Quantum Computing and Quantum Parallelism Dan C. Marinescu and Gabriela M. Marinescu
Download ReportTranscript Quantum Computing and Quantum Parallelism Dan C. Marinescu and Gabriela M. Marinescu
Quantum Computing and Quantum
Parallelism
Dan C. Marinescu and Gabriela M. Marinescu
School of Computer Science
University of Central Florida
Orlando, Florida 32816, USA
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.
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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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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Technological limits – density and speed
For the past two decades we have enjoyed Gordon
Moore’s 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
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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 Heissenberg’s uncertainty
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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.
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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.
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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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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.
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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.
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Energy consumption of a logic circuit
E
S
Speed of individual logic gates
Heat removal for a circuit with densely packed
logic gates poses tremendous challenges.
(b)
(a)
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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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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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
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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.
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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.
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Heissenberg’s uncertainty principle
The position and the momentum of a quantum
particle cannot be determined with arbitrary
precision.
X PX h / 2
h=1.054 10-34 J second reduced Planck’s constant
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Max Born’s 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”.
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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.
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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.
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Milestones in quantum physics (cont’d)
1926 - Erwin Schrödinger Schrödinger’s 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.
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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,.
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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.
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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.
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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.
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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
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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.
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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?
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D1
D3
Incident beam of light
D5
Detector D1
D7
Reflected beam
Beam splitter
Transmitted beam
Detector D2
D2
(a)
(b)
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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?
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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.
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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:
Circularly polarized light the electric field vector moves along a
circle in a plane perpendicular to the direction of propagation:
vertical/horizontal polarization
diagonal: 45/135 deg polarization
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.
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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.
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The puzzling observations
1.
2.
3.
4.
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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|
0
|
1
(a)
E
S
S
(b)
(c)
intensity = I
A
E
B
S
intensity = I/2
A
C
B
E
S
intensity = 0
(d)
E
A
intensity = I/8
(e)
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A mathematical model to describe the state
of a quantum system
0 0 1 1
| 0 , 1 | are complex numbers
| 0 | | 1 | 1
2
2
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Superposition and uncertainty
In this model a state 0 1
0
1
is a superposition of two basis states, “0” and “1” or
(Dirac’s notation)
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:
2
| 0 | the probability of observing the outcome 1
2
| 1 | the probability of observing the outcome 0
| 0 | | 1 | 1
2
2
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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 > and
| 1>.
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.
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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.
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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?
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Multiple measurements in different bases
black
hard
white
soft
(a)
(b)
black
hard
white
black
soft
white
(c)
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Measurements in multiple bases
|
>
|
|
>
|
>
1
1
1
0
0
0
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|
>
41
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.
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The superposition probability rule
If an event may occur in two or more indistinguishable
ways
For classical systems Bayes rules:
P( B j | A)
P( A | B j ) P( B j )
P( A | B ) P( B )
i
i
i
In quantum mechanics the probability amplitude
of the
event is the sum of the probability amplitudes of each case
considered separately (sometimes known as Feynman’s rule).
An experiment illustrating the superposition probability
rule.
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Reflecting mirror U
Source S1
Detector D1
direction1
BS1
BS2
direction2
Detector D2
Source S2
Reflecting mirror L
(a)
| 0 > (| t >)
| 0 > (| t >)
V
V
+q
O
(b)
1
1
|1>
(| r >)
+q
+q
|1>
(| r >)
-q
direction1
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(c)
direction2
44
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.
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direction1
direction2
S1
S1
D1
BS1
BS2
T
T
+q
D1
U
BS1
+q
BS2
R
+q
R
+q
(b) The RR case: the probability
L
amplitude is (+q)(+q).
(a) - The TT case: the probability
amplitude is (+q)(+q).
S1
S1
R
BS1
T
U
-q
BS2
T
BS1
R
+q
+q
BS2
+q
D2
L
D2
(d) The RT case: the probability
amplitude is (+q)(+q).
(c) - The TR case: the
probability amplitude is (+q)(-q).
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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..
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Detector D1
Reflecting mirror U
Beam splitter
Source
Reflecting mirror L
Detector D2
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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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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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.
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Frontier(s)…from Webster’s 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.
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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.
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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 today’s
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.
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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.
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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.
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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,…
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Large problem state space
From black hole thermodynamics – a system enclosed by
a surface with area A has N(A) observable states with
N ( A) e
3
Ac / 4 hG
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
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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.
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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.
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Quantum key distribution setup
Alice and Bob connected via two communication
channels:
Alice sends photons via the quantum channel to Bob.
A photon may be prepared with
a classical one, and
a quantum one.
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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Vertical
Horizontal
45 deg
Vertical/Horizontal (VH)
135 deg
Diagonal (DG)
(a)
(b)
Quantum communication channel
Source of
polarized
photons
Quantum wiretap
Photon
separation
system
Eve
Classical wiretap
Alice
Classical communication channel
Bob
(c)
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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.
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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.
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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 (cont’d)
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
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 vertical
polarization (V), and
a $0$ as a photon with horizontal (H) polarization;
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 (cont’d)
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 (cont’d)
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 (cont’d)
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.
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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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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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 | 0> and |1>.
four- dimensional Hilbert space | 00>, |01>, |10>, and |11>.
eight-dimensional Hilbert space | 000>, |001>, |010>,
|011>, |100>, |101>, |110>, |101>, and |111>.
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The tensor product of two vectors in a twodimensional Hilbert space
0 0 1 1
0 0 1 1
0 0
0 0 0 1
1 1 1 0
1 1
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Two vectors in a four-dimensional Hilbert space
00 00 01 01 10 10 11 11
00 00 01 01 10 10 11 11
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The outer product of two vectors in a fourdimensional Hilbert space
00
01
00
10
11
00 00
01 00
10 00
11 00
00 01
01 01
10 01
11 01
01
00 10
0110
10 10
1110
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10
11
00 11
0111
00 11
1111
73
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.
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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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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One qubit
Mathematical abstraction
Vector in a two dimensional complex vector space
(Hilbert space)
Dirac’s notation
ket
bra
|
column vector
row vector
bra dual vector (transpose and complex conjugate)
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State description
0
0
V
q0 = q
1
45o
O
q1 = q
q0
1
1
O
(a)
V
30o
1
q1
(b)
San Mallo, June 2004
77
|0>
z
|ψ
r
x
y
b
|1>
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78
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
0 0 1 1
can be in state | 0 or in state | 1 or in any other state
that is a linear combination of the basis state
When we measure the qubit we find it
in state | 0 with probability
in state | 1 with probability
| 0 |2
| 1 | 2
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79
0
0
Superposition states
1
(a) One bit
1
Basis (logical) state 0
Basis (logical) state 1
(b) One qubit
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80
Qubit measurement
0
p0
p1
1
Possible states of one qubit before
the measurement
The state of the qubit after
the measurement
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81
Two qubits
Represented as vectors in a 2-dimensional Hilbert
space with four basis vectors
00 , 01 , 10 , 11
When we measure a pair of qubits we decide that the
system it is in one of four states
00 , 01 , 10 , 11
with probabilities
| 00 | , | 01 | , | 10 | , | 11 |
2
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2
2
2
82
Two qubits
00 00 01 01 10 10 11 11
| 00 | | 01 | | 10 | | 11 | 1
2
2
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2
2
83
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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84
Measuring two qubits (cont’d)
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
| 00 | | 01 |
2
1 with probability
| 10 | | 11 |
2
2
2
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85
Measuring two qubits (cont’d)
0I
Call
measure
I
Call 1
measure
I
0
the post-measurement state when we
the first qubit and find it to be 0.
the post-measurement state when we
the first qubit and find it to be 1.
00 00 01 01
| 00 | | 01 |
2
2
I
1
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10 10 11 11
| 10 | | 11 |
2
2
86
Measuring two qubits (cont’d)
II
0
0II
Call
measure
II
Call 1
measure
the post-measurement state when we
the second qubit and find it to be 0.
the post-measurement state when we
the second qubit and find it to be 1.
00 00 10 10
| 00 | | 10 |
2
2
II
1
San Mallo, June 2004
01 01 11 11
| 01 | | 11 |
2
2
87
Bell states - a special state of a pair of qubits
If
1
00 11
2
and
01 10 0
When we measure the first qubit we get the post
I
measurement state
I
| 11
| 00
1
0
When we measure the second qubit we get the post
mesutrement state II | 00 1II | 11
0
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88
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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89
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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90
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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91
Physical embodiment of a qubit (cont’d)
A two-level atom in an optical cavity.
Two internal states of an ion in a trap.
Others
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92
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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93
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
s=+1/2 and
s=-1/2
bosons - spin one particles. The spin quantum numbers of
bosons can be
s=+1,
s=0, and
s=-1
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94
The Stern-Gerlach experiment with
hydrogen atoms
Screen
N
Source of
hydrogen
atoms
S
Magnet
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95
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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96
Sz
Rn(0)
1
h
2
-
1
h
2
Rn(180)
(a)
(b)
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97
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 anticorrelated state (spin up and spin down).
We measure particle 1 and particle 2 and set them in an anticorrelated state.
Then particle 1 ends up in the same state particle 3 was
initially.
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98
Initial state
particle1
particle2
particle3
Entangle particle2 and particle3
Particle2 and particle3 in an anti-symmetric entangled state
particle1
particle2
particle3
Separate the entangled particles
New York
particle1
London
particle2
particle3
Entangle particle1 and particle2
Measure the entangled system
(particle1, particle2)
New York
particle1
London
particle2
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particle3
99
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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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100
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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101
NOT gate
x
x
0
1
y
1
0
z = (x) AND (y)
x
0
0
1
1
y
0
1
0
1
z
0
0
0
1
z = (x) NAND (y)
x
0
0
1
1
y
0
1
0
1
z
1
1
z = (x) OR (y)
x
0
0
1
1
y
0
1
0
1
z
0
1
1
1
z = (x) NOR (y)
x
0
0
1
1
y
0
1
0
1
z
1
0
0
0
z = (x) XOR (y)
x
0
0
1
1
y
0
1
0
1
z
0
1
1
0
y = NOT(x)
x
AND gate
y
x
NAND gate
y
x
OR gate
y
x
NOR gate
y
x
XOR gate
y
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1
0
102
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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103
0 0 1 1
G
g11 g12
gate
GOne-qubit
g 21 g 22
0 g11
1 g 21
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0 1
'
0
'
1
g12 0
g 22 1
104
One qubit gates
g11
G
g 21
g12
g 22
g11
T
G
g12
*
*
g11g11 g 21g 21
G G *
*
g
g
g
22 g 21
12 11
g 21
g 22
*
g11
G *
g12
g
g
*
21
*
22
g g g g
I
g g g g
*
11 12
*
12 12
San Mallo, June 2004
*
21 22
*
22 22
105
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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106
Identity transformation, Pauli matrices, Hadamard
1 0
0 I
0 1
0 0 1 1
0 1
1 X
1 0
1 0 0 1
i1 0 i 0 1
0 i
2 Y
i 0
0 0 1 1
1 0
3 Z
0 1
1 1 1
H
2 1 1
0
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0 1
2
1
0 1
2
107
Two-qubit gates
0 0 1 1
0 0 1 1
g11 g12 g13
g 22 g 23
g 21CNOT
G
g 31 g 32 g 33
g
41 g 42 g 43
g14
g 24
g 34
g 44
VCNOT
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108
Tensor products and outer products
1
1 1 0
00 | 0 | 0
0 0 0
0
1
1
0
0
00 00 1 0 0 0
0
0
0
0
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0 0 0
0 0 0
0 0 0
0 0 0
109
The two input qubits of a two qubit gates
0 0 1 1
0 0 1 1
VCNOT
0 0
0 0 0 1
1 1 1 0
1 1
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110
A two qubit gate: CNOT
Control input
Target input
Two inputs
Control
Target
+
O
+
O
addition modulo 2
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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111
VCNOT
GCNOT
1
0
0
0
WCNOT GCNOT VCNOT
CNOT
0
1
0
0
0
0
0
1
0
0
1
0
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112
The transfer matrix of the CNOT gate
GCNOT 00 00 01 01 10 11 11 10
GCNOT
1
0
0
0
0
1
0
0
0
0
0
1
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0
0
1
0
113
The transformation performed by CNOT gate
WCNOT GCNOT VCNOT
WCNOT
1
0
0
0
0 0 0 0 0 0 0
1 0 0 0 1 0 1
11
0 0 1 1 0
0 1 0 11 1 0
WCNOT 0 0 00 0 1 01 11 10 10 11
WCNOT 0 0 (0 0 1 1 ) 1 1 (1 0 0 1 )
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114
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.
WCNOT 0 0 (0 0 1 01 ) 1 1 (1 0 0 1 )
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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115
Three qubit gates
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116
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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117
Three
qubit
gate
a
a
a
b
a
a'
b
b
b
a
b
b'
0
0
1
1
c
c'
a
Input
b
c
a'
Output
b' c'
a
Input
b
c
a'
Output
b' c'
a
Input
b
c
a'
Output
b' c'
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
1
00
0
1
1
1
0
1
0
1
0
0
1
0
1
0
0
1
0
0
1
0
1
0
1
1
1
0
0
1
0
0
1
1
0
1
1
0
1
1
1
1
1
1
1
0
1
0
0
1
1
0
1
0
0
1
0
1
1
1
0
1
1
0
1
0
1
1
1
1
1
1
1
1
(a)
(b)
(c)
0
bc
1
c
b
bc
0
c
c
c'
c
c'
a=0
a’= bc & b’ = bc
(d)
a=1
& b=0
San Mallo, June 2004
(e)
a’= c
& b’ = c
118
The transformation performed by the
Fredkin gate
|000>
|001>
|010>
|011>
|100>
|101>
|110>
|111>
|000>
|001>
|010>
|101>
|100>
|011>
|111>
|110>
flip the two target qubits when the control qubit is 1
flip the two target qubits when the control qubit is 1
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119
The transfer matrix of the Fredkin gate
GFredkin | 000 000 | | 001 001 | | 010 010 | | 011 101 |
| 100 100 | | 101 011 | | 110 111 | | 111 110 |
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120
The transfer matrix of the Fredkin gate
GFredkin
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
San Mallo, June 2004
0
0
0
0
0
0
0
1
121
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.
San Mallo, June 2004
122
The transformation performed by the Toffoli
gate
|000>
|001>
|010>
|011>
|100>
|101>
|110>
|111>
|000>
|001>
|010>
|001>
|100>
|101>
|111> when both control qubits are 1
|110> then the target qubit is flipped
San Mallo, June 2004
123
The transfer matrix of the Toffoli gate
GToffoli | 000 000 | | 001 001 | | 010 010 | | 011 011 |
| 100 100 | | 101 101 | | 110 111 | | 111 110 |
San Mallo, June 2004
124
The transfer matrix of a Toffoli gate
GToffoli
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
San Mallo, June 2004
0
0
0
0
0
0
1
0
125
Toffoli gate is universal. It may emulate an
AND and a NOT gate
a
a
a
a
1
1
b
b
b
b
b
b
+ ab
c O
1
0
b
c
(a)
NAND(ab)
(b)
San Mallo, June 2004
(c)
126
Controlled H gate
H
H
(a)
(b)
San Mallo, June 2004
127
Generic one qubit controlled gate
|c>
|t>
|c>
U
|t>
A
(a)
B
C
(b)
San Mallo, June 2004
128
|c0>
|c0>
|c1>
|c1>
|c2>
|c2>
|c3>
|c3>
|c4>
|c4>
|c5>
|c5>
| w0>=| 0 >
| 0>
| w1>=| 0 >
| 0>
| w2>=| 0 >
| 0>
| w3>=| 0 >
| 0>
| w4>=| 0 >
| 0>
C
o
n
t
r
o
l
Target
|t>
U
San Mallo, June 2004
129
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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
San Mallo, June 2004
130
Problem formulation
Consider a circuit to compute f(x).
We wish to compute the values of f(x) for the 2n
values of x using
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
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
San Mallo, June 2004
131
Problem formulation (cont’d)
Consider first the case n=1, x takes only two values
x=0
x=1
We’ll 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
San Mallo, June 2004
132
0
f(0)
1
0
f(0)
1
f(1)
f(1)
2T
(a)
T
(b)
|x>
|x>
Uf
|y>
| y > O+ f(x) >
T
(c)
San Mallo, June 2004
133
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
|x>
|x>
Uf
|y>
| y o+ f(x )>
The function f(x) is hardwired in the circuit
San Mallo, June 2004
134
A quantum circuit to compute f(x) (cont’d)
If the second input is zero then the transformation
done by the circuit is
|x>
|x>
Uf
| f(x )>
|0>
San Mallo, June 2004
135
A quantum circuit (cont’d)
We apply the first qubit through a Hadamad gate.
0 1
|0>
2
H
|0>
Uf
|0>
0
f (0) 1 f (1)
2
The resulting sate of the circuit is
0 1
0 f(
)
2
The output state contains information about f(0) and f(1).
San Mallo, June 2004
136
|x>
|x>
|x>
Uf
|y>
Uf
| y O+ f(x) >
(a)
|0>
|x>
|0>
| f(x) >
(b)
H
Uf
|0>
(c)
San Mallo, June 2004
137
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 |0> and we apply a
Walsh-Hadamard transformation.
San Mallo, June 2004
138
|x>
(m-dimensional)
|y>
(k-dimensional)
|x>
Uf
| y O+ f(x) >
(n=m+k)-dimensional)
San Mallo, June 2004
139
H0
0 1
2
( H H H ) 000
Uf(
1
2n
2 n 1
x ,0 )
x 0
1
2n
1
2
1
n
2
n
[( 0 1 ) ( 0 1 ) ( 0 1 )]
2 n 1
x
x 0
2 n 1
U
x 0
f
( x,0 )
San Mallo, June 2004
1
2 n 1
2n
x 0
x, f ( x ) )
140
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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
San Mallo, June 2004
141
Deutsch’s 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
f ( 0) f (1)
f ( 0) f (1)
San Mallo, June 2004
142
0
f(0)
1
0
f(0)
1
f(1)
f(1)
2T
(a)
T
(b)
|x>
|x>
Uf
|y>
| y > O+ f(x) >
T
(c)
San Mallo, June 2004
143
A quantum circuit to solve Deutsch’s problem
|0>
H
|x>
|x>
H
Uf
|1>
H
0
|y>
1
| y > +O f(x)
2
San Mallo, June 2004
3
144
0
1 0 1
0 0 1
0 1 0
0
1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
G1 H H
2 1 1
2 1 1 2 1 1 1 1
1 1 1 1
1 1 1 1 0
1
1 1 1 1 1 1 1 1
1 G1 0
2 1 1 1 1 0
2 1
1 1 1 1 0
1
0 1 0 1
1
1 ( 00 01 10 11 )
2
2
2
x
0 1
2
y
0 1
2
San Mallo, June 2004
145
y f ( x)
0 1
2
f ( x)
0 f ( x) 1 f ( x)
y f ( x) (1)
2
f ( x)
f ( x) 1 f ( x)
2
0 1
0 1
2
y f ( x)
0 1
2
2
if
f ( x) 0
if
f ( x) 1
San Mallo, June 2004
146
x
y
0 1
1
0
1
0
1
1 1
2
2 2 1
2
1
2 x ( y f ( x))
1
0 1 0 1
1 1
1
2
2
2
1
1
0 1 0 1 1 1
2
2 2 1
1
2 x ( y f ( x))
1
0 1 0 1
1 1
1
2
2
2
1
San Mallo, June 2004
2
0 1
if
f (0) f (1) 0
if
f (0) f (1) 1
if
f (0) 0, f (1) 1
if
f (0) 1, f (1) 0
147
2
1
1 1 if
2 1
1
x ( y f ( x))
1
1 1
if
2 1
1
San Mallo, June 2004
f (0) f (1)
f (0) f (1)
148
1
1 1 1 1 0 1 0
G3 H I
2 1 1 0 1
2 1
0
3 G3 2
1
1 0
2 1
0
1
1 0
2 1
0
0
1
0
1
0
1
0
1
0 1
0 1 1 1
1 0 2 1
0 1 1
1 0 1
0 1 1 1
1 0 2 1
0 1 1
1
0 1
1 0
0 1
1
0
1
0
1
0
1
0 1
1 1
0
2 0
2
0
0
0 1
1 0
1
2 1
2
1
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if
f (0) f (1)
if
f (0) f (1)
149
Evrika!!
By measuring the first output qubit qubit we are able to
determine
f (0) f (1) performing a single evaluation.
3 f (0) f (1)
0 if
f (0) f (1)
1 if
0 1
2
f (0) f (1)
f (0) f (1)
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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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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151
Quantum circuit to create Bell states
stage 1
|a>
|a>
H
H
stage 2
|a’>
|V>
|b>
|b>
I
(a)
|W>
|b’>
(b)
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152
Pair of entangled qubits
particle
1
particle
2
particle
3
Caroll
Bob
Alice
particle
1
particle
2
particle
3
Quantum
Channel
CNOT
particle 1 - target qubit
particle 3 - control qubit
The measurement on
the pair (1&3) changes
the state of particle 2 to
one of four states: S1,
S2, S3, S4
iY
Receive from Alice
results of measurements
00 01
10
11
Measurement
particle 3 - measured
particle 1 - unchanged
I
Send to Bob results of
measurement
00 01
10
11
X
Z
Z
Classical
Channel
Particle 2 is in the same
state as particle 3
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153
Pair of entangled qubits
Alice’s
qubit
Bob’s
qubit
Caroll
Bob’s
qubit
Alice’s
qubit
I
Z
X
00
01
10
iY
11
Alice’s modified
but still entangled qubit
Quantum
channel
Qubit from
Alice
Alice
CNOT
First
qubit
Second
qubit
0
H
0
1
1
Bob
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154
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. Deutsch’s Problem
IX. Bell States, Teleportation, and Dense Coding
X.
Summary
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155
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 (cont’d)
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 (cont’d)
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 (cont’d)
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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159
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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