In classical computers, information is represented by binary digits 0 or 1. These are called bits. Computer hardware understands the 1-bit as an electrical current flowing through a wire in a transistor. 0-bit is the absence of an electrical current in a wire.Computer decodes the 1 or 0 bits into words, images or videos etc.

Quantum bits or qubits are similar to bits in classical computing with 2 measurable states called the 0 and 1 states. An important distinction from classical bits, is that, quantum bits can be in a superposition state of these 0 and 1 states. Certain computations that would normally need to be performed on 0 or 1 seperately on a classical computer can now be completed in a single operation using a qubit on a quantum computer. Intituitively, this could make the computer much faster than classical computers.A single qubit is in a superposition of 2 classical bits, when a qubit is measured, the measurement will result in one classical bit of information. This is due to quantum property.

Mathematical Representation of Qubits:

 In quantum mechanics and quantum conmputing,  bra-ket notation is used to denote quantum states. The notation uses angle brackets, ⟨ and ⟩ , and a vertical bar |  to construct "bras" and "kets". This notation is also known as Dirac notation.

A bra is of the form ⟨ f | and ket is of the form  |v⟩.The state of a qubit is  enclosed in the right half of an angled bracket  called the  "ket"${\displaystyle ⟨f}$.

A qubit, |ψ ⟩ , could be in a |0⟩ or |1⟩ state or even a superposition of both |0⟩ and |1⟩. This is represented as shown below.
 
                            |ψ ⟩ = α|0⟩  + β|1⟩  

 α and β are called the amplitudes of the states. Amplitudes are generally complex numbers and it allows us to mathematically represent all of the possible superpositions.

Amplitude will return the probability of finding a particle in that specific state when performing measurement. Therefore, this is important.

     
 Quantum state of flipping a coin:

     In quantum theory, the quantum state of flipping a coin can be written as a superposition of heads and tails. Using heads as      | 1 ⟩     and tails as  | 0⟩ , the quantum state of the coin is

                     | coin ⟩ = 1 / √2  (  | 1 ⟩  +     | 0⟩   )

The amplitude of  | 1 ⟩ is β = 1 / √2, so | β | ² =  (1 / √2) ²   =  1 / 2 .  So the probability is 0.5 or 50%.