The NOT gate, represented by the X-gate, is a fundamental quantum gate used to flip the state of a qubit. It is often referred to as a bit-flip gate. When applied to a qubit in the |0⟩ state, it changes it to the |1⟩ state, and vice versa. Mathematically, if |ψ⟩ represents the qubit's state, applying the X-gate results in X|ψ⟩ = |¬ψ⟩.
The Hadamard gate, denoted as the H-gate, is used to create superposition states. When applied to a qubit in the |0⟩ state, it transforms it into an equal superposition of |0⟩ and |1⟩ states. Mathematically, H|0⟩ = (|0⟩ + |1⟩) / √2. It's a crucial gate in quantum algorithms like Grover's and quantum teleportation.
The CNOT gate, or controlled-X gate, is an entangling gate that acts on two qubits: a control qubit and a target qubit. If the control qubit is in the |1⟩ state, it applies the X-gate to the target qubit; otherwise, it does nothing. It creates entanglement between the two qubits and is essential in building quantum circuits for various applications.
The Toffoli gate is a three-qubit gate that acts as a controlled-controlled-X gate. It applies the X-gate to the target qubit only if both control qubits are in the |1⟩ state. It is used in quantum computing for implementing reversible classical logic operations.
The SWAP gate exchanges the states of two qubits. When applied, it swaps the state of the first qubit with the state of the second qubit. This gate is fundamental in quantum algorithms and quantum error correction codes.
The Pauli X gate is another name for the NOT gate, as mentioned earlier. It flips the state of a qubit, changing |0⟩ to |1⟩ and vice versa. It's named after Wolfgang Pauli, a pioneer in quantum physics.
The Pauli Y gate is a quantum gate that introduces complex phase shifts along with a bit-flip operation. It can be thought of as a combination of bit-flip and phase-flip gates. Mathematically, it is represented as Y|0⟩ = i|1⟩ and Y|1⟩ = -i|0⟩. It's essential in quantum error correction and quantum algorithms.
The Pauli Z gate is a gate that introduces a phase flip without changing the qubit's state. It leaves |0⟩ unchanged and transforms |1⟩ to -|1⟩. Mathematically, Z|0⟩ = |0⟩ and Z|1⟩ = -|1⟩. It's used for measuring the phase of a qubit.
These quantum gates are fundamental building blocks in quantum computing, enabling the manipulation and transformation of qubit states to perform various quantum algorithms and computations.