# 2. Qubits and Quantum States

Syllabus : The Qubit , Complex Numbers, Vector Space, Linear combination of vectors, Linear dependent and indepent vectors . Basis Set and dimension ,Inner products, Outer products, Dirac Notation, Boolean Algebra, Logic Gates and Quantum Information Processing

1. The Qubit

A qubit is a quantum bit, the fundamental unit of information in a quantum computer. In contrast to classical bits, which can only exist in one of two states (0 or 1), a qubit can exist in a superposition of both states simultaneously. This property of superposition is one of the fundamental features of quantum mechanics and allows quantum computers to perform certain calculations exponentially faster than classical computers.

Qubits can be physically implemented using a variety of quantum systems, such as the spin of a single electron, the polarization of a single photon, or the energy levels of a superconducting circuit. The state of a qubit can be described by a quantum state vector, which is a mathematical object that encodes the probabilities of measuring the qubit in different states.

In addition to superposition, qubits also exhibit another quantum property called entanglement. Entanglement is a correlation between two or more qubits that arises from their quantum mechanical nature and can be used to perform certain quantum algorithms and communication protocols.

A qubit is different from an ordinary bit in several fundamental ways:

1. Superposition: Unlike a classical bit, which can only exist in one of two definite states (0 or 1), a qubit can exist in a superposition of both states simultaneously. This means that a qubit can be in a state that is a linear combination of 0 and 1, with coefficients that can be complex numbers.
2. Measurement: When a classical bit is measured, it collapses into either the 0 or 1 state with certainty. In contrast, when a qubit is measured, the probability of it collapsing into the 0 or 1 state is determined by the coefficients of its superposition. This means that a single qubit can be used to store more than one bit of classical information.
3. Entanglement: Two or more qubits can be entangled, meaning that their states are correlated in a way that cannot be explained by classical physics. This property of entanglement is a unique feature of quantum mechanics and has the potential to enable new forms of computation and communication.
4. Fragility: Qubits are much more fragile than classical bits and can be easily disturbed by environmental noise and interference. This makes it challenging to build a large-scale quantum computer, which requires careful control and isolation of the qubits.

These differences between qubits and classical bits are what make quantum computing a fundamentally different paradigm of computing. While classical computers are based on the manipulation of bits with definite states, quantum computers are based on the manipulation of qubits with superpositions and entanglement.

2. Complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and b is the imaginary part, and i is the imaginary unit, which is equal to the square root of -1.

Here’s an example of a complex number:

3 + 4i

In this example, the real part is 3 and the imaginary part is 4i. To perform arithmetic with complex numbers, we treat the real and imaginary parts separately and then combine them at the end. For example, to add two complex numbers, we add the real parts and add the imaginary parts separately, like this:

(3 + 4i) + (2 + 5i) = (3 + 2) + (4 + 5)i = 5 + 9i

So the sum of (3 + 4i) and (2 + 5i) is 5 + 9i.