What are the basic quantum gates (Hadamard, Pauli, etc.)?

Quantum gates are fundamental operations used to manipulate qubits in quantum computing. The most common basic gates include the Hadamard, Pauli (X, Y, Z), and CNOT gates. Each performs specific transformations on qubit states, enabling superposition, entanglement, and other quantum effects. Unlike classical bits, qubits can exist in superpositions, and gates act on these probabilistic states through mathematical operations represented by matrices. Understanding these gates is essential for building quantum algorithms.

The Hadamard gate (H) creates superposition, a core quantum phenomenon. When applied to a qubit in state |0⟩ or |1⟩, it transforms it into an equal superposition: (|0⟩ + |1⟩)/√2 or (|0⟩ − |1⟩)/√2, respectively. For example, in quantum algorithms like Grover’s search, the Hadamard initializes qubits into a superposition to explore multiple solutions simultaneously. The Pauli gates include X, Y, and Z. The Pauli-X gate acts like a classical NOT gate, flipping |0⟩ to |1⟩ and vice versa. Pauli-Z introduces a phase flip (multiplying |1⟩ by -1), while Pauli-Y combines both flipping and phase changes. These gates are often used in error correction and as building blocks for more complex operations.

Two-qubit gates like the CNOT (Controlled-NOT) enable entanglement, a key quantum resource. CNOT flips the target qubit if the control qubit is |1⟩. For instance, applying CNOT to |0⟩ (control) and |1⟩ (target) leaves them unchanged, but applying it to |1⟩ and |1⟩ flips the target to |0⟩. This gate is critical for algorithms like quantum teleportation and Shor’s factoring algorithm. Other important gates include the Phase (S) and T gates, which introduce finer phase shifts for tasks like quantum Fourier transforms. Together, these gates form a toolkit for designing circuits that leverage quantum mechanics to solve problems intractable for classical computers.