A review paper for the Quantum Computing topic, done as a final project for the Solid State Physics course on 2014

1. 1
Quantum Computing
Abdelrahman Othman, Abdelrahman Zayed, Omar Elzaafarany,Amr Elsaid, Marwan Salem
Abstract—Quantum computers are one of the most
significant promising topics in both researches and practical
applications nowadays; we will introduce some introductory
concepts such as: Superposition, Entanglement and turing
machine. Then we will take a look at the main differences
between the quantum bits and normal bits. Then we will discuss
one of the main challenges in the field known as quantum
decoherence, Then illustrating briefly the process of the quantum
computing and We will end up with some important applications
such as searching, Cryptography and Steganography.
I. INTRODUCTION
he world top tech companies are moving towards a dead
end as they are following Moore’s law by doubling the
number of transistors on a chip for constant cost once
every two years, and keeping the footprint constant but now in
order that Moore’s law remains applicable we have to deal with
the atomic scale so Quantum effects are beginning to interfere in
the functioning of the new electronic devices. Our new paradigm
is based on the idea of using quantum mechanics to perform
computations instead of classical physics. We would move into a
new era of quantum computing where we can perform unlimited
operations. But to understand it we must revisit the quantum
physics rules specially the Schrödinger’s equation as there is
always uncertainty of the particle's position, speed and
momentum that's why we had to find a new way to define its
behavior we call it "Quantum state". This quantum state is
affected by mechanics phenomena, such as superposition
and entanglement, to perform operations on data. A theoretical
model of quantum computation is the quantum Turing machine,
also known as the universal quantum computer.
II. QUANTUM COMPUTING FLOW
III. QUANTUM SUPERPOSITION
Quantum superposition is an important principle in
quantum mechanics it says that the electron can be in all
possible states but when looking at it we only get one state
because we have had disturbed it. Paul Dirac gave a good
example to describe it "if we consider the superposition of two
states, A and B, such that there exists an observation which,
when made on the system in state A, is certain to lead to one
particular result, a say, and when made on the system in state
B is certain to lead to some different result, b say. What will
be the result of the observation when made on the system in
the superposed state? The answer is that the result will be
sometimes a and sometimes b, according to a probability law
depending on the relative weights of A and B in the
superposition process. It will never be different from both a
and b [i.e, either a or b]. The intermediate character of the
state formed by superposition thus expresses itself through the
probability of a particular result for an observation being
intermediate between the corresponding probabilities for the
original states, not through the result itself being intermediate
between the corresponding results for the original states.
it has been confirmed by some experiments like double
slits experiment; briefly its about a light beam get across two
slits so if a photon get to cross the slits it will actually get
across the two slits at the same time but if we tried to detect it
we will distribute the system and it will get through one slit
only that is because when we try to detect it we change its
state and it takes a new state.
So if we have a quantum bit it either be a |1> or |0> or
superposition between both, that’s actually one of the most
important concepts of quantum computer.
IV. QUANTUM ENTANGLEMENT
Quantum entanglement is phenomena occurs when a pair
of particles are generated such that they share their quantum
state; measurement on them are found to be correlated as if we
have in an orbital two electrons their total spin equal zero and
we took them from the orbital and separated them and we
know that one of them spins in the direction of clockwise then
we know for sure that the other one is spinning anticlockwise.
we can get two entangled particles say we have two
electrons in an orbital we can separate the form the atom and
we separate them from each other another experiment called
Spontaneous parametric down-conversion used to entangled
photons ,scientists reached entanglement for photons,
electrons, molecules the size of buck balls.
Quantum entanglement was firstly discussed by Albert
Einstein in 1935 in paper known as EPR paradox as an
experiment was held showed that the quantum theory is not
complete, Einstein described it as a spooky action; the first
one to call it "entanglement" was Schrödinger a lot of work
was done by Bell, Freedman and Clauser led us to an
agreement with quantum mechanics.
So if we have a qubit and we want to read its state as in
quantum computer we can’t look at it directly so we will
change its state. Scientists used quantum entanglements in
quantum computer as without looking to the qubit itself we
can read its state by knowing the state of the particle which is
entangled.
Quantum
computers
Quantum
circuits
Quantum
gates
Qubits

2. 2
V. BITS VS QUBITS (CLASSICAL VS QUANTUM )
In a classical computer the needed space to store a number is
2.^n where n is the number of bits in the number so if we have
a digital register holding 2 bits of data then the current state
probability is evenly distributed over the following state
distributions: 00,01,10,11. But if we have a quantum computer
the 2 qubits would be represented as a four bit vector which is
known as a Ket and each qubit will have a superposition of all
these states so the state at any time could be defined as
a|00>+b|01>+c|10>+d|11> where a.^2 is the probability of
measuring the |00> state. As vectors has a magnitude and a
direction then the phase difference matters when operating on
more than a qubit. Measuring the qubit's state collapses it to
one of the basic states of the classical bit.
VI. QUANTUM DECOHERENCE
It's an irreversible process happens due to the interference
with the surrounding environment which causes the system to
decohere. If the error was small there are error correction
algorithms that could be used to slow down the decoherence
time so that more processes could be done on the Qubit. But
the error correction algorithms come with the cost of using a
huge number of qubits as the number required for factoring
integers in shor's algorithm is increasingly polynomial and the
time of computation increases as well
VII. INSIDE A QUANTUM COMPUTER
1. The quantum bit (qubit) | ψ > :
It is the unit of quantum
information.
It is classically analog. It
has two basis states |0> and
|1> or in any linear
combination of these two
states, called a
superposition:
| ψ > = a|0> + b|1>
Where a and b are complex
numbers and are called
probability amplitudes,
Since |0> and |1> are the two
possible states, And
according to the probability
distribution function a2
+b2
=1. The qubit is described as
a vector in a two-
dimensional Hilbert space or
as a vector in block sphere.
Where ϑ and φ are angles that determine the position of the
qubit vector direction in Bloch sphere. The phase angle φ is
related to quantum mechanics interference.
Quantum bits (Qubits) can be Trapped ions, superconductors and
photonics.
If we have one Qubit (its state can only be 1 or 0 or
superposition of both), what’s the significance of its phase?
Really there is not, but what if we have two Qubits, now
during computation we can have two wave functions (out of
each Qubit), so there could be an interference between these
two waves (constructive or destructive), and here appears the
phase shift between the two waves (it determines the type of
interference , nλ for constructive interference and (n + )λ for
destructive interference, where n is an arbitrary integer), and
that’s consequently affects the superposition of the two states
of each Qubit and so affects the probability of the state of
finding each Qubit at, which can help us in leading the
computation towards the right answer (by enhancing the right
probabilities –needed for the right answer- of each Qubit) and
so away from the wrong answer.
Quantum Measurement: The act of convergence of a
Quantum probability wave to one of its choices using a
detector, It is impossible to detect a qubit's state without
performing a measurement. When we try to detect a qubit’s
state, The state of the qubit turns to one of the basis states. It
turns to state |0> with probability α2
and to state |1 > with the
probability β2
, so we must have α2
+ β2
=1.
2.The quantum register
It is a physical system that stores qubits. The quantum register
is described as a vector in a multidimensional Hilbert space
represented by the tensor product of its qubits states. In case of
it contains two qubits, the relation is
where:
General formula:
where:

3. 3
3. QUANTUM GATES
Quantum gates are operators of the Hilbert space that act on
qubits and quantum registers and change their states by
rotating their corresponding state vectors.
The input |q1> to these gates can be one or two or three qubits. The action of
a quantum gate G on a qubit can be represented by
the formula:
The Hadamard gate (H) The phase shift gate (Φ)
If the qubit is in a basis state
the H gate brings it to a
state superposition in which
the probability to measure
both
basis states equals to 0.5.
acts on one qubit and changes
its phase
angle φ
Quantum Not Gate:
It acts linearly.
It takes the state α|0>+β|1> to the corresponding state α|1>+β|0>
This linear behavior is a general property of the Quantum
mechanics.
The gates H, Φ and CNOT are the universal quantumGates of
the quantum computation and any quantum computation can
be carried out using combinations of these gates.
Quantum gates class VS Classical gate class:
Toffoli gate: A universal
reversible gate that can
simulate any classical
gate.
It can be easily seen that
putting b equal to 0 we
have the Nand result of a
and c.
Since 1 xor (a.c) = -(a.c) = a Nand c.
we have obtained the Nand gate using this toffoli gate.And
since the Nand gate is a Universal gate for classic gates, we
obtain all the classic gates using Toffoli gate.
4. No Cloning Theorem
As there is only one constrain in quantum gates which is
unitarily. U+
U=I
So there is no universal quantum gate can copy every input
qubit.
5.Quantum Computation in Artificial Intelligence:
Our brain can be regarded as a computer and our
consciousness as computation. So if we succeed to prove that
our brain handles quantum type transformation somewhere
into its neutral network, Then there could be biological or
chemical natures of quantum computers in the future.
The phase shift gate (Φ) The controlled–NOT (CNOT)
acts on one qubit and
changes its phase
angle φ
The upper
qubit, c , is the control qubit
and the lower qubit, t , is
the target qubit. The gate acts
on the target qubit and
reverses its state only if the
control qubit is in state 1 .
The controlled–NOT
(CNOT)
The controlled-controlled-
NOT gate(CCNOT)
The upper
qubit, c , is the control qubit
and the lower qubit, t , is
the target qubit. The gate
acts on the target qubit and
reverses its state only if the
control qubit is in state 1 .
The two upper qubits
2 c and 1 c are the control
qubits and the lower qubit,
t , is the target qubit. The gate
acts on the target qubit
and reverses its state only if
the control qubits are both in
state 1 .

4. 4
VIII. APPLICATIONS
The two most promising uses for Quantum computers are
quantum search and quantum factoring.
A. Quantum Search Applications: Quantum search is much
faster than classical search according to quantum parallelism
by performing operations on qubits; many values can be
processed by one calculation.
NASA Quantum computer: One of the applications NASA
will use its quantum computers for is the Kepler search for
exoplanets (which are planets outside the Solar System).
NASA astronomers use their various telescopes to look at light
curves to understand whether any noticeable dimming
represents a potential exoplanet as it moves across its host star
which is a massive search problem. NASA's qubit machine is
more than just a prototype; it's actually ready to do some
work.
B. Quantum Factoring Applications: Although quantum
search is impressive, quantum factoring algorithms pose a
legitimate, considerable threat to security. This is because the
most common form of Internet security, public key
cryptography, relies on certain math problems (like factoring
numbers that are hundreds of digits long) being effectively
impossible to solve. Quantum algorithms can perform this task
exponentially faster than the best known classical strategies,
rendering some forms of modern cryptography powerless to
stop a quantum code-breaker.
1) Steganography: In a classical computer: It's a method of
hiding data inside a media file as a picture or a movie without
being observed. In a Quantum computer: RGB colors are
represented as r, g and phase angle theta of the Qubit in the
quantum system. There is a challenge of hiding the secret
information inside the
picture as it mustn't be
observed by someone
who doesn't know the
existence of the secret
information and the
efficiency of the
steganography is
determined by: amount
of hidden data,
difficulty of detection
and difficulty of
removal. So there is a range of the angle theta that shouldn't
exceed in order not to be observed in the image.
Method of Steganography:
based on the entanglement of
the qubits a qubit holds the
image data, a qubit holds the
Key, and a qubit holds the
secret data.
Operation: the number of the key qubits must equal the
number of the message qubits then each qubit of the secret key
is entangled with a qubit in the key then they are entangled
with a qubit in the image and they are formed as a node graph
on the surface of the picture and in order to be decrypted the
receiver must have the place of the first node on the surface
usually it's given to him using a secure classical channel.
2) MIT solving systems of linear equations:
MIT researchers present a new algorithm that could bring the
same type of efficiency as in cryptography to systems of linear
equations, whose solution is crucial to image processing,
video processing, signal processing, robot control, weather
modeling, genetic analysis and population analysis.
IX. CONCLUSION
Quantum computing still most of it is just researches and
simulations and most of the complicated systems still just
designs and couldn't be implemented in todays labs. 2013
witnessed a breakthrough in the field of quantum computing
when Kastrenakes, Jacob could break the world record on
entanglement of approximately 3 billion qubits for 39 minutes
at room temparature while the previous record was 2
seconds.In 2014 news was leaked about a project called
Penetrating hard targets by which the NSA seeks
developing a new quantum computing capability for
cryptographic purposes. The field is very promising and many
scientists expect that it will be used widely for cryptographic
purposes and simulations very soon.
REFERENCES
 Introduction to the World of Quantum Computers by :
Sina Jafarpour
 Introduction to quantum computers by: Pedram Khalili
Amiri
 Hide Secrets Using the Power of Quantum Computers
by: Gabriela Mogos Al.I.Cuza University
 Quantum Computers: Registers, Gates and Algorithms
by: Paul Isaac Hagouel and Ioannis G. Karafyllidis
 A tale of two qubits
 A call with a MIT computer scientist
 NASA Quantum Computer
 MIT algorithm
 NSA encryption breaking