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Qubit Representation
  • Last Updated : 23 Nov, 2020

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 :

\begin{bmatrix}\alpha\\\beta\end{bmatrix}

where



\alpha, \beta

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. 

| \alpha | ^2 +| \beta |^2 = 1

State 0 is represented as :

 \begin{bmatrix}1\\0\end{bmatrix}

State 1 is represented as:

 \begin{bmatrix}0\\1\end{bmatrix}

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.

|0\rangle\ \ =\begin{bmatrix}   1 \\ 0 \end{bmatrix} and \ \ |1\rangle\ \ =\begin{bmatrix}   0 \\ 1 \end{bmatrix}

It also has a dual form written as:

\langle0|\ \ =\begin{bmatrix}   1 \ 0 \end{bmatrix} and \ \ \langle1|\ \ =\begin{bmatrix}   0 \ 1 \end{bmatrix}

Hence, any arbitrary state can be represented as: 

|\Psi\rangle\ \ =\begin{bmatrix}   \alpha \\ \beta \end{bmatrix} or \ |\Psi\rangle\ \ =\alpha |0\rangle + \beta |1\rangle

Some other symbols used are:

|+\rangle\ \ = \frac{1}{\sqrt2}(|0\rangle+|1\rangle) \ \ and \ \ |-\rangle\ \ = \frac{1}{\sqrt2}(|0\rangle-|1\rangle)

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

|i\rangle\ \ = \frac{1}{\sqrt2}(|0\rangle+i|1\rangle) \ \ and \ \ |-i\rangle\ \ = \frac{1}{\sqrt2}(|0\rangle-i|1\rangle)

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

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