Bit and Qubit

Bit

A bit is a binary digit. It is the smallest increment of data on a computer i.e.; it is used to represent information in classical computers. A bit can hold two values, either 0 or 1. 0 corresponds to the electrical values are off and 1 corresponds to on. Bit is the basic unit of information.

Single Qubit

In quantum computing, information is stored and processed using qubits, which can exist in a superposition of the basis states \( |0\rangle \) and \( |1\rangle \). A single-qubit state is written as:

\( |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \quad\text{where}\quad |\alpha|^2 + |\beta|^2 = 1 \)

Multi and Qubits

When multiple qubits are combined, the system is described using the tensor product of individual qubit states. For example, a two-qubit system can be in a superposition of four computational basis states:

\( |\psi\rangle = \alpha_{00} |00\rangle + \alpha_{01} |01\rangle + \alpha_{10} |10\rangle + \alpha_{11} |11\rangle \)

The squared magnitudes of the coefficients \( |\alpha_i|^2 \) represent the probabilities of obtaining the corresponding basis state upon measurement.

General n-Qubit System

In general, an \( n \)-qubit quantum state is represented as a vector in a \( 2^n \)-dimensional Hilbert space:

\( |\psi\rangle = \sum_{i=0}^{2^n - 1} \alpha_i |i\rangle \)
• \( |i\rangle \) are the computational basis states.
• \( \alpha_i \) are complex amplitudes satisfying \( \sum_i |\alpha_i|^2 = 1 \).

This exponential growth of the state space with the number of qubits is a key reason why quantum computers have the potential to outperform classical computers for certain computational tasks.

Classification between Classical bits and Quantum bits