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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

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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 +b

2

=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 α

2+ β

2=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:

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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 .

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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