Quantum Gates and Quantum Circuits

Definition¹

For $n \in \mathbb{N}$, we call the following [unitary operator] $G$ a quantum gate or a $n$qubit gate.

$$G : \left( \mathbb{C}^{2} \right)^{\otimes n} \to \left( \mathbb{C}^{2} \right)^{\otimes n}$$

The composition of quantum gates is called a quantum circuit. Here, $\otimes$ is the tensor product of vector spaces.

Explanation

This is the quantum computer’s definition of gates and circuits as seen in classical computers. Since unitary operators are reversible, it is always possible to compute the inputs from the outputs in a quantum computer made up of quantum circuits.

Types

• Pauli gates
• Hadamard gate $H$
• Phase gate $R_{\theta}$