Key Quantum Gates

Quantum gates are the building blocks of quantum computing, manipulating qubits to perform complex calculations. Each gate, like the Hadamard or CNOT, plays a unique role in creating superposition, entanglement, and controlling qubit states for powerful algorithms.

1. Hadamard (H) gate

   • Creates superposition by transforming a qubit from |0⟩ or |1⟩ to an equal probability of both states.
   • The matrix representation is (1/√2) * [[1, 1], [1, -1]].
   • Essential for quantum algorithms, enabling parallelism in computations.

2. Pauli-X gate (NOT gate)

   • Flips the state of a qubit: |0⟩ becomes |1⟩ and |1⟩ becomes |0⟩.
   • Represented by the matrix [[0, 1], [1, 0]].
   • Acts as a fundamental building block for constructing more complex quantum circuits.

3. Pauli-Y gate

   • Combines a bit flip and a phase flip, transforming |0⟩ to i|1⟩ and |1⟩ to -i|0⟩.
   • Its matrix form is [[0, -i], [i, 0]].
   • Important for introducing phase shifts in quantum states.

4. Pauli-Z gate

   • Applies a phase flip to the |1⟩ state, leaving |0⟩ unchanged.
   • Represented by the matrix [[1, 0], [0, -1]].
   • Useful for controlling the phase of qubits in quantum algorithms.

5. Phase (S) gate

   • Introduces a phase shift of π/2 (90 degrees) to the |1⟩ state.
   • Its matrix representation is [[1, 0], [0, i]].
   • Often used in conjunction with other gates to manipulate qubit phases.

6. T gate

   • Applies a phase shift of π/4 (45 degrees) to the |1⟩ state.
   • Represented by the matrix [[1, 0], [0, e^(iπ/4)]].
   • Important for achieving precise control over qubit states in quantum circuits.

7. CNOT (Controlled-NOT) gate

   • A two-qubit gate that flips the second qubit (target) if the first qubit (control) is |1⟩.
   • Its matrix form is a 4x4 matrix: [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]].
   • Crucial for creating entanglement between qubits.

8. SWAP gate

   • Exchanges the states of two qubits.
   • Its matrix representation is a 4x4 matrix that swaps the basis states: [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]].
   • Useful for rearranging qubits in quantum circuits.

9. Toffoli (CCNOT) gate

   • A three-qubit gate that flips the third qubit (target) if the first two qubits (controls) are both |1⟩.
   • Its matrix is a 8x8 matrix, maintaining the states of all other qubits.
   • Important for error correction and implementing classical logic in quantum circuits.

10. Controlled-Z (CZ) gate

    • Applies a phase flip to the target qubit if the control qubit is |1⟩.
    • Represented by the matrix [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]].
    • Essential for creating entangled states and manipulating qubit phases in quantum algorithms.