In quantum computing, the fundamental unit of information is the qubit, which can exist in a superposition of states, unlike the classical bit, which can only be in one state at a time.

A qubit is mathematically represented as a unit vector in a two-dimensional Hilbert space, denoted as $|\alpha\rangle$, where $\alpha$ is a complex number.

Superposition is the fundamental property of quantum mechanics that allows a qubit to exist in a combination of states simultaneously.

Entanglement is the process of linking two or more qubits such that they become correlated, meaning the state of one qubit cannot be described independently of the others.

Decoherence is the process where a quantum system loses its quantum properties due to interaction with the environment, causing it to behave classically.

Quantum gates are the fundamental building blocks of quantum algorithms, which are unitary operators that manipulate the state of qubits.

The phase gate is used to rotate a qubit around the Bloch sphere by a specific angle, allowing for controlled manipulation of the qubit's state.

The CNOT gate is used to entangle two qubits, creating a correlation between their states.

Shor's algorithm is a quantum algorithm that can efficiently factor large integers, which is a challenging task for classical computers.

Grover's algorithm is a quantum algorithm that can efficiently search an unsorted database, providing a quadratic speedup compared to classical algorithms.

The Deutsch-Jozsa algorithm is used to determine if a function is constant or balanced, providing a significant speedup compared to classical algorithms.

Simon's algorithm is used to find the period of a function, providing a significant speedup compared to classical algorithms.

Decoherence is the primary challenge in implementing quantum computers, as it causes the loss of quantum properties due to interaction with the environment.

Quantum error correction is the technique used to mitigate the effects of decoherence in quantum computers by detecting and correcting errors that occur during quantum operations.

Quantum computing technology is still in its early stages of development, with ongoing research and advancements in hardware and software.