This document provides an introduction to quantum computing and quantum information theory. It discusses how technological limitations of conventional computing motivate the development of quantum computing. The key laws of quantum mechanics that enable quantum computing are introduced, including superposition, entanglement, and the Heisenberg uncertainty principle. The document explains how quantum bits (qubits) can represent more than the two states of classical bits, and how quantum gates operate on qubits. It provides examples of one-qubit gates like the Hadamard gate. The potential for quantum computers to massively scale parallelism through quantum effects like entanglement is also summarized.

1. Introduction to
Quantum Computing and
Quantum Information Theory

2. Contents
I.
Computing and the Laws of Physics
II. A Happy Marriage; Quantum Mechanics &
Computers
III. Qubits and Quantum Gates
IV. Quantum Parallelism
V. Summary

3. Technological limits
For the past two decades we have enjoyed Gordon Moore’s
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

4. 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.

5. 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.

6. Contents
I.
Computing and the Laws of Physics
II. A Happy Marriage; Quantum Mechanics &
Computers
III. Qubits and Quantum Gates
IV. Quantum Parallelism
V. Summary

7. A happy marriage…
The two greatest discoveries of the 20-th century
quantum mechanics
stored program computers
produced quantum computing and quantum information
theory

8. 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
smallest possible discrete unit of any physical
property
Quantum mechanics is a mathematical model of the physical
world

9. 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.''

10. 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.

11. Deterministic versus probabilistic photon behavior

12. A mathematical model to describe the state of a
quantum system
ψ = α 0 0 + α1 1
| α 0 , α1 |
are complex numbers
| α 0 | + | α1 | = 1
2
2

13. Superposition and uncertainty
In this model a state
ψ = α 0 0 + α1 1
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.
2 is the probability of observing the outcome “1”

|α0 |
2 is the probability of observing the outcome “0”

|α |
1
| α 0 |2 + | α 1 |2 = 1

14. 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).

15. 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.

16. Contents
I.
Computing and the Laws of Physics
II. A Happy Marriage; Quantum Mechanics &
Computers
III. Qubits and Quantum Gates
IV. Quantum Parallelism
V. Summary

17. 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
ψ
can be in state
α
α1
| 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
 in state
| 0〉 with probability
| 1〉with probability
| α0 |2
| α1 |2

19. Other states of a qubit
1
1
0 +
1
2
2
1
3
0 +
1
2
2

20. 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.

23. 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
| α 00 | , | α 01 | , | α10 | , | α11 |
2
2
2
2

24. Two qubits
ψ = α 00 00 + α 01 01 + α10 10 + α11 11
| α 00 | + | α 01 | + | α10 | + | α11 | = 1
2
2
2
2

25. 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.

26. Measuring two qubits (cont’d)
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
| α 00 |2 + | α 01 |2
| α10 |2 + | α11 |2

27. Measuring two qubits (cont’d)
Call
ψ
I
0
the post-measurement state when we measure
the first qubit and find it to be 0.
I
Call ψ 1 the post-measurement state when we measure
the first qubit and find it to be 1.
ψ
I
0
=
α 00 00 + α 01 01
| α 00 | + | α 01 |
2
2
ψ
I
1
=
α10 1 0 + α11 11
| α10 | + | α11 |
2
2

28. Measuring two qubits (cont’d)
Call
ψ
II
0
the post-measurement state when we measure
the second qubit and find it to be 0.
II
Call ψ 1 the post-measurement state when we measure
the second qubit and find it to be 1.
ψ 0II =
α 0 0 00 + α10 10
| α 00 | + | α10 |
2
2
ψ
II
1
=
α 01 01 + α11 11
| α 01 | + | α11 |
2
2

29. Bell states - a special state of a pair of
qubits
If
1
α 00 = α11 =
and
2
α 01 = α10 = 0
When we measure the first qubit we get the post
I
I
measurement state
ψ =| 11〉
ψ 0 =| 00〉
1
When we measure the second qubit we get the post
mesutrement state
II
ψ II =| 11〉
ψ 0 =| 00〉
1

30. 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.

31. 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".

32. 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 halfinteger 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.

33. 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

34. 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.

35. 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.

36. 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 |h> and |v>

37. 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

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

40. ψ = α 0 0 + α1 1
ϕ = α 0 +α 1
'
0
 g11
G=
g
 21
ϕ =Gψ
g12 

g 22 

′
 α 0   g11
 =
α′   g
 1   21
g12  α 0 
 
g 22  α1 
 
'
1

41. 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.

42. One qubit gates
 g11
G =
g
 21
g12 

g 22 

 g11
G =
g
 12
T
*
*
 g11 g11 + g 21 g 21
G +G =  *
 g g + g* g
22 21
 12 11
g 21 

g 22 

*
 g11
+
G = *
g
 12
g g + g g 22 
=I
g g + g g 22 

*
11 12
*
12 12
*
21
*
22
*
g 21 

*
g 22 


43. Identity transformation, Pauli matrices,
Hadamard
1 0
δ0 = I = 
0 1



ϕ = α 0 0 + α1 1
0 1
δ1 = X = 
1 0



ϕ = α1 0 + α 0 1
0 − i
δ2 = Y = 
i 0 



1 0 
δ3 = Z = 
 0 − 1



1 1 1 

H=
 1 − 1

2

ϕ = −iα1 0 + iα 0 1
ϕ = α 0 0 − α1 1
ϕ = α0
0 +1
2
+ α1
0 −1
2

44. 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.

45. VCNOT = ψ ⊗ φ
GCNOT
1

0
=
0

0

WCNOT = GCNOT VCNOT
0
1
0
0
0
0
0
1
0

0
1

0


46. The two input qubits of a two qubit
gates
ψ = α 0 0 + α1 1
φ = β 0 0 + β1 1
VCNOT
α 0 β 0 


 α 0   β 0   α 0 β1 
= ψ ⊗ φ =  ⊗  = 
α   β  α β 
 1  1  1 0
α β 
 1 1

47. Two qubit gates
GCNOT = 00 00 + 01 01 + 10 11 + 11 10
GCNOT
1

0
=
0

0

0
1
0
0
0
0
0
1
0

0
1

0


48. Two qubit gates
WCNOT = GCNOT VCNOT
WCNOT
1

0
=
0

0

0 0 0  α 0 β 0   α 0 β 0 

 

1 0 0  α 0 β1   α 0 β1 
 α β  =  α β 
0 0 1
 1 0   1 1 
0 1 0  α1β1   α1β 0 

 

WCNOT = α 0 β 0 00 + α 0 β1 01 + α1β1 10 + α1β 0 11
WCNOT = α 0 0 ( β 0 0 + β1 1 ) + α1 1 ( β1 0 + β 0 1 )

49. 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).
WCNOT = α 0 0 ( β 0 0 + β1 01 ) + α1 1 ( β1 0 + β 0 1 )
The CNOT gate is reversible. The control qubit is
replicated at the output and knowing it we can reconstruct
the target input qubit.

50. 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

52. 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.

53. Toffoli gate is universal. It may
emulate an AND and a NOT gate

54. Controlled H gate

55. Generic one qubit controlled gate

57. Contents
I.
Computing and the Laws of Physics
II. A Happy Marriage; Quantum Mechanics &

Computers
III. Qubits and Quantum Gates
IV. Quantum Parallelism
V. Summary

58. 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

59. A quantum circuit (cont’d)
If the second input is zero then the transformation done by
the circuit is

60. A quantum circuit (cont’d)
Now apply the first qubit through a Hadamad gate.
0 +1
2
0
The resulting sate of the circuit is

0 f(
0 +1
)
f (0) + 1 f (1)
2
2
The output state contains information about f(0) and f(1).

62. 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 |0> and we apply a WelshHadamard transformation.


63. Advantages
• Much more powerful
• Faster
• Smaller
• Improvements to science
• Can improve on practical personal electronics

64. • what we are waiting for….
Christmas!

65. Difficulties
• Hard to control quantum particles
• Lots of heat
• Expensive
• Difficult to build
• Not enough is known about quantum mechanics

66. Future Work
• Silicon Quantum Computer
• It may become technology sooner than we expect
• New algorithms and communication
• Maximum exploitation
• Simulate other quantum systems.

67. Contents
I.
Computing and the Laws of Physics
II. A Happy Marriage; Quantum Mechanics &
Computers
III. Qubits and Quantum Gates
IV. Quantum Parallelism
V. Summary

68. 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.

69. 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.

70. Final remarks (cont’d)
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.

71. Gallary

77. D-Wave's Quantu

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

79. Thank You