Key Quantum Gates to Know for Quantum Computing

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.