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