Introduction to Quantum Computing and Quantum Information Theory

1
Introduction to Quantum Computing andQuantum
Information Theory

• Dan C. Marinescu and Gabriela M. Marinescu
• Computer Science Department
• University of Central Florida
• Orlando, Florida 32816, USA

2
Acknowledgments

• The material presented is from the book
• Lectures on Quantum Computing
• by Dan C. Marinescu and Gabriela M. Marinescu
• Prentice Hall, 2004
• Work supported by National Science Foundation
  grants MCB9527131, DBI0296107,ACI0296035, and
  EIA0296179.

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Contents

• I. Computing and the Laws of Physics
• II. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

4
Technological limits

• For the past two decades we have enjoyed Gordon
  Moores law. But all good things may come to an
  end
• We are limited in our ability to increase
• the density and
• the speed of a computing engine.
• 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 Heissenberg uncertainty

5
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 a liner
  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.
• Even if this limit is reduced say 100-fold we
  shall see a 10 (ten) times increase in the amount
  of power needed by devices operating at a speed
  103 times larger than the sped of today's devices.

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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.

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Talking about the heat

• 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. The heat dissipated grows
  as the cube of the radius of the device.
• To prevent the destruction of the engine we have
  to remove the heat through a surface surrounding
  the device. Henceforth, our ability to remove
  heat increases as the square of the radius while
  the amount of heat increases with the cube of the
  size of the computing engine.

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Contents

• I. Computing and the Laws of Physics
• II. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

10
A happy marriage

• The two greatest discoveries of the 20-th century
• quantum mechanics
• stored program computers
• produced quantum computing and quantum
  information theory

11
Quantum Quantum mechanics

• Quantum is a Latin word meaning some quantity. In
  physics it is 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. For
  example, on a segment of a line there are
  infinitely many points, the segment consists of a
  continuum of points. This means that we can cut
  the segment in half, and then cut each half in
  half, and continue the process indefinitely.
• Quantum mechanics is a mathematical model of the
  physical world

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Heissenberg uncertainty principle

• Heisenberg uncertainty principle says we cannot
  determine both the position and the momentum of a
  quantum particle with arbitrary precision.
• In his Nobel prize lecture on December 11, 1954
  Max Born says about this fundamental principle of
  Quantum Mechanics ... It 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.''

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A revolutionary approach to computing and
communication

• We need to consider a revolutionary rather than
  an evolutionary approach to computing.
• Quantum theory does not play only a supporting
  role by prescribing the limitations of physical
  systems used for computing and communication.
• Quantum properties such as
• uncertainty,
• interference, and
• entanglement
• form the foundation of a new brand of theory,
  the quantum information theory where
  computational and communication processes rest
  upon fundamental physics.

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Milestones in quantum physics

• 1900 - Max Plank presents the black body
  radiation theory the quantum theory is born.
• 1905 - Albert Einstein develops the theory of the
  photoelectric effect.
• 1911 - Ernest Rutherford develops the planetary
  model of the atom.
• 1913 - Niels Bohr develops the quantum model of
  the hydrogen atom.
• 1923 - Louis de Broglie relates the momentum of a
  particle with the wavelength
• 1925 - Werner Heisenberg formulates the matrix
  quantum mechanics.
• 1926 - Erwin Schrodinger proposes the equation
  for the dynamics of the wave function.

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Milestones in quantum physics (contd)

• 1926 - Erwin Schrodinger and Paul Dirac show the
  equivalence of Heisenberg's matrix formulation
  and Dirac's algebraic one with Schrodinger's wave
  function.
• 1926 - Paul Dirac and, independently, Max Born,
  Werner Heisenberg, and Pasqual Jordan obtain a
  complete formulation of quantum dynamics.
• 1926 - John von Newmann introduces Hilbert spaces
  to quantum mechanics.
• 1927 - Werner Heisenberg formulates the
  uncertainty principle.

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Milestones in computing and information theory

• 1936 - Alan Turing dreams up the Universal
  Turing Machine, UTM.
• 1936 - Alonzo Church publishes a paper asserting
  that every function which can be regarded as
  computable can be computed by an universal
  computing machine''.
• 1945 - ENIAC, the world's first general purpose
  computer, the brainchild of J. Presper Eckert
  and John Macauly becomes operational.
• 1946 - A report co-authored by John von Neumann
  outlines the von Neumann architecture.
• 1948 - Claude Shannon publishes A Mathematical
  Theory of Communication.
• 1953 - The first commercial computer, UNIVAC I.

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Milestones in quantum computing

• 1961 - Rolf Landauer decrees that computation is
  physical and studies heat generation.
• 1973 - Charles Bennet studies the logical
  reversibility of computations.
• 1981 - Richard Feynman suggests that physical
  systems including quantum systems can be
  simulated exactly with quantum computers.
• 1982 - Peter Beniof develops quantum mechanical
  models of Turing machines.
• 1984 - Charles Bennet and Gilles Brassard
  introduce quantum cryptography.
• 1985 - David Deutsch reinterprets the
  Church-Turing conjecture.
• 1993 - Bennet, Brassard, Crepeau, Josza, Peres,
  Wooters discover quantum teleportation.
• 1994 - Peter Shor develops a clever algorithm for
  factoring large numbers.

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Deterministic versus probabilistic photon behavior
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The puzzling nature of light

• If we start decreasing the intensity of the
  incident light we observe the granular nature of
  light. Imagine that we send a single photon.
  Then either detector D1 or detector D2 will
  record the arrival of a photon.
• If we repeat the experiment involving a single
  photon over and over again we observe that each
  one of the two detectors records a number of
  events.
• Could there be hidden information, which controls
  the behavior of a photon? Does a photon carry a
  gene and one with a transmit'' gene continues
  and reaches detector D2 and another with a
  reflect'' gene ends up at D1?

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The puzzling nature of light (contd)

• Consider now a cascade of beam splitters. As
  before, we send a single photon and repeat the
  experiment many times and count the number of
  events registered by each detector.
• According to our theory we expect the first beam
  splitter to decide the fate of an incoming
  photon the photon is either reflected by the
  first beam splitter or transmitted by all of
  them. Thus, only the first and last detectors in
  the chain are expected to register an equal
  number of events.
• Amazingly enough, the experiment shows that all
  the detectors have a chance to register an event.

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State description
22
A mathematical model to describe the state of a
quantum system
are complex numbers
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Superposition and uncertainty

• In this model a state
• is a superposition of two basis states, 0
  and 1
• 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.
• is the probability of observing
  the outcome 1
• is the probability of observing
  the outcome 0

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Multiple measurements
25
Measurements in multiple bases
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Measurements of superposition states

• The polarization of a photon is described by a
  unit vector on a two-dimensional space with
  basis 0 gt and 1gt.
• Measuring the polarization is equivalent to
  projecting the random vector onto one of the two
  basis vectors.
• Source S sends randomly polarized light to the
  screen the measured intensity is I.
• The filter A with vertical polarization is
  inserted between the source and the screen an the
  intensity of the light measured at E is about
  I/2.
• Filter B with horizontal polarization is inserted
  between A and E. The intensity of the light
  measured at E is now 0.
• Filter C with a 45 deg. polarization is inserted
  between A and B. The intensity of the light
  measured at E is about 1 / 8.

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The superposition probability rule

• If an event may occur in two or more
  indistinguishable ways then the probability
  amplitude of the event is the sum of the
  probability amplitudes of each case considered
  separately (sometimes known as Feynmann rule).

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The experiment illustrating the superposition
probability rule

• In certain conditions, 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.

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A photon coincidence experiment
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A glimpse into the world of quantum computing and
quantum information theory

• Quantum key distribution
• Exact simulation of systems with a very large
  state space
• Quantum parallelism

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Quantum key distribution

• To ensure confidentiality, data is often
  encrypted. The most reliable encryption
  techniques are based upon one time pads whereby
  the encryption key is used for one session only
  and then discarded.
• Thus, there exists the need for reliable and
  effective methods for the distribution of the
  encryption keys. The problem rests on the
  physical difficulty to detect the presence of an
  intruder when communicating through a classical
  communication channel.
• To date, secure and reliable methods for
  cryptographic key distribution have largely
  eluded the cryptographic community in spite of
  considerable research effort and ingeniousness.

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Quantum key distribution setup

• Alice and Bob are connected via two communication
  channels, a quantum and a classical one. Eve
  eavesdrops on both.
• The photons prepared by Alice may have
  vertical/horizontal (VH) or diagonal polarization
  (DG).
• The photons with vertical/horizontal (VH)
  polarization may be used to transmit binary
  information as follows a photon with vertical
  polarization may transmit a 1 while one with a
  horizontal polarization may transmit a 0.
• Similarly, those with diagonal (DG) polarization
  may transmit binary information, 1 encoded as a
  photon with 45 deg. polarization, and 0 encoded
  as a photon with a 135 deg. polarization.
• Bob uses a calcite crystal to separate photons
  with different polarization. Shown is the case
  when 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.

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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.

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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.

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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.

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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.

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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.

42
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.

43
Quantum computers

• In quantum systems the amount of parallelism
  increases exponentially with the size of the
  system, thus with the number of qubits. This
  means that the price to pay for an exponential
  increase in the power of a quantum computer is a
  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.

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Contents

• I. Computing and the Laws of Physics
• II. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

45
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)

46
Ortonormal basis
47
One qubit
are complex numbers
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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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Other states of a qubit
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A different basis for one qubit
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Qubit measurement
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The Boch sphere representation of one qubit

• A qubit in a superposition state is represented
  as a vector connecting the center of the Bloch
  sphere with a point on its periphery.
• The two probability amplitudes can be expressed
  using Euler angles.

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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

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Two qubits
58
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.

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Measuring two qubits (contd)

• What if we observe only the first qubit, what
  conclusions can we draw?
• We expect that 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

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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.

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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.

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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

• Entanglement is an elegant, almost exact
  translation of the German term Verschrankung used
  by Schrodinger who was the first to recognize
  this quantum effect.
• 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.
• The important property of an entangled pair is
  that the measurement of one particle influences
  the state of the other particle. Einstein called
  that Spooky
• action at a distance".

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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 ,
  which may assume integer and half-integer values.
  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, in other words the
  spin is quantized.

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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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Physical embodiment of a qubit

• The electron with tow independent spin values,
  1/2 and -1/2
• The photon, with tow independent polarizations,
  horizonatla and vertical

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The spin of the electron

• The electron has spin s 1 /2 and the spin
  projection can assume the values ½ referred
  to as spin up, and -1/2 referred to as spin
  down.

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Light and photons

• Light is a form of electromagnetic radiation the
  wavelength of the radiation in the visible
  spectrum varies from red to violet.
• Light can be filtered by selectively absorbing
  some color ranges and passing through others.
• A polarization filter is a partially transparent
  material that transmits light of a particular
  polarization.

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Photons

• Photons differ from the spin 1/2 electrons in two
  ways
• (1) they are massless and
• (2) have spin 1.
• A photon is characterized by its
• vector momentum (the vector momentum determines
  the frequency) and
• polarization.
• In the classical theory light is described as
  having an electric field which oscillates either
  vertically, the light is x-polarized, or
  horizontally, the light is y-polarized in a plane
  perpendicular to the direction of propagation,
  the z-axis.
• The two basis vectors are hgt and vgt

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Vertically and horizontally polarized photons
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Polarization filters

• A polarization filter is a partially transparent
  material that transmits light of a particular
  polarization.
• If we set the axis of a polarization filter to
  let pass y-polarized light, then all photons in
  the state vgt will be absorbed in the filter and
  only the photons in state hgt will pass through.
  If the axis of the polarization filter is set to
  let pass x-polarized light, then all photons in
  state hgt will be absorbed and only photons in
  state v gt will pass through.
• When the polarization filter is set at angle with
  respect to the coordinate system of the incoming
  beam of light the emerging photons are in a
  superposition state

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The effect of a polarization filter
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Communication with entangles particles

• Even when separated two entangled particles
  continue to interact with one another.
• Particle 2 and particle 3 in an anti-correlated
  state (spin up and spin down).
• Then if 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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Classical gates

• Implement Boolean functions.
• Are not reversible (invertible). We cannot
  recover the input knowing the output.
• This means that there is an iretriviable 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

• 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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One qubit gates
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Identity transformation, Pauli matrices, Hadamard
85
Tensor products and outer products
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CNOT a two qubit gate

• Two inputs
• Control
• Target
• The control qubit is transferred to the output as
  is.
• The target qubit
• Unaltered if the control qubit is 0
• Flipped if the control qubit is 1.

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The two input qubits of a two qubit gates
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Two qubit gates
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Two qubit gates
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Final comments on the CNOT gate

• CNOT preserves the control qubit (the first and
  the second component of the input vector are
  replicated in the output vector) and 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).
• The 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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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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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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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. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

100
A quantum circuit

• 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 (contd)

• If the second input is zero then the
  transformation done by the circuit is

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A quantum circuit (contd)

• Now 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. In a classical system we can
  compute the truth table in one time step with 2n
  gate arrays running in parallel, or we need 2n
  time steps with a single gate array.
• We start with n qubits, each in state 0gt and we
  apply a Welsh-Hadamard transformation.

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Contents

• I. Computing and the Laws of Physics
• II. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

108
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. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

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Quantum circuit to create Bell states
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Contents

• I. Computing and the Laws of Physics
• II. A Happy Marriage Quantum Mechanics
• Computers
• III. Qubits and Quantum Gates
• IV. Quantum Parallelism
• V. Deutschs Algorithm
• VI. Bell States, Teleportation, and Dense Coding
• VII. Summary

122
Final remarks

• 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.
• 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 and
  knowledge in this information age.

123
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.

124
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.

125
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.

126
Summary

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