Bonaparte Quantum Logic

Other Gates

In quantum computing, a small set of gates can be combined to approximate any unitary operation on a quantum system. Such a set is called universal. The {H, T, CNOT} gate set is one of the most common universal gate sets used in quantum algorithms. Additionally, gates like the S gate and Controlled-Z (CZ) gate provide useful phase operations and controlled operations that complement the universal set.

Universal Gate Set

HTX

Figure B: Quantum circuit representation of the universal gate set {H, T, CNOT}.

Universal Gate Components:

Hadamard (H)

Creates superposition, mapping |0⟩ → (|0⟩ + |1⟩) / √2.

T Gate

Also known as the π/8 gate, applies a phase shift to |1⟩, crucial for universality.

CNOT

A two-qubit entangling gate, flips the target qubit if the control is |1⟩.

Why This Set is Universal:

  • Single-qubit gates: H and T gates together can approximate any single-qubit rotation to arbitrary precision
  • Multi-qubit entanglement: CNOT provides the ability to create entanglement between qubits
  • Completeness: Any quantum algorithm can be decomposed into a sequence of these three gates
  • Fault-tolerance: This gate set is particularly useful for fault-tolerant quantum computing implementations
  • Approximation: Complex gates can be approximated by combining H, T, and CNOT gates in specific sequences

Extra Gates:

Controlled-Z (CZ)

Applies a phase flip (Z rotation) to the target qubit when control qubit is |1⟩. Useful for phase-based quantum algorithms.

S Gate

Applies a π/2 phase shift to |1⟩, also known as the phase gate. Related to the Z gate by S = √Z.