Gate Decomposition

Gate decomposition breaks complex gates into simpler, more fundamental gates. Useful for implementing gates not native on specific hardware or for circuit optimization.

Any single-qubit unitary can be decomposed as:

$$U = e^{i\alpha} R_Z(\beta) R_X(\gamma) R_Z(\delta)$$

This is the Z-X-Z decomposition. Alternative decompositions use different rotation axes.

$$U = e^{i\alpha} R_X(\beta) R_Z(\gamma) R_X(\delta)$$

(X-Z-X decomposition)

CNOT can decompose arbitrary two-qubit unitaries:

$$U_{2Q} = (A \otimes B) \text{CNOT} (C \otimes D) \text{CNOT} (E \otimes F)$$

requires up to 3 CNOTs and 12 single-qubit gates. This is the KAK decomposition.

Toffoli in terms of Hadamards, CNOTs, T gates (~6 CNOTs):

     ┌───┐
q_0: ┤ H ├──■───
     └───┘┌─┴─┐
q_1: ─────┤ X ├
          └───┘

(Simplified; full Toffoli decomposition is longer)

SWAP in terms of CNOTs (3 CNOTs):

q_0: ──■────────■──
     ┌─┴─┐┌────┐└─┬─┘
q_1: ┤ X ├┤ CX ├──■──
     └───┘└────┘

• Fewer native gates: decomposition uses fewer gate types but may require more total gates
• Shorter depth: reorder decomposed gates to minimize circuit depth
• Hardware constraints: decompose to match hardware's native gate set

Quantum compiler tools (Qiskit, Cirq, Q#) handle decomposition automatically.