 Controlled arbitrary rotation
27 Preparing for logical gates Page 76
28 Arbitrary single qubit operation in terms of z and y rotations (page 75)
29 Preparing for logical gates Page 76
30 Controlled arbitrary rotation
31 Notation Computational basis states (page 22) A quantum circuit operating on n qubits acts in a 2 n -dimensional complex Hilbert space. The computational basis state are product states x,...,x n > where x i =,. For example a two qubit state x > x 2 > in the computational basis is expressed as > = > + > + > + d > Where we have the basis states for the tensor product of the x > x 2 > basis states
32 .4.3 Deutsch s algorithm Calculates f ( ) f () Read Deutsch-Josza algorithm before Görans lectures on Thursday
33 Hadamard gates, exercise 4.6
34 Exercise 4.6, page Matrix for H-gate on x 2 Matrix for H-gate on x 2 2
35 H-gate is unitary Matrix multiplication H H=I 2 2
36 Exercise 4.2
37 Multiplying the Hadamard gates We can just extend exercise 4.6 by multiplying the two Hadamard gates 2 2 2
38 Exercise 4.2 Multiplying Hadamard and CNOT gates give Answer: a CNOT gate where the first bit is the control bit 2 2 What operation is this matrix carrying out?
39 Two control bits
40 Many control bits
41 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
42 .3.6 Bell states (EPR states) An entangled state basis
43 .3.7 Quantum teleportation Quantum teleportation
44 Principle of deferred measurement Measurements followed by classically controlled operations can always be replaced by conditional quantum operations Why is Alice not transmitting information faster than the speed of light through the measurment now when no classical operations need to be carried out by Bob?
45 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
46 Universal quantum gates (4.5) A set of gates is universal if any unitary operation can be approximated to arbitrary accuracy by quantum circuits only involving these gates
47 CNOT, Hadamard and T-gates form a universal set. An arbitrary unitary operator can be expressed as a product of unitary operators acting on two computational basis states ( subsection 4.5.). 2. An arbitrary unitary operator acting on two computational basis states can be expressed as a product of single qubit operations and CNOT gates. (page 92-93) (subsection 4.5.2, exercise 4.39)
48 Exercise 4.93, Page 93
49 CNOT, Hadamard and T-gates form a universal set. An arbitrary unitary operator can be expressed as a product of unitary operators acting on two computational basis states ( subsection 4.5.). 2. An arbitrary unitary operator acting on two computational basis states can be expressed as a product of single qubit operations and CNOT gates. (page 92-93) (subsection 4.5.2, exercise 4.39) 3. Single qubit operations may be approximated to arbitrary accuracy using Hadamard and T gates (subsection 4.5.3)
50 Single qubit operations may be approximated to arbitrary accuracy using H- and T-gates (subsection 4.5.3) From the corrected version of Eq. 4.3 page 76 we see that (exercise 4.): If m and n are non parallel vectors in three dimensions any arbitrary single qubit unitary operation, U may be written as U e i R n ( ) Rm( ) Rn ( 2) Rm( 2)... It is shown on page 96 that rotations of arbitrary angles can be carried out around two different axes using combinations of H- and T-gates
51 Approximating arbitrary unitary gates (4.5.3 & 4.5.4) In order to approximate a quantum circuit consisting of m CNOT and single qubit gates with an accuracy e, only about O[m*log(m/e)] gate operations are required (Solovay-Kitaev theorem). This does not sound too bad! The problem is that of the order 2 2n gates are required to implement an arbitrary n-qubit unitary operation (see page 9-93), thus this is a computationally hard problem.
52 Quantum Computational Complexity Relation to classical complexity classes BPP is thought to be subset of BQP, enabling QC to solve some problems more efficient than classical computers P BPP BQP PSPACE BQP is a subset of PSPACE a) Any problem consuming polynomial time can consume a maximal polynomial amount of space b) A problem inefficiently solvable regarding time might still be solved efficiently regarding space (usage of the same set of gates)
53 Complexity classes
54 The power of quantum computation NP complete problems are a subgroup of NP If any NP-complete problem has a polynomial time solution then P=NP Factoring is not known to be NP-complete Quantum computers are known to solve all problems in P efficiently but cannot solve problems outside PSPACE efficiently If quantum computers are proved to be more efficient than classical computers P=/=PSPACE Quantum computers are (supposedly) experimentally realizable, thus this is not just a mathematical curiosity but also tells us something about how the limits and power of computation in general
55 Quantum computing changes the landscape of computer science QC algorithms do not violate the Church- Turing thesis: any algorithmic process can be simulated using a Turing machine QC algorithms challenge the strong version of the Church-Turing thesis If an algorithm can be performed at any class of hardware, then there is an equivalent efficient algorithm for a Turing machine
56 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
57 The quantum circuit model of computation Properties: Quantum computer may be a hybrid of quantum and classical resources to maximize efficiency A quantum circuit operates on n qubits spanning a 2 n dimensional state space. The product states of the form x,x 2,..,x n > ;x i ={,} are the computational basis states Basic steps: Preparation Computation Measurement
58 The quantum circuit model of computation Maybe it would be better that the computational basis states are entangled Maybe the measurements should not be carried out in the computational basis It is not known whether the quantum circuit model constitute an optimum quantum computer language
59 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement 4.5 Universal quantum gates 4.6 Circuit model summary 4.7 Simulation of quantum systems
60 The quantum-mechanical computation of one molecule of methane requires 42 grid points. Assuming that at each point we have to perform only elementary operations, and the computation is performed at the extremely low temperature T=3-3 K, we would still have to use all the energy produced on Earth during the last century Page 24
61 Quantum simulation Feynman 982 Chemistry (large molecules, reactions) Biology (even larger molecules) Storing the amplitudes of 5 connected qubits may be possible on the supercomputers in the second half of the 2st century provided Moore s law continues to hold Simulation of the properties of new synthesized molecules
62 Simulation of quantum systems i d dt H With n qubits there are 2 n different differential equations that must be solved The equation above has the formal solution i Ht ( t) e ()
63 Simulation of quantum systems While quantum computers are hoped to solve general calculations efficiently, another field in which they are hoped to succeed is the simulation of specific physical systems described by a Hamiltonian. However the physical system must be approximated efficiently, to do so: y()> y ()> Approximating Approximating the the Operator state Discretizing The continuous of the function differential is discretised equations to begins arbitrary with choosing precision an using appropriate a finite set Dt. of It basis has to vectors fulfill the demands set by the maximum Exp[-iHt] error. y(t)> c(x) (x)dx ' The approximation of the differential operator is a cthree kk step process. k The set of basis vectors must be chosen such that for First, any given if possible time, separate the approximated H into a set state of has Hamiltonians, to be equal to H the i, which original act state on a within maximal a given constant error number tolerance of particles. (next neighbor interaction, etc). Secondly, write the effect of H on the system as time evolves as a product U of the effect of the individual y (t)> H i. Thirdly, the effect of H i is written in terms of a quantum circuit.
64 Simulation of quantum systems Approximating the Operator The evolution operator is of the form Exp[ iht] If By possible varying H n is we broken can obtain down a into product a sum representation of local interactions of the evolution operator within any given error H Exp (Trotter formula) H k 2 i( Hk H2) Dt Exp ihdtexp ih 2Dt O( Dt ) If the sub Hamiltonians do not commute it follows that Now that H is broken down into a product of local Hamiltonians which may be written on a quantum circuit form (Exercise 4.5). The product of the circuit corresponds to a unitary operator U. Exp[ i(h H...)t] Exp[ ih t]exp[ ih t]... U 2 2 ih t Exp ih 2 t Exp ih n We now introduce the Trotter formula Exp[ i( A ( t t Exp... ( t Dt) f ) B) jt ] U U ( t ( t) lim ) ; t n f Exp iat / nexp ibt / n jdt n
65 Exercise 4.5, page 2
66 Chapter 4, Quantum circuits 4. Quantum algorithms 4.2 Single qubit operations 4.3 Controlled operations 4.4 Measurement
67 Chapter 4, Quantum circuits 4.5 Universal quantum gates Single qubit and CNOT gates are universal A discrete set of universal operators Approximating arbitrary unitary gates is hard Quantum computational complexity 4.6 Circuit model summary 4.7 Simulation of quantum systems
Single 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 informationQuantum Computation. Michael A. Nielsen. University of Queensland
Quantum Computation Michael A. Nielsen University of Queensland Goals: 1. To eplain the quantum circuit model of computation. 2. To eplain Deutsch s algorithm. 3. To eplain an alternate model of quantum
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 informationSeminar 1. Introduction to Quantum Computing
Seminar 1 Introduction to Quantum Computing Before going in I am also a beginner in this field If you are interested, you can search more using: Quantum Computing since Democritus (Scott Aaronson) Quantum
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 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 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 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 information1 Quantum Circuits. CS Quantum Complexity theory 1/31/07 Spring 2007 Lecture Class P - Polynomial Time
CS 94- Quantum Complexity theory 1/31/07 Spring 007 Lecture 5 1 Quantum Circuits A quantum circuit implements a unitary operator in a ilbert space, given as primitive a (usually finite) collection of gates
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 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 informationShort introduction to Quantum Computing
November 7, 2017 Short introduction to Quantum Computing Joris Kattemölle QuSoft, CWI, Science Park 123, Amsterdam, The Netherlands Institute for Theoretical Physics, University of Amsterdam, Science Park
More informationIntroduction to Quantum Information Processing
Introduction to Quantum Information Processing Lecture 6 Richard Cleve Overview of Lecture 6 Continuation of teleportation Computation and some basic complexity classes Simple quantum algorithms in the
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 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 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 informationGrover s algorithm. We want to find aa. Search in an unordered database. QC oracle (as usual) Usual trick
Grover s algorithm Search in an unordered database Example: phonebook, need to find a person from a phone number Actually, something else, like hard (e.g., NP-complete) problem 0, xx aa Black box ff xx
More informationLecture 22: Quantum computational complexity
CPSC 519/619: Quantum Computation John Watrous, University of Calgary Lecture 22: Quantum computational complexity April 11, 2006 This will be the last lecture of the course I hope you have enjoyed the
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 informationQuantum Computing. Thorsten Altenkirch
Quantum Computing Thorsten Altenkirch Is Computation universal? Alonzo Church - calculus Alan Turing Turing machines computable functions The Church-Turing thesis All computational formalisms define the
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 informationb) (5 points) Give a simple quantum circuit that transforms the state
C/CS/Phy191 Midterm Quiz Solutions October 0, 009 1 (5 points) Short answer questions: a) (5 points) Let f be a function from n bits to 1 bit You have a quantum circuit U f for computing f If you wish
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Petros Wallden Lecture 7: Complexity & Algorithms I 13th October 016 School of Informatics, University of Edinburgh Complexity - Computational Complexity: Classification
More informationIntroduction to Quantum Computation
Chapter 1 Introduction to Quantum Computation 1.1 Motivations The main task in this course is to discuss application of quantum mechanics to information processing (or computation). Why? Education:Asingleq-bitisthesmallestpossiblequantummechanical
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 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 informationSimon s algorithm (1994)
Simon s algorithm (1994) Given a classical circuit C f (of polynomial size, in n) for an f : {0, 1} n {0, 1} n, such that for a certain s {0, 1} n \{0 n }: x, y {0, 1} n (x y) : f (x) = f (y) x y = s with
More informationIntroduction to Quantum Information, Quantum Computation, and Its Application to Cryptography. D. J. Guan
Introduction to Quantum Information, Quantum Computation, and Its Application to Cryptography D. J. Guan Abstract The development of quantum algorithms and quantum information theory, as well as the design
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 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 information2.0 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 informationShort Course in Quantum Information Lecture 5
Short Course in Quantum Information Lecture 5 Quantum Algorithms Prof. Andrew Landahl University of New Mexico Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html
More informationQuantum gate. Contents. Commonly used gates
Quantum gate From Wikipedia, the free encyclopedia In quantum computing and specifically the quantum circuit model of computation, a quantum gate (or quantum logic gate) is a basic quantum circuit operating
More information6.080/6.089 GITCS May 6-8, Lecture 22/23. α 0 + β 1. α 2 + β 2 = 1
6.080/6.089 GITCS May 6-8, 2008 Lecturer: Scott Aaronson Lecture 22/23 Scribe: Chris Granade 1 Quantum Mechanics 1.1 Quantum states of n qubits If you have an object that can be in two perfectly distinguishable
More informationThe Deutsch-Josza Algorithm in NMR
December 20, 2010 Matteo Biondi, Thomas Hasler Introduction Algorithm presented in 1992 by Deutsch and Josza First implementation in 1998 on NMR system: - Jones, JA; Mosca M; et al. of a quantum algorithm
More informationQuantum Information Processing and Diagrams of States
Quantum Information and Diagrams of States September 17th 2009, AFSecurity Sara Felloni sara@unik.no / sara.felloni@iet.ntnu.no Quantum Hacking Group: http://www.iet.ntnu.no/groups/optics/qcr/ UNIK University
More information*WILEY- Quantum Computing. Joachim Stolze and Dieter Suter. A Short Course from Theory to Experiment. WILEY-VCH Verlag GmbH & Co.
Joachim Stolze and Dieter Suter Quantum Computing A Short Course from Theory to Experiment Second, Updated and Enlarged Edition *WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA Contents Preface XIII 1 Introduction
More informationQuantum Computing Lecture 8. Quantum Automata and Complexity
Quantum Computing Lecture 8 Quantum Automata and Complexity Maris Ozols Computational models and complexity Shor s algorithm solves, in polynomial time, a problem for which no classical polynomial time
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 informationIntroduction to Quantum Algorithms Part I: Quantum Gates and Simon s Algorithm
Part I: Quantum Gates and Simon s Algorithm Martin Rötteler NEC Laboratories America, Inc. 4 Independence Way, Suite 00 Princeton, NJ 08540, U.S.A. International Summer School on Quantum Information, Max-Planck-Institut
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 information. An introduction to Quantum Complexity. Peli Teloni
An introduction to Quantum Complexity Peli Teloni Advanced Topics on Algorithms and Complexity µπλ July 3, 2014 1 / 50 Outline 1 Motivation 2 Computational Model Quantum Circuits Quantum Turing Machine
More informationFourier Sampling & Simon s Algorithm
Chapter 4 Fourier Sampling & Simon s Algorithm 4.1 Reversible Computation A quantum circuit acting on n qubits is described by an n n unitary operator U. Since U is unitary, UU = U U = I. This implies
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 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 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 Computing. Joachim Stolze and Dieter Suter. A Short Course from Theory to Experiment. WILEY-VCH Verlag GmbH & Co. KGaA
Joachim Stolze and Dieter Suter Quantum Computing A Short Course from Theory to Experiment Second, Updated and Enlarged Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA Preface XIII 1 Introduction and
More informationPh 219b/CS 219b. Exercises Due: Wednesday 4 December 2013
1 Ph 219b/CS 219b Exercises Due: Wednesday 4 December 2013 4.1 The peak in the Fourier transform In the period finding algorithm we prepared the periodic state A 1 1 x 0 + jr, (1) A j=0 where A is the
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 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 informationBasics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2014 Mika Hirvensalo Basics on quantum information 1 of 49 Brief
More informationα x x 0 α x x f(x) α x x α x ( 1) f(x) x f(x) x f(x) α x = α x x 2
Quadratic speedup for unstructured search - Grover s Al- CS 94- gorithm /8/07 Spring 007 Lecture 11 01 Unstructured Search Here s the problem: You are given an efficient boolean function f : {1,,} {0,1},
More informationC/CS/Phys C191 Quantum Gates, Universality and Solovay-Kitaev 9/25/07 Fall 2007 Lecture 9
C/CS/Phys C191 Quantum Gates, Universality and Solovay-Kitaev 9/25/07 Fall 2007 Lecture 9 1 Readings Benenti, Casati, and Strini: Quantum Gates Ch. 3.2-3.4 Universality Ch. 3.5-3.6 2 Quantum Gates Continuing
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 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 information- Why aren t there more quantum algorithms? - Quantum Programming Languages. By : Amanda Cieslak and Ahmana Tarin
- Why aren t there more quantum algorithms? - Quantum Programming Languages By : Amanda Cieslak and Ahmana Tarin Why aren t there more quantum algorithms? there are only a few problems for which quantum
More informationThe Solovay-Kitaev theorem
The Solovay-Kitaev theorem Maris Ozols December 10, 009 1 Introduction There are several accounts of the Solovay-Kitaev theorem available [K97, NC00, KSV0, DN05]. I chose to base my report on [NC00], since
More informationShor s Algorithm. Polynomial-time Prime Factorization with Quantum Computing. Sourabh Kulkarni October 13th, 2017
Shor s Algorithm Polynomial-time Prime Factorization with Quantum Computing Sourabh Kulkarni October 13th, 2017 Content Church Thesis Prime Numbers and Cryptography Overview of Shor s Algorithm Implementation
More informationQuantum Circuits and Algorithms
Quantum Circuits and Algorithms Modular Arithmetic, XOR Reversible Computation revisited Quantum Gates revisited A taste of quantum algorithms: Deutsch algorithm Other algorithms, general overviews Measurements
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 informationSimulating classical circuits with quantum circuits. The complexity class Reversible-P. Universal gate set for reversible circuits P = Reversible-P
Quantum Circuits Based on notes by John Watrous and also Sco8 Aaronson: www.sco8aaronson.com/democritus/ John Preskill: www.theory.caltech.edu/people/preskill/ph229 Outline Simulating classical circuits
More informationCOMPARATIVE ANALYSIS ON TURING MACHINE AND QUANTUM TURING MACHINE
Volume 3, No. 5, May 2012 Journal of Global Research in Computer Science REVIEW ARTICLE Available Online at www.jgrcs.info COMPARATIVE ANALYSIS ON TURING MACHINE AND QUANTUM TURING MACHINE Tirtharaj Dash
More informationClassical Verification of Quantum Computations
Classical Verification of Quantum Computations Urmila Mahadev UC Berkeley September 12, 2018 Classical versus Quantum Computers Can a classical computer verify a quantum computation? Classical output (decision
More informationQuantum Algorithms Lecture #3. Stephen Jordan
Quantum Algorithms Lecture #3 Stephen Jordan Summary of Lecture 1 Defined quantum circuit model. Argued it captures all of quantum computation. Developed some building blocks: Gate universality Controlled-unitaries
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 informationQuantum Computer. Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study
Quantum Computer Jaewan Kim jaewan@kias.re.kr School of Computational Sciences Korea Institute for Advanced Study KIAS (Korea Institute for Advanced Study) Established in 1996 Located in Seoul, Korea Pure
More informationQuantum Information Processing with Liquid-State NMR
Quantum Information Processing with Liquid-State NMR Pranjal Vachaspati, Sabrina Pasterski MIT Department of Physics (Dated: May 8, 23) We demonstrate the use of a Bruker Avance 2 NMR Spectrometer for
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 informationCS286.2 Lecture 8: A variant of QPCP for multiplayer entangled games
CS286.2 Lecture 8: A variant of QPCP for multiplayer entangled games Scribe: Zeyu Guo In the first lecture, we saw three equivalent variants of the classical PCP theorems in terms of CSP, proof checking,
More informationDiscrete Quantum Theories
Discrete Quantum Theories Andrew J. Hanson 1 Gerardo Ortiz 2 Amr Sabry 1 Yu-Tsung Tai 3 (1) School of Informatics and Computing (2) Department of Physics (3) Mathematics Department Indiana University July
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 informationON THE ROLE OF THE BASIS OF MEASUREMENT IN QUANTUM GATE TELEPORTATION. F. V. Mendes, R. V. Ramos
ON THE ROLE OF THE BASIS OF MEASREMENT IN QANTM GATE TELEPORTATION F V Mendes, R V Ramos fernandovm@detiufcbr rubens@detiufcbr Lab of Quantum Information Technology, Department of Teleinformatic Engineering
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 information6.896 Quantum Complexity Theory 30 October Lecture 17
6.896 Quantum Complexity Theory 30 October 2008 Lecturer: Scott Aaronson Lecture 17 Last time, on America s Most Wanted Complexity Classes: 1. QMA vs. QCMA; QMA(2). 2. IP: Class of languages L {0, 1} for
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 informationQuantum Computing. Quantum Computing. Sushain Cherivirala. Bits and Qubits
Quantum Computing Bits and Qubits Quantum Computing Sushain Cherivirala Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited
More informationBaby's First Diagrammatic Calculus for Quantum Information Processing
Baby's First Diagrammatic Calculus for Quantum Information Processing Vladimir Zamdzhiev Department of Computer Science Tulane University 30 May 2018 1 / 38 Quantum computing ˆ Quantum computing is usually
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 informationquantum mechanics is a hugely successful theory... QSIT08.V01 Page 1
1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation What is quantum mechanics good for? traditional historical perspective: beginning of 20th century: classical physics fails
More information1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation
QSIT09.V01 Page 1 1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation What is quantum mechanics good for? traditional historical perspective: beginning of 20th century: classical
More informationLower Bounds on the Classical Simulation of Quantum Circuits for Quantum Supremacy. Alexander M. Dalzell
Lower Bounds on the Classical Simulation of Quantum Circuits for Quantum Supremacy by Alexander M. Dalzell Submitted to the Department of Physics in partial fulfillment of the requirements for the degree
More informationQuantum Computational Complexity
Quantum Computational Complexity John Watrous Institute for Quantum Computing and School of Computer Science University of Waterloo, Waterloo, Ontario, Canada. arxiv:0804.3401v1 [quant-ph] 21 Apr 2008
More informationTeleportation-based approaches to universal quantum computation with single-qubit measurements
Teleportation-based approaches to universal quantum computation with single-qubit measurements Andrew Childs MIT Center for Theoretical Physics joint work with Debbie Leung and Michael Nielsen Resource
More informationLecture 6: Quantum error correction and quantum capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde The quantum capacity theorem is one of the most important theorems in quantum hannon theory. It is a fundamentally quantum theorem
More informationX row 1 X row 2, X row 2 X row 3, Z col 1 Z col 2, Z col 2 Z col 3,
1 Ph 219c/CS 219c Exercises Due: Thursday 9 March 2017.1 A cleaning lemma for CSS codes In class we proved the cleaning lemma for stabilizer codes, which says the following: For an [[n, k]] stabilizer
More informationGreat Ideas in Theoretical Computer Science. Lecture 28: A Gentle Introduction to Quantum Computation
5-25 Great Ideas in Theoretical Computer Science Lecture 28: A Gentle Introduction to Quantum Computation May st, 208 Announcements Please fill out the Faculty Course Evaluations (FCEs). https://cmu.smartevals.com
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 informationIs Entanglement Sufficient to Enable Quantum Speedup?
arxiv:107.536v3 [quant-ph] 14 Sep 01 Is Entanglement Sufficient to Enable Quantum Speedup? 1 Introduction The mere fact that a quantum computer realises an entangled state is ususally concluded to be insufficient
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 informationQuantum information and quantum computing
Middle East Technical University, Department of Physics January 7, 009 Outline Measurement 1 Measurement 3 Single qubit gates Multiple qubit gates 4 Distinguishability 5 What s measurement? Quantum measurement
More informationPhysics ; CS 4812 Problem Set 4
Physics 4481-7681; CS 4812 Problem Set 4 Six problems (six pages), all short, covers lectures 11 15, due in class 25 Oct 2018 Problem 1: 1-qubit state tomography Consider a 1-qubit state ψ cos θ 2 0 +
More informationIntroduction to Quantum Information Hermann Kampermann
Introduction to Quantum Information Hermann Kampermann Heinrich-Heine-Universität Düsseldorf Theoretische Physik III Summer school Bleubeuren July 014 Contents 1 Quantum Mechanics...........................
More informationTHE RESEARCH OF QUANTUM PHASE ESTIMATION
THE RESEARCH OF QUANTUM PHASE ESTIMATION ALGORITHM Zhuang Jiayu 1,Zhao Junsuo,XuFanjiang, QiaoPeng& Hu Haiying 1 Science and Technology on Integrated Information System Laboratory, Institute of Software
More informationInformation quantique, calcul quantique :
Séminaire LARIS, 8 juillet 2014. Information quantique, calcul quantique : des rudiments à la recherche (en 45min!). François Chapeau-Blondeau LARIS, Université d Angers, France. 1/25 Motivations pour
More informationQUANTUM COMPUTING. Part II. Jean V. Bellissard. Georgia Institute of Technology & Institut Universitaire de France
QUANTUM COMPUTING Part II Jean V. Bellissard Georgia Institute of Technology & Institut Universitaire de France QUANTUM GATES: a reminder Quantum gates: 1-qubit gates x> U U x> U is unitary in M 2 ( C
More informationQuantum NP - Cont. Classical and Quantum Computation A.Yu Kitaev, A. Shen, M. N. Vyalyi 2002
Quantum NP - Cont. Classical and Quantum Computation A.Yu Kitaev, A. Shen, M. N. Vyalyi 2002 1 QMA - the quantum analog to MA (and NP). Definition 1 QMA. The complexity class QMA is the class of all languages
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 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 informationChallenges in Quantum Information Science. Umesh V. Vazirani U. C. Berkeley
Challenges in Quantum Information Science Umesh V. Vazirani U. C. Berkeley 1 st quantum revolution - Understanding physical world: periodic table, chemical reactions electronic wavefunctions underlying
More informationHow quantum computation gates can be realized in terms of scattering theory approach to quantum tunneling of charge transport
ISSN: 2347-3215 Volume 3 Number 3 (March-2015) pp. 62-66 www.ijcrar.com How quantum computation gates can be realized in terms of scattering theory approach to quantum tunneling of charge transport Anita
More information