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Quantum Computing and Quantum Parallelism

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Title: Quantum Computing and Quantum Parallelism


1
Quantum Computing and Quantum Parallelism
  • Dan C. Marinescu and Gabriela M. Marinescu
  • School of Computer Science
  • University of Central Florida
  • Orlando, Florida 32816, USA

2
Acknowledgments
  • The material presented is from the book
  • Approaching Quantum Computing
  • by Dan C. Marinescu and Gabriela M. Marinescu
  • ISBN 013145224X, Prentice Hall, July 2004.
  • Work supported by National Science Foundation
    grants MCB9527131, DBI0296107,ACI0296035, and
    EIA0296179.

3
Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

4
Technological limits density and speed
  • For the past two decades we have enjoyed Gordon
    Moores law ?the speed doubles every 18 months.
  • But all good things may come to an end
  • We are limited in our ability to increase
  • the density of solid-state circuits ? due to
  • power dissipation and
  • quantum effects.
  • the speed of a computing device ? due to
  • density

5
Technological limits reliability
  • Reliability will also be affected
  • to increase the speed we need increasingly
    smaller circuits (light needs 1 ns to travel 30
    cm in vacuum)
  • smaller circuits ? systems consisting only of a
    few particles subject to Heissenbergs
    uncertainty

6
Energy/operation
  • If there is a minimum amount of energy dissipated
    to perform an elementary operation, then to
    increase the speed, thus the number of operations
    performed each second, we require at least a
    linear increase of the amount of energy
    dissipated by the device.
  • The computer technology vintage year 2000
    requires some 3 x 10-18 Joules per elementary
    operation.

7
The effect of increasing the speed upon the power
consumption
  • Assume that
  • the minimum amount of energy dissipated to
    perform an elementary operation is reduced
    100-fold (this may not be technologically
    feasible)
  • the speed of a solid state device is increased
    1,000 fold
  • Then we shall see a 10 (ten) fold increase in the
    amount of power needed by a solid state device
    operating at a 1,000 times higher speed.

8
Power dissipation, circuit density, and speed
  • In 1992 Ralph Merkle from Xerox PARC calculated
    that a 1 GHz computer operating at room
    temperature, with 1018 gates packed in a volume
    of about 1 cm3 would dissipate 3 MW of power.
  • A small city with 1,000 homes each using 3 KW
    would require the same amount of power
  • A 500 MW nuclear reactor could only power some
    166 such circuits.

9
Heat generation
  • The heat produced by a super dense computing
    engine is proportional with the number of
    elementary computing circuits, thus, with the
    volume of the engine.
  • If the devices are densely packed in a sphere of
    radius r the heat dissipated grows as the cube of
    the radius.

10
Heat removal
  • If the devices are densely packed in a sphere of
    radius r, then the surface of the sphere is
    proportional with the square of the radius.
  • To prevent the destruction of the engine we have
    to remove the heat through a surface surrounding
    the device.
  • Our ability to remove heat increases as the
    square of the radius while the amount of heat
    increases with the cube of the radius of the
    computing device.

11
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12
Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

13
A happy marriage
  • Quantum computing and quantum information theory
    ?a product of a happy marriage between two of the
    greatest scientific achievements of the 20th
    century
  • quantum mechanics
  • stored program computers

14
Quantum
  • Quantum ? Latin word meaning some quantity.
  • In physics used with the same meaning as the word
    discrete in mathematics, i.e., some quantity or
    variable that can take only sharply defined
    values as opposed to a continuously varying
    quantity.
  • The concepts continuum and continuous are known
    from geometry and calculus.

15
Quantum mechanics
  • Quantum mechanics is a mathematical model of the
    physical world.
  • Quantum properties such as
  • uncertainty,
  • interference, and
  • entanglement
  • do not have a correspondent in classical
    physics.

16
Heissenbergs uncertainty principle
  • The position and the momentum of a quantum
    particle cannot be determined with arbitrary
    precision.
  • h1.054 10-34 J second ? reduced Plancks
    constant

17
Max Borns Nobel prize lecture, Dec. 11, 1954
  • ... Quantum Mechanics shows that not only the
    determinism of classical physics must be
    abandoned, but also the naive concept of reality
    which looked upon atomic particles as if they
    were very small grains of sand. At every instant
    a grain of sand has a definite position and
    velocity. This is not the case with an electron.
    If the position is determined with increasing
    accuracy, the possibility of ascertaining its
    velocity becomes less and vice versa.

18
Quantum theory and computing and communication
  • Quantum theory
  • Does not play only a supporting role by
    prescribing the limitations of physical systems
    used for computing and communication
  • It provides a revolutionary rather than an
    evolutionary approach to computing and
    communication.

19
Milestones in quantum physics
  • 1900 - Max Plank ? black body radiation theory
    the foundation of quantum theory.
  • 1905 - Albert Einstein ? the theory of the
    photoelectric effect.
  • 1911 - Ernest Rutherford ? the planetary model
    of the atom.
  • 1913 - Niels Bohr ? the quantum model of the
    hydrogen atom.
  • 1923 - Louis de Broglie ? relates the momentum of
    a particle with the wavelength.
  • 1925 - Werner Heisenberg ? the matrix quantum
    mechanics.

20
Milestones in quantum physics (contd)
  • 1926 - Erwin Schrödinger ? Schrödingers equation
    for the dynamics of the wave function.
  • 1926 - Erwin Schördinger and Paul Dirac ? show
    the equivalence of Heisenberg's matrix
    formulation and Dirac's algebraic one with
    Schrödinger's wave function.
  • 1926 - Paul Dirac and, independently, Max Born,
    Werner Heisenberg, and Pascual Jordan ? obtain a
    complete formulation of quantum dynamics.
  • 1926 - John von Newmann ?introduces Hilbert
    spaces to quantum mechanics.
  • 1927 - Werner Heisenberg ? the uncertainty
    principle.

21
Milestones in computing and information theory
  • 1936 - Alan Turing ? the Universal Turing
    Machine, UTM.
  • 1936 - Alonzo Church ? every function which
    can be regarded as computable can be computed by
    an universal computing machine''.
  • 1945 J. Presper Eckert and John Macauly ?
    ENIAC, the world's first general purpose
    computer.
  • 1946 - John von Neumann ? the von Neumann
    architecture.
  • 1948 - Claude Shannon ? A Mathematical Theory
    of Communication.
  • 1953 - UNIVAC I ? the first commercial computer,.

22
Milestones in quantum computing the pioneers
  • 1961 - Rolf Landauer ?computation is physical
    studies heat generation.
  • 1973 - Charles Bennet ? logical reversibility of
    computations.
  • 1981 - Richard Feynman ? physical systems
    including quantum systems can be simulated
    exactly with quantum computers.
  • 1982 - Peter Beniof ? develops quantum
    mechanical models of Turing machines.

23
Milestones in quantum computing
  • 1984 - Charles Bennet and Gilles Brassard ?
    quantum cryptography.
  • 1985 - David Deutsch ?reinterprets the
    Church-Turing conjecture.
  • 1993 - Bennet, Brassard, Crepeau, Josza, Peres,
    Wooters ? quantum teleportation.
  • 1994 - Peter Shor ?a clever algorithm for
    factoring large numbers.

24
Can we observe quantum effects with simple
experimental setups?
  • Experiments with light beams.
  • Beam splitters and cascaded beam-splitters.
  • Photon polarization and an experiment with
    polarization filters.
  • Multiple measurements indifferent bases
  • A photon coincidence experiment
  • What do we notice
  • Non-deterministic behavior
  • Strange effects that cannot be explained using
    classical models of physics.

25
Can we construct a mathematical model to explain
the results of the experiments?
  • The model of photon behavior
  • non-deterministic
  • captures superposition effects
  • captures the effect of the measurement process
  • superposition probability rule

26
Light
  • Light ? a form of electromagnetic radiation.
  • The electric and magnetic field
  • oscillate in a plane perpendicular to the
    direction of propagation and
  • are perpendicular to each other.
  • The dual, wave and corpuscular, nature of light
  • Diffraction phenomena ? can only be explained
    assuming a wave-like behavior
  • The photoelectric effect ? corpuscular/granular
    nature of light. The light consists of quantum
    particles called photons.

27
Beam splitters deterministic versus
probabilistic photon behavior
  • Beam splitter? a half silvered mirror. Part of an
    incident beam of light is transmitted and part is
    reflected.
  • What happens when we send a single photon to a
    beam splitter?

28
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29
A single beam splitter
  • Either detector D1 or detector D2 will record
    the arrival of a photon
  • How do we explain this behavior? ?
    probabilistic/genetic model?
  • We repeat the experiment involving a single
    photon over and over again ? D1 and D2 record
    about the same number of events.
  • Does a photon carry a gene?
  • one with a transmit gene ? D2
  • one with a reflect'' gene ? D1?

30
Cascaded beam splitters
  • The experiment? we send a single photon, repeat
    the experiment many times, and count the number
    of events registered by each detector.
  • If the gene theory is true ? the photon is either
    reflected by the first beam splitter or
    transmitted by all of them. Only the first and
    last detectors in the chain are expected to
    register events (each one of them should register
    an equal number of events).
  • The experiment shows ? all detectors have an
    equal chance to register an event.

31
The polarization of light
  • Is given by the electric field vector
  • Linearly polarized light? the electric filed
    oscillates along any straight line in a plane
    perpendicular to the direction of propagation
  • vertical/horizontal polarization
  • diagonal 45/135 deg polarization
  • Circularly polarized light ?the electric field
    vector moves along a circle in a plane
    perpendicular to the direction of propagation
  • Right-hand polarization ?counterclockwise
    rotation
  • Left-hand polarization ? clockwise rotation
  • Elliptically polarized light ?the electric field
    vector moves along an ellipse in a plane
    perpendicular to the direction of propagation.

32
An experiment with polarization analyzers/filters
  • A polarization analyzer or polarized filter ? A
    partially transparent material that transmits
    light of a particular polarization.
  • We perform an experiment involving
  • A source S of linearly polarized light of
    intensity I.
  • A screen E where we measure the intensity of
    incoming beam of light.
  • There types of polarization filters
  • A ? vertical polarization
  • B ? horizontal polarization
  • C ? a 45 degree polarization
  • Each photon has a random orientation of the
    polarization vector.

33
The puzzling observations
  • Without any filter the measured intensity is I.
  • When we introduce a vertically polarized filter
    between the source and the screen the measured
    intensity is I/2.
  • When we introduce a horizontally polarized filter
    between the vertically polarized filter and the
    screen the measured intensity is 0.
  • When we introduce a 45 deg polarized filter
    between the vertically polarized filter and the
    horizontally polarized filter the measured
    intensity is I/8.

34
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35
A mathematical model to describe the state of a
quantum system
are complex numbers
36
Superposition and uncertainty
  • In this model a state
  • is a superposition of two basis states, 0
    and 1 or
  • and (Diracs notation)
  • This state is unknown before we make a
    measurement.
  • After we perform a measurement the system is no
    longer in an uncertain state but it is in one of
    the two basis states
  • ?the probability of observing
    the outcome 1
  • ?the probability of observing
    the outcome 0

37
The measurement of superposition states
  • The polarization of a photon is described by a
    unit vector on a two-dimensional vector space
    with basis
  • 0 gt and
  • 1gt.
  • Measuring the polarization is equivalent to
    projecting the random vector onto one of the two
    basis vectors.
  • Thus after a measurement each photon is forced to
    choose between one of the two basis states.

38
Does the model explain the results?
  • When filter A with vertical polarization is
    inserted between the source S and the screen E
    all photons are forced to choose between vertical
    and horizontal polarization. About half of them
    reach E because they choose vertical polarization
    ? the measured intensity is about I/2.
  • When filter B with horizontal polarization is
    inserted between A and E then none of the
    incoming photons (all have horizontal
    polarization) reach E? the measured intensity is
    0.

39
A puzzling question
  • Why when filter C with a 45 deg. polarization is
    inserted between A and B, the measured intensity
    is intensity is about I / 8?

40
Multiple measurements in different bases
41
Measurements in multiple bases
42
The answer to the puzzling question in the
polarization filters experiment
  • When we insert C, the 45 deg filter we force a
    measurement in a new base (45/135 degree).
  • About half of the I/2 photons with vertical
    polarization
  • (emerging from filter A) pass through filter B
    and exit with a 45 degree polarization.
  • Then these I/4 photons are measured again in new
    basis (Vertical/Horizontal) and about half of
    them choose a horizontal polarization. They pass
    through filter B.
  • Thus, the intensity of the measured light is now
    I/8.

43
The superposition probability rule
  • If an event may occur in two or more
    indistinguishable ways
  • For classical systems Bayes rules
  • In quantum mechanics ?the probability amplitude
    of the event is the sum of the probability
    amplitudes of each case considered separately
    (sometimes known as Feynmans rule).
  • An experiment illustrating the superposition
    probability rule.

44
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45
The experiment
  • We observe experimentally that
  • a photon emitted by S1 is always detected by D1
    and never by D2 and
  • one emitted by S2 is always detected by D2 and
    never by D1.
  • A photon emitted by one of the sources S1 or S2
    may take one of four different paths shown on the
    next slide, depending whether
  • it is transmitted, or
  • reflected
  • by each of the two beam splitters.

46
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47
A photon coincidence experiment
  • One source emits two photons simultaneously into
    two separate beams.
  • Each beam is directed by a reflecting mirror to
    one of two beam splitters.
  • There are two detectors. We never observe a
    coincidence..

48
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49
Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

50
The new frontier in computing and communication
  • Applications of quantum computing and quantum
    information theory
  • Exact simulation of systems with a very large
    state space.
  • Quantum algorithms based upon quantum
    parallelism.
  • Quantum key distribution.

51
Frontier(s)from Websters unabridged dictionary.
  • The part of a settled or civilized country
    nearest to an unsettled or uncivilized region.
  • Any new or incompletely investigated field of
    learning or thought.

52
What is a Quantum computer?
  • A device that harnesses quantum physical
    phenomena such as entanglement and superposition.
  • The laws of quantum mechanics differ radically
    from the laws of classical physics.
  • The unit of information, the qubit can exist as
    a 0, or 1, or, simultaneously, as both 0 and 1.

53
Does quantum computing represent the frontiers
of computing?
  • Is it for real? Can we actually build quantum
    computers? - Very likely, but it will take
    some time.
  • If so, what would a quantum computer allow us to
    do that is either unfeasible or impractical with
    todays most advanced systems? Exact
    simulation of physical systems, among other
    things.
  • Once we have quantum computers do we need new
    algorithms? Yes, we need quantum
    algorithms.
  • Is it so different from our current thinking that
    it requires a substantial change in the way we
    educate our students? Yes, it does.

54
Quantum computers now and then
  • All we have at this time is a 7 qubit quantum
    computer able to compute the prime factors of a
    small integer, 15.
  • To break a code with a key size of 1024 bits
    requires more than 3000 qubits and 108 quantum
    gates.

55
Approximate computer simulation of physical
systems
  • Eniac and the Manhattan project. The first
    programs to run, simulation of physical
    processes.
  • Computer simulation new approach to scientific
    discovery, complementing the two well established
    methods of science experiment and theory.
  • Approximate simulation based upon a model that
    abstracts some properties of interest of a
    physical system.

56
Exact simulation of physical systems
  • How far do we want to go at the microscopic
    level? Molecular, atomic, quantum? - All of the
    above.
  • What about cosmic level? - Yes, of course.
  • Is it important? - - Yes (Feynman,
    1981) .
  • Who will benefit?
  • Natural sciences ? physics, chemistry, biology,
    astrophysics, cosmology,.
  • Applications ? nanotechnology, smart materials,
    drug design,

57
Large problem state space
  • From black hole thermodynamics a system
    enclosed by a surface with area A has N(A)
    observable states with
  • c 3 x1010 cm/sec
  • h 1.054 x 10-34 Joules/second
  • G 6.672 x 10-8 cm3 g-1 sec-2
  • For an object with a radius of 1 Km ? N(A)
    e80

58
Quantum Parallelism
  • In quantum systems the amount of parallelism
    increases exponentially with the size of the
    system, thus with the number of qubits. For
    example, a 21 qubit quantum computer is twice as
    powerfulas as a 20 qubit
  • An exponential increase in the power of a quantum
    computer requires linear increase in the amount
    of matter and space needed to build the larger
    quantum computing engine.
  • A quantum computer will enable us to solve
    problems with a very large state space.

59
Quantum key distribution
  • To ensure confidentiality, data is often
    encrypted.
  • We need for reliable methods for the distribution
    of the encryption keys.
  • The problem ? physical difficulty to detect the
    presence of an intruder when communicating
    through a classical communication channel.

60
Quantum key distribution setup
  • Alice and Bob connected via two communication
    channels
  • a classical one, and
  • a quantum one.
  • Alice sends photons via the quantum channel to
    Bob. A photon may be prepared with
  • vertical/horizontal (VH) or
  • diagonal polarization (DG).
  • Alice and Bob also exchange messages over the
    classical channel.
  • Eve eavesdrops on both communication channels.

61
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62
Information encoding
  • A photon may be used to transmit information on a
    quantum communication channel.
  • The classical binary information may be encoded
    as follows
  • We can use a photon with vertical/horizontal
    (VH) polarization
  • 1 ? a photon with vertical polarization
  • 0 ? a photon with a horizontal polarization.
  • Alternatively we may use a photon with diagonal
    (DG) polarization
  • 1 ? a photon with 45 deg. polarization, and
  • 0 ? a photon with a 135 deg. polarization.

63
The measurements of photons sent over the quantum
communication channel
  • Bob uses a calcite crystal to separate photons
    with different polarization. The crystal is set
    up to separate vertically polarized photons from
    the horizontally polarized ones. To perform a
    measurement in the DG basis the crystal is
    oriented accordingly.
  • Whenever Eve eavesdrops on the quantum
    communication channel she performs a measurement
    thus, she alters the state of the photons.

64
The quantum key distribution algorithm of Bennett
and Brassard (BB84)
  • Alice selects n, the approximate length of the
    encryption key. Alice generates two random
    strings a and b, each of length (4 )n. By
    choosing sufficiently large Alice and Bob can
    ensure that the number of bits kept is close to
    2n with a very high probability.
  • A subset of length n of the bits
  • in string a will be used as the encryption key
    and
  • the bits in string b will be used by Alice to
    select the basis (VH) or (DG) for each photon
    sent to Bob.

65
BB84 (contd)
  • Alice encodes the binary information in string a
    based upon the corresponding values of the bits
    in string b.
  • For example, if the i-th bit of string b is
  • 1 then Alice selects Vertical-Horizontal (VH)
    polarization. If VH is selected, then
  • a 1 in the i-th position of string a is sent as
    a photon with vertical polarization (V), and
  • a 0 as a photon with horizontal (H)
    polarization
  • 0 then Alice selects Diagonal (DG) polarization.
    If DG is selected, then
  • a 1 in the i-th position of string a is sent as
    a photon with a 45 deg. polarization, and
  • a 0 as a photon with 135 deg. polarization.
  • Both Alice and Bob use the same encoding
    convention for each of the bases.

66
BB84 (contd)
  • In turn, Bob picks up randomly (4 )n bits to
    form a string b. He uses one of the two basis
    for the measurement of each incoming photon in
    string a based upon the corresponding value of
    the bit in string b.
  • For example, a 1 in the i-th position of b
    implies that the i-th photon is measured in the
    DG basis, while a 0 requires that the photon is
    measured in the VH basis.
  • As a result of this measurement Bob constructs
    the string a.

67
BB84 (contd)
  • Bob uses the classical communication channel to
    request the string b and Alice responds on the
    same channel with b. Then Bob sends Alice string
    b on the classical channel.
  • Alice and Bob keep only those bits in the set a,
    a for which the corresponding bits in the set
    b, b are equal. Let us assume that Alice and
    Bob keep only 2n bits.

68
BB84 (contd)
  • Alice and Bob perform several tests to determine
    the level of noise and eavesdropping on the
    channel. The set of 2n bits is split into two
    sub-sets of n bits each.
  • One sub-set will be the check bits used to
    estimate the level of noise and eavesdropping,
    and
  • The other consists of the \it data bits used
    for the quantum key.
  • Alice selects n check bits at random and sends
    the positions and values of the selected bits
    over the classical channel to Bob. Then Alice and
    Bob compare the values of the check bits. If more
    than say t bits disagree then they abort and
    re-try the protocol.

69
Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

70
Hilbert space
  • A vector space over the field of complex numbers
    with an inner product (a norm).
  • In mathematics ? Hilbert spaces are
    infinite-dimensional.
  • In quantum mechanics ? finite-dimensional Hilbert
    spaces.
  • The basis vectors in a
  • two-dimensional Hilbert space ? 0gt and 1gt.
  • four- dimensional Hilbert space ? 00gt, 01gt,
    10gt, and 11gt.
  • eight-dimensional Hilbert space ? 000gt, 001gt,
    010gt, 011gt, 100gt, 101gt, 110gt, 101gt, and
    111gt.

71
The tensor product of two vectors in a
two-dimensional Hilbert space
72
Two vectors in a four-dimensional Hilbert space
73
The outer product of two vectors in a
four-dimensional Hilbert space
74
State space dimension of classical and quantum
systems
  • Individual state spaces of n particles combine
    quantum mechanically through the tensor product.
    If X and Y are vectors, then
  • their tensor product X Y is also a vector,
    but its dimension is
  • dim(X) x dim(Y)
  • while the vector product X x Y has dimension
  • dim(X)dim(Y).
  • For example, if dim(X) dim(Y)10, then the
    tensor product of the two vectors has dimension
    100 while the vector product has dimension 20.

75
Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

76
One qubit
  • Mathematical abstraction
  • Vector in a two dimensional complex vector space
    (Hilbert space)
  • Diracs notation
  • ket ? column
    vector
  • bra ? row vector
  • bra ? dual vector (transpose and complex
    conjugate)

77
State description
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79
A bit versus a qubit
  • A bit
  • Can be in two distinct states, 0 and 1
  • A measurement does not affect the state
  • A qubit
  • can be in state or in state or in
    any other state that is a linear combination of
    the basis state
  • When we measure the qubit we find it
  • in state with probability
  • in state with probability

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81
Qubit measurement
82
Two qubits
  • Represented as vectors in a 2-dimensional Hilbert
    space with four basis vectors
  • When we measure a pair of qubits we decide that
    the system it is in one of four states
  • with probabilities

83
Two qubits
84
Measuring two qubits
  • Before a measurement the state of the system
    consisting of two qubits is uncertain (it is
    given by the previous equation and the
    corresponding probabilities).
  • After the measurement the state is certain, it is
  • 00, 01, 10, or 11 like in the case of a
    classical two bit system.

85
Measuring two qubits (contd)
  • What if we observe only the first qubit, what
    conclusions can we draw?
  • We expect the system to be left in an uncertain
    sate, because we did not measure the second qubit
    that can still be in a continuum of states. The
    first qubit can be
  • 0 with probability
  • 1 with probability

86
Measuring two qubits (contd)
  • Call the post-measurement state when we
    measure the first qubit and find it to be 0.
  • Call the post-measurement state when we
    measure the first qubit and find it to be 1.

87
Measuring two qubits (contd)
  • Call the post-measurement state when we
    measure the second qubit and find it to be 0.
  • Call the post-measurement state when we
    measure the second qubit and find it to be 1.

88
Bell states - a special state of a pair of qubits
  • If and
  • When we measure the first qubit we get the
    post measurement state
  • When we measure the second qubit we get the
    post mesutrement state

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This is an amazing result!
  • The two measurements are correlated, once we
    measure the first qubit we get exactly the same
    result as when we measure the second one.
  • The two qubits need not be physically constrained
    to be at the same location and yet, because of
    the strong coupling between them, measurements
    performed on the second one allow us to determine
    the state of the first.

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Entanglement (Vërschrankung)
  • Discovered by Schrödinger.
  • An entangled pair is a single quantum system in a
    superposition of equally possible states.
  • The entangled state contains no information about
    the individual particles, only that they are in
    opposite states.
  • Einstein called entanglement Spooky action at a
    distance".

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Physical embodiment of a qubit
  • The photon ? with tow independent polarizations,
    horizonatal and vertical.
  • The electron ? with two independent spin values,
    1/2 and -1/2.
  • Quantum dots ?
  • Small devices that contain a tiny droplet of free
    electrons.
  • They are fabricated in semiconductor materials
    and have typical dimensions between nanometres to
    a few microns.
  • The size and shape of these structures and
    therefore the number of electrons they contain,
    can be precisely controlled a quantum dot can
    have anything from a single electron to a
    collection of several thousands.
  • The binary information is encoded as the
    presence/absence of electrons.

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Physical embodiment of a qubit (contd)
  • A two-level atom in an optical cavity.
  • Two internal states of an ion in a trap.
  • Others

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The spin
  • In quantum mechanics the intrinsic angular
    moment, the spin, is quantized and the values it
    may take are multiples of the rationalized Planck
    constant.
  • The spin of an atom or of a particle is
    characterized by the spin quantum number s.
  • The spin quantum number s may assume integer and
    half-integer values.
  • The spin is quantized ? for a given value of s
    the projection of the spin on any axis may assume
    2s 1 values ranging from - s to s by unit
    steps.

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More about the spin
  • There are two classes of quantum particles
  • fermions - spin one-half particles such as the
    electrons. The spin quantum numbers of fermions
    can be
  • s1/2 and
  • s-1/2
  • bosons - spin one particles. The spin quantum
    numbers of bosons can be
  • s1,
  • s0, and
  • s-1

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The Stern-Gerlach experiment with hydrogen atoms
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The spin of the electron
  • The electron has spin s 1 /2
  • The spin projection can assume the values
  • ½ ? spin up, and
  • -1/2 ? spin down.

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Communication with entangles particles
  • Even when separated two entangled particles
    continue to interact with one another.
  • Basic idea. Consider three particles
  • Two particles (particle 2 and particle 3)? in an
    anti-correlated state (spin up and spin down).
  • We measure particle 1 and particle 2 and set them
    in an anti-correlated state.
  • Then particle 1 ends up in the same state
    particle 3 was initially.

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Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

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Classical gates
  • Implement Boolean functions.
  • Are not reversible (invertible). We cannot
    recover the input knowing the output.
  • This means that there is an irretrievable loss of
    information.

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One qubit gates
  • Transform an input qubit into an output qubit
  • Characterized by a 2 x 2 matrix with complex
    coefficients

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One qubit gates
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One qubit gates
  • I ? identity gate leaves a qubit unchanged.
  • X or NOT gate? transposes the components of an
    input qubit.
  • Y gate.
  • Z gate ? flips the sign of a qubit.
  • H ? the Hadamard gate.

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Identity transformation, Pauli matrices, Hadamard
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Two-qubit gates
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Tensor products and outer products
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The two input qubits of a two qubit gates
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A two qubit gate CNOT
  • Two inputs
  • Control
  • Target
  • The control qubit ? transferred to the output as
    is.
  • The target qubit ?
  • Unaltered if the control qubit is 0
  • Flipped (0 ? 1 and 1 ? 0) if the control qubit is
    1.

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The transfer matrix of the CNOT gate
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The transformation performed by CNOT gate
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The output of CNOT gate
  • CNOT preserves the control qubit ?the first and
    the second component of the input vector are
    replicated in the output vector.
  • CNOT flips the target qubit ?the third and fourth
    component of the input vector become the fourth
    and respectively the third component of the
    output vector.
  • CNOT gate is reversible. The control qubit is
    replicated at the output and knowing it we can
    reconstruct the target input qubit.

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Three qubit gates
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The Fredkin gate
  • Three input and three output qubits
  • One control
  • Two target
  • When the control qubit is
  • 0 ? the target qubits are replicated to the
    output
  • 1 ? the target qubits are swapped

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The transformation performed by the Fredkin gate
  • 000gt ? 000gt
  • 001gt ? 001gt
  • 010gt ? 010gt
  • 011gt ? 101gt flip the two target qubits
    when the control qubit is 1
  • 100gt ? 100gt
  • 101gt ? 011gt flip the two target qubits
    when the control qubit is 1
  • 110gt ? 111gt
  • 111gt ? 110gt

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The transfer matrix of the Fredkin gate
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The transfer matrix of the Fredkin gate
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The Toffoli gate
  • Three input and three output qubits
  • Two control
  • One target
  • When both control qubit
  • are 1 ? the target qubit is flipped
  • otherwise the target qubit is not changed.

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The transformation performed by the Toffoli gate
  • 000gt ? 000gt
  • 001gt ? 001gt
  • 010gt ? 010gt
  • 011gt ? 001gt
  • 100gt ? 100gt
  • 101gt ? 101gt
  • 110gt ? 111gt when both control qubits are
    1
  • 111gt ? 110gt then the target qubit is
    flipped

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The transfer matrix of the Toffoli gate
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The transfer matrix of a Toffoli gate
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Toffoli gate is universal. It may emulate an AND
and a NOT gate
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Controlled H gate
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Generic one qubit controlled gate
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Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

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Problem formulation
  • Consider a circuit to compute f(x).
  • x is a binary n-tuple ? there are 2n possible
    values of x.
  • a gate array is used to compute f(x) in one time
    step
  • We wish to compute the values of f(x) for the 2n
    values of x using
  • a classical circuit
  • a quantum circuit
  • Classical system
  • Sequential computation using a single gate array
    ? we need 2n time steps
  • Parallel computation using 2n gate arrays ? we
    need a single time step

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Problem formulation (contd)
  • Consider first the case n1, x takes only two
    values
  • x0
  • x1
  • Well show that the output of a quantum circuit
    is a superposition of f(0) and f(1)
  • Both values, f(0) and f(1) are available
  • But..once we measure the output of the quantum
    circuit we can obtain only one of them

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A quantum circuit to compute f(x)
  • Given a function f(x) we can construct a
    reversible quantum circuit consisting of Fredking
    gates only, capable of transforming two qubits as
    follows
  • The function f(x) is hardwired in the circuit

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A quantum circuit to compute f(x) (contd)
  • If the second input is zero then the
    transformation done by the circuit is

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A quantum circuit (contd)
  • We apply the first qubit through a Hadamad gate.
  • The resulting sate of the circuit is
  • The output state contains information about f(0)
    and f(1).

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Quantum parallelism
  • The output of the quantum circuit contains
    information about both f(0) and f(1). This
    property of quantum circuits is called quantum
    parallelism.
  • Quantum parallelism allows us to construct the
    entire truth table of a quantum gate array having
    2n entries at once.
  • We start with n qubits, each in state 0gt and we
    apply a Walsh-Hadamard transformation.

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Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

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Deutschs problem
  • Consider a black box characterized by a transfer
    function that maps a single input bit x into an
    output, f(x). It takes the same amount of time,
    T, to carry out each of the four possible
    mappings performed by the transfer function f(x)
    of the black box
  • f(0) 0
  • f(0) 1
  • f(1) 0
  • f(1) 1
  • The problem posed is to distinguish if

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A quantum circuit to solve Deutschs problem
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Evrika!!
  • By measuring the first output qubit qubit we are
    able to determine performing
    a single evaluation.

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Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

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Quantum circuit to create Bell states
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Contents
  • I. Computing and the Laws of Physics
  • II. The Strange World of Quantum Mechanics
  • III. Quantum Computing and Communication
  • IV Hilbert Spaces and Tensor Products
  • V. Qubits
  • VI. Quantum Gates and Quantum Circuits
  • VII. Quantum Parallelism
  • VIII. Deutschs Problem
  • IX. Bell States, Teleportation, and Dense
    Coding
  • X. Summary

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Final remarks
  • The growing interest in quantum computing and
    quantum information theory is motivated by the
    incredible impact this discipline could have on
    how
  • we store,
  • process, and
  • transmit data.
  • A tremendous progress has been made in the area
    of quantum computing and quantum information
    theory during the past decade. Thousands of
    research papers, a few solid reference books, and
    many popular-science books have been published in
    recent years in this area.

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Final remarks (contd)
  • Computer and communication systems using quantum
    effects have remarkable properties.
  • Quantum computers enable efficient simulation of
    the most complex physical systems we can
    envision.
  • Quantum algorithms allow efficient factoring of
    large integers with applications to cryptography.
  • Quantum search algorithms speedup considerably
    the process of identifying patterns in apparently
    random data.
  • We can guarantee the security of our quantum
    communication systems because eavesdropping on a
    quantum communication channel can always be
    detected.

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Final remarks (contd)
  • It is true that we are years, possibly decades
    away from actually building a quantum computer
  • requiring little if any power at all,
  • filling up the space of a grain of sand, and
  • computing at speeds that are unattainable today
    even by covering tens of acres of floor space
    with clusters made from tens of thousands of the
    fastest processors built with current state of
    the art solid state technology.

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Final remarks (contd)
  • All we have at the time of this writing is a
    seven qubit quantum computer able to compute the
    prime factors of a small integer, 15.
  • Building a quantum computer faces tremendous
    technological and theoretical challenges.
  • At the same time, we witness a faster rate of
    progress in quantum information theory where
    applications of quantum cryptography seem ready
    for commercialization. Recently, a successful
    quantum key distribution experiment over a
    distance of some 100 km has been announced.

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Summary
  • Quantum computing and quantum information theory
    is truly an exciting field.
  • It is too important to be left to the physicists
    alone.
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