Title: Quantum Computing and Quantum Parallelism
1Quantum Computing and Quantum Parallelism
- Dan C. Marinescu and Gabriela M. Marinescu
- School of Computer Science
- University of Central Florida
- Orlando, Florida 32816, USA
2Acknowledgments
- 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.
3Contents
- 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
4Technological 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
5Technological 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
6Energy/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.
7The 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.
8Power 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.
9Heat 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.
10Heat 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.
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12Contents
- 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
13A 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
14Quantum
- 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.
15Quantum 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.
16Heissenbergs 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 -
-
17Max 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.
18Quantum 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. -
19Milestones 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.
20Milestones 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.
21Milestones 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,.
22Milestones 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.
23Milestones 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.
24Can 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.
25Can 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
26Light
- 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.
27Beam 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?
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29A 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?
30Cascaded 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.
31The 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.
32An 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.
33The 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.
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35A mathematical model to describe the state of a
quantum system
are complex numbers
36Superposition 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
37The 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.
38Does 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.
39A 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?
40Multiple measurements in different bases
41Measurements in multiple bases
42The 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.
43The 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.
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45The 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.
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47A 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..
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49Contents
- 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
50The 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.
51Frontier(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.
52What 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.
53Does 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.
55Approximate 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.
56Exact 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,
57Large 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
58Quantum 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.
59Quantum 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.
60Quantum 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.
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62Information 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.
63The 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.
64The 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.
65BB84 (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.
66BB84 (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.
67BB84 (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.
68BB84 (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.
69Contents
- 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
70Hilbert 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.
71The tensor product of two vectors in a
two-dimensional Hilbert space
72Two vectors in a four-dimensional Hilbert space
73The outer product of two vectors in a
four-dimensional Hilbert space
74State 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.
75Contents
- 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
76One 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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79A 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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81Qubit measurement
82Two 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
83Two qubits
84Measuring 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.
85Measuring 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
86Measuring 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.
87Measuring 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.
88Bell 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
89This 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.
90Entanglement (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".
91Physical 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.
92Physical embodiment of a qubit (contd)
- A two-level atom in an optical cavity.
- Two internal states of an ion in a trap.
- Others
93The 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.
94More 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
95The Stern-Gerlach experiment with hydrogen atoms
96The 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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98Communication 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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100Contents
- 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
101Classical 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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103One qubit gates
- Transform an input qubit into an output qubit
- Characterized by a 2 x 2 matrix with complex
coefficients
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105One qubit gates
106One 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.
107Identity transformation, Pauli matrices, Hadamard
108Two-qubit gates
109Tensor products and outer products
110The two input qubits of a two qubit gates
111A 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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113The transfer matrix of the CNOT gate
114The transformation performed by CNOT gate
115The 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.
116Three qubit gates
117The 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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119The 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
120The transfer matrix of the Fredkin gate
121The transfer matrix of the Fredkin gate
122The 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.
123The 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
124The transfer matrix of the Toffoli gate
125The transfer matrix of a Toffoli gate
126Toffoli gate is universal. It may emulate an AND
and a NOT gate
127Controlled H gate
128Generic one qubit controlled gate
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130Contents
- 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
131Problem 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
132Problem 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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134A 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
135A quantum circuit to compute f(x) (contd)
- If the second input is zero then the
transformation done by the circuit is
136A 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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138Quantum 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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141Contents
- 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
142Deutschs 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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144A quantum circuit to solve Deutschs problem
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150Evrika!!
- By measuring the first output qubit qubit we are
able to determine performing
a single evaluation.
151Contents
- 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
152Quantum circuit to create Bell states
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155Contents
- 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
156Final 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.
157Final 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.
158Final 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.
159Final 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.
160Summary
- Quantum computing and quantum information theory
is truly an exciting field. - It is too important to be left to the physicists
alone.