Qubit Representation

Similar to a bit in classical computing, a qubit is a basic building block of Quantum Computers. It represents the superposition of states 0 and 1. What we mean by superposition is that the given state is in a linear combination of state 0 and state 1.

It is a common misconception that qubit is in one state 0 or 1, we just don’t know until we measure it. However, qubit always exists in a state between 0 and 1 (inclusive), the act of measuring brings it to either of the states.

Matrix Representation : A qubit is represented as a complex vector of size 2. Generally represented as :

where

are amplitudes of states 0 and 1 respectively or we can say the probability to be in state 0 and 1 respectively. This vector is normalized i.e. 

State 0 is represented as :

State 1 is represented as:

Quantum states |0> and |1> form orthogonal basis also called computational basis or canonical basis.

Dirac’s Notation : It is a shorthand notation for a qubit. A vector is represented using a ket.

It also has a dual form written as:

Hence, any arbitrary state can be represented as: 

Some other symbols used are:

|+> and |-> are called Hadamard’s basis. These are also orthogonal to each other.

Bloch’s Representation : Bloch’s sphere is a geometric representation of quantum states where different points in the surface of the unit radius sphere represent various quantum states. A qubit can be represented in a 3D space as a vector of unit length connecting points in the surface of the Bloch’s sphere and it’s center.

Bloch’s Qubit Representation