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The beginning of the twentieth century was a mile stone in the history of science which redirect all the fundamental understanding. Beginning with the change in the perspective of how we think of the physical reality was changed by the theory of relativity and the understanding of universe by the introduction of quantum mechanics, which tells us the behavior of particles. Then there comes the Uncertainty principle, which turn around the understanding of elemental particle theories and measurements. Richard Feynman in the 1960s  proposed many aspects of quantum electrodynamics, the study of the way in which electrons behave in presence of electromagnetic radiation, through photons. Feynman along with others began to investigate the possibility of handling information using the simple quantum mechanical processes, considering the representation of binary numbers in accordance with the quantum states of two-state quantum system. This was a giant undertaking, in the scene that the scientists actually trying to emulate quantum mechanical process using a quantum computation device rather than a normal classical computer. During a conference co-hosted by IBM and MIT, Feynman challenged the computer science community to develop a computing device using the principles of quantum mechanics.

 First ever practical and commercially available computing device "ENIAC" (Wikipedia)

In 1985, David Deutsch from Oxford university published an article describing the possibility of a universal quantum computing device. He explained and proved that if a two-state quantum system could be made to evolve by means of a set of simple yet powerful operations, it can be used to simulate any physical system. This operation was called quantum gates because of its similarity with binary logic gates. Then in 1994, Peter Shor, currently a professor at Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts, USA, proposed and developed an algorithm using quantum gates to find the prime factors of large number, which used to take enormous amount of time by  classical computer. This algorithm was a valuable leap in quantum computing, because many of the normal encryption schemes uses large number, of which, finding factors is extremely difficult. This opened an enormous opportunity in the scientific community in the field of computing. There were so many problems associated with the creation of quantum states. In 1995 a group of researchers from Caltech and The National Institute of Standard and Technology (NIST) jointly developed methods to create quantum states using magnetic fields, which allow ions or particles to isolate from environmental influences. This was a giant leap in the quantum technology even though the states are very unstable.

Richard Feynman (Wikipedia)

In early 2000s a group of scientists demonstrated the Shor's algorithm with a time span of 48 hrs using a quantum computer. As the time continues more and more technological and algorithmic developments have been made which eventually made better understanding of how to build a real quantum system. Then a group of researchers lead by Robert J. Schoelkopf invented a way to create superconducting qubits called transom qubits. It is a charged qubits and most of the todays research is focused on technology related to transom qubits. Major changes happened in the last few years, companies like IBM, Microsoft, Google and startups like rigeti, D-wave developing state-of-the-art quantum computers for commercial applications. In 2019 January, IBM released its first quantum computer for commercial applications. The efforts are still on to remove all the hurdles in quantum computing and researchers are pushing the limits to make a universal quantum machine to fulfill all the needs of the masses.

Today's computers are very good at fulfilling our needs. We are enjoying the benefits of today's classical computers. It is entertaining us, connecting us and even help us in each and every moment of life in one way or another. Classical computers are helping us to process huge amount of data in this data driven world and make the world is a better place. From medicine to finance, engineering and so many other areas, computers and computer knowledge is an essential for a human being to survive. The power of the computers are increasing day by day and it will continue to increase as we develop further. 

However, there are limitations to the classical computers. There are problems that cannot be addressed with classical computers. In other words, there are problems that cannot be addressed using a classical computer. There are many complex system in science which are so complex and massive that we cannot do it with even the powerful supercomputers using classical computing technology. This leads the scientists to seek new methods and approaches to tackle with this limitation of classical computers.

We need systems with computational power with the size and the complexity of the system itself. The best example we can choose is the simulation of atoms and molecules. Everyone knows that simulating an atom or a molecule is very complex due to the complex quantum mechanical calculation involved. The behavior of an electron inside an atom or molecule is very complex to compute, which is very essential in understanding the inner mechanism of that molecule.

The simulation of chemistry and simulation of even small scale macro molecules which are essential for life on earth is very complex for a classical computer. Nature is quantum mechanical and classical computer is very limited in simulating quantum mechanical aspects. So it is essential to use the object which obeys the same natural laws to simulate or understand the nature. A quantum computer can easily simulate these complex systems so that we can understand it. The implications this leads to is immense, opportunities in new drugs, new materials, understanding catalytic interaction in chemical reactions complex molecular interactions which are impossible to calculate using today's computing technology.

In this digital age, as we all know the information is digitally handled as bits, that is zeros and ones. Each of this information chunks are called "bit". For last five decades, we have been playing with this so called bit and today all of our digital life, even real life is connected to these two numbers, 0 and,1 called bits. Each of this bit can be intimated to the states of an electric switch. The on state of the switch can be considered as 1 and off state is 0. Analogs to this classical approach, we can materialize the representation of information in quantum computing using units called quantum bits or abbreviated as qubits. The quantum version of classical bits. Qubit is a two-state or two-level quantum mechanical representation of information. The fundamental difference between a qubit and classical bit is that, classical bit can only be in either one of its states or level whereas qubit can be in coherent superposition of both states/levels. The term qubit was originally introduced by Benjamin Schumacher, an American theoretical physicist working in the area of quantum information theory, in his 1995 paper, Quantum Coding published in Physical Review A. The paper deals with the compression of quantum information created by a qubit, and it is called Schumacher compression.

Qubit Vs Bit

The classical information processed as bits using a CPU is implemented by one of the two levels of a low  DC voltage.  This binary logic is represented by two voltage states in different logic families, for example, the voltage level for TTL family for logical 0 is

and for logical 1 is

. Similarly, for CMOS family of logic, 0 is 0.5 V and 1 is 2 V_cc. For all these logic families, information is represented by either 0 or 1 or a combination of both. A qubit or quantum bit is used in quantum computing and it is different from normal bits. Consider representing data using an atom. The ground state of electron in the atom can be taken as state 

and the excited state of the atom to another energy level can be taken as 

.  In this format, a qubit can simultaneously represent the states 

and

. Using this property, if we use 4 qubits to represent information, the qubit can represent all the data simultaneously and it is 16 times faster than a normal computer. Table 1—the table represents quantum equivalent of classical computers. A quantum computer with 300 qubit can replace all the supercomputers on the planet

Quantum Bits	Equivalent Classical Bit
1	1
10	1024
50	1.125899907E17
300	2.037035976E90

Qubit Realization Techniques

There are various proposals that have been made to physically implement a qubit. Many researches are going on to find a good method to realize qubit. A quantum mechanical system with two-levels can be used as a qubit.  We can also use multilevel systems provided if they have two levels that can be decouple  from other levels. Researchers are successfully implemented systems that can be approximated two-levels with various degrees of accuracy. The various technologies and techniques to produce a qubit is will serve various purposes in a practical quantum machine. The following list will be an incomplete set of technologies practically implemented by researchers:

Physics	Terminology	Information
Coherent state of light	Squeezed Light	Quadrature	Early	Late
Photon	Number of Photons	Fock State	Vacuum	Single Photon State
Electron	Spin/Number	Spin/Charge	Up/No Charge	Down/Charge
Josephson Junction	Superconducting Charge/Flux/Phase Qubit	Charge/Current/Energy	Uncharged Island/Clockwise Current/Ground State	Charged Island/Counter Clockwise Current/First Excited State

Quantum mechanics deals with the smallest elementary particles and their interactions. Quantum computing is utilizes two special quantum mechanical phenomena called superposition and entanglement.

Superposition

The quantum mechanical super position is much like the classical wave superposition. Any two or more quantum sates can be added or superposed together to form a valid quantum states. In other words, a quantum state can be represented by the sum of another two or more different states. This is a property of the solutions of Schrödinger equation (Equation 1). Schrödinger equation is linear in nature and any combination of the solutions of a linear equation is also a solution.

We can observe superposition phenomena in nature, the interference pattern observed from electron double-slit experiment is similar to that of observed by diffraction of classical waves. This pattern is the result of superposition.

Quantum Mechanical Representation of Qubit

A quantum mechanical state is generally represented by Dirac notations also known as "bra-ket" notations.  There are infinitely many states are possible for a qubit, but we represent these states by two basis vectors or states. The vectors are

and

, pronounced as "ket-1" and "ket-0". These vectors are called orthonormal basis states,

, together they called computational basis. These vectors are span the two-dimensional linear vector or Hilbert space of qubit.

A pure qubit can be represented by the linear combination of basis states

as

Where 

and 

are the probability amplitude in complex numbers. The probability of outcome "0" is 

and probability of outcome "1" is

and will be constrained by

. We use Bloch sphere to represent a qubit graphically, Fig. 2.

Figure 2: Bloch Sphere representation of qubit

A qubit have four degrees of freedom due to the involvement of complex numbers in probability  amplitudes. This will be reduced to three due to constraint

. Further we reduce the degrees of freedom by using changing the coordinates. Probability amplitudes are rewritten into the following form using coordinate change,

For a single qubit,  due to the  insignificance of

, and we can choose

to be real, the general state is further reduced to,

Where the term

is the physically significant relative phase. On the Bloch sphere, the north and south poles represents the classical bits and any point on the sphere will represent the concurrent state of a qubit. This polar axis is arbitrarily chosen.

This chapter gives an idea about the historical background about quantum computing and related technologies. An attempt is made to explain the benefits of using quantum computer and how it is going to reshape our future. This chapter will help us to explore about the building block of quantum computer - qubit, the origin of the word, meaning, and  brief mathematical representation.

The next chapter will give a detailed idea about Q# language. The aspects discussed in detail will be Q# Constructs, Q# Control flow, I/O conventions and basic setup for visual studio for programming in Q#.