# Mathematics for Quantum Computing

Quantum computing stands at the forefront of technological innovation, promising to revolutionize the way we solve complex problems that are beyond the reach of classical computers. With the potential to exponentially speed up computations and tackle challenges in fields like cryptography, optimization, and drug discovery, quantum computing has captured the imagination of scientists, researchers, and technology enthusiasts alike.

However, understanding quantum computing requires a firm grasp of the underlying mathematical principles. The realm of quantum mechanics, upon which quantum computing is built, relies heavily on mathematical concepts and tools to describe and manipulate quantum states, quantum gates, and quantum algorithms.

This page describes essential mathematic concepts require to build a good understanding of quantum computing concept.

### Quantum Mechanics

Quantum mechanics is a branch of physics that explores the behavior of matter and energy on the smallest scales, where classical physics principles no longer apply. It is a revolutionary theory that describes the fundamental nature of particles, such as electrons and photons, and their interactions.

Wave Particle Duality: Quantum mechanics introduces the concept of wave-particle duality, suggesting that particles can exhibit both wave-like and particle-like properties.

Superposition: Quantum mechanics also introduces the principle of superposition, stating that quantum systems can exist in multiple states simultaneously until observed or measured. The quantum system exists in both states until observed or measured. Any measurement of a properties of a quantum system or a particle causes wave function collapse into measured state.

Quantum Entanglement: Quantum mechanics introduces the idea of quantum entanglement, where two or more particles can become correlated in such a way that the state of one particle is instantaneously connected to the state of the others, regardless of their spatial separation. It is a phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others.

Measurement of physical properties such as position, momentum, spin, polarization performed on entangled particles are found be perfectly corelated. For example if entangled particles are generate in such a way that total spin if zero and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. Any measurement of a particle properties affect the entangled system as a whole.

### Mathematical postulates of quantum mechanics

1. The state (or wavefunction) of individual quantum systems is described by unit vectors living in separate complex Hilbert spaces.
2. The probability of measuring a system in a given state is given by the modulus squared of the inner product of the output state and the current state of the system. This is known as Bornâ€™s rule. Immediately after the measurement, the wavefunction collapses into that state.
3. Quantum operations are represented by unitary operators on the Hilbert space.
4. The Hilbert space of a composite system is given by the tensor product (aka Kronecker product) of the separate, individual Hilbert spaces.
5. Physical observables are represented by the eigenvalues of a Hermitian operator on the Hilbert Space.