Quantum gates are the fundamental operations in quantum computing, analogous to logic gates in classical computing. They manipulate qubits, the basic units of quantum information, to perform computations. By stringing together these gates in a specific sequence, we create quantum circuits, which are the blueprints for quantum algorithms.

Unlike classical bits that can only be 0 or 1, qubits can exist in a superposition of both states simultaneously. This superposition is represented as a linear combination of the basis states $|0\rangle$ and $|1\rangle$: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, where $\alpha$ and $\beta$ are complex probability amplitudes such that $|\alpha|^2 + |\beta|^2 = 1$. The values $|\alpha|^2$ and $|\beta|^2$ represent the probabilities of measuring the qubit in the state $|0\rangle$ or $|1\rangle$, respectively.

Single-qubit gates operate on a single qubit, transforming its quantum state. These operations are represented by unitary matrices, ensuring that the quantum state remains normalized and reversible. Key single-qubit gates include:

Multi-qubit gates are essential for creating entanglement and performing complex computations. Entanglement is a quantum phenomenon where the states of two or more qubits are linked, regardless of the distance between them. The most common multi-qubit gate is the Controlled-NOT (CNOT) gate.

Other important multi-qubit gates include the Controlled-Z (CZ) gate, which applies a Z gate to the target qubit only if the control qubit is $|1\rangle$, and the Toffoli (CCNOT) gate, a three-qubit gate that flips the target qubit if both control qubits are $|1\rangle$. These gates are universal, meaning any quantum computation can be constructed using combinations of them.

A quantum circuit is a sequence of quantum gates applied to a set of qubits. It's a visual representation of a quantum algorithm. The circuit starts with qubits in a defined initial state (usually $|0\rangle$ for all qubits), applies a series of single- and multi-qubit gates, and ends with measurements that collapse the quantum states into classical bits. The order and type of gates determine the computation performed.

This simple circuit demonstrates the creation of a Bell state. The Hadamard gate on Qubit 1 creates a superposition. The CNOT gate then entangles Qubit 1 (control) with Qubit 2 (target). Measuring the qubits will yield correlated results (either both 0 or both 1).

Understanding quantum gates and circuits is fundamental to developing quantum algorithms for tasks like Shor's algorithm for factoring large numbers, Grover's algorithm for searching unsorted databases, and quantum simulation for materials science and drug discovery. The ability to precisely control and manipulate qubits through gates is the core of quantum computation's power.